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+RESEARCH ARTICLE
+Random forests, sound symbolism and
+Poke´mon evolution
+Alexander James KilpatrickID1☯*, Aleksandra C´ wiek2☯, Shigeto Kawahara3
+1 International Communication, Nagoya University of Commerce and Business, Nagoya, Aichi, Japan,
+2 Department Leibniz-Zentrum Allgemeine Sprachwissenschaft, Berlin, Germany, 3 Institute of Cultural and
+Linguistic Studies, Keio University, Tokyo, Japan
+☯ These authors contributed equally to this work.
+* [email protected]
+Abstract
+This study constructs machine learning algorithms that are trained to classify samples using
+sound symbolism, and then it reports on an experiment designed to measure their under-
+standing against human participants. Random forests are trained using the names of Poke´-
+mon, which are fictional video game characters, and their evolutionary status. Poke´mon
+undergo evolution when certain in-game conditions are met. Evolution changes the appear-
+ance, abilities, and names of Poke´mon. In the first experiment, we train three random forests
+using the sounds that make up the names of Japanese, Chinese, and Korean Poke´mon to
+classify Poke´mon into pre-evolution and post-evolution categories. We then train a fourth
+random forest using the results of an elicitation experiment whereby Japanese participants
+named previously unseen Poke´mon. In Experiment 2, we reproduce those random forests
+with name length as a feature and compare the performance of the random forests against
+humans in a classification experiment whereby Japanese participants classified the names
+elicited in Experiment 1 into pre-and post-evolution categories. Experiment 2 reveals an
+issue pertaining to overfitting in Experiment 1 which we resolve using a novel cross-valida-
+tion method. The results show that the random forests are efficient learners of systematic
+sound-meaning correspondence patterns and can classify samples with greater accuracy
+than the human participants.
+Introduction
+Natural language processing (NLP) is a field of study that combines computational linguistics
+and artificial intelligence and is concerned with giving computers the ability to understand
+language in much the same way humans can. The present study tests whether an NLP algo-
+rithm can classify samples using sound symbolism, which has been a largely overlooked feature
+of human language in NLP. While in modern linguistics, the relationship between sound and
+meaning is generally assumed to be arbitrary [1], a growing number of studies have revealed
+systematic relationships between sounds and meanings, some of which hold cross-linguisti-
+cally. For example, speakers of many languages tend to associate words containing [i] with
+PLOS ONE
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+1 / 27
+a1111111111
+a1111111111
+a1111111111
+a1111111111
+a1111111111
+OPEN ACCESS
+Citation: Kilpatrick AJ, C´wiek A, Kawahara S (2023)
+Random forests, sound symbolism and Poke´mon
+evolution. PLoS ONE 18(1): e0279350. https://doi.
+org/10.1371/journal.pone.0279350
+Editor: Maki Sakamoto, The University of Electro-
+Communications, JAPAN
+Received: July 12, 2022
+Accepted: December 6, 2022
+Published: January 4, 2023
+Peer Review History: PLOS recognizes the
+benefits of transparency in the peer review
+process; therefore, we enable the publication of
+all of the content of peer review and author
+responses alongside final, published articles. The
+editorial history of this article is available here:
+https://doi.org/10.1371/journal.pone.0279350
+Copyright: © 2023 Kilpatrick et al. This is an open
+access article distributed under the terms of the
+Creative Commons Attribution License, which
+permits unrestricted use, distribution, and
+reproduction in any medium, provided the original
+author and source are credited.
+Data Availability Statement: All data, scripts and
+an explanation of their implementation are available
+at the following OSF repository. https://osf.io/
+pe24w/?view_only=
+02e9327a7bd54b9280b57434a90ed83a
+
+small objects, while words containing [a] are typically associated with larger objects [2–5].
+Humans understand certain sound symbolic associations in infancy and these associations are
+said to scaffold language development and facilitate word learning [6–9]. It is therefore impor-
+tant for any NLP algorithm to understand sound symbolism if its goal is to understand lan-
+guage in the same way that humans can. This study is concerned with the random forest
+algorithm (further RF: [10]), which is an ensemble method machine learning algorithm typi-
+cally applied to classification and regression tasks. It builds upon recent research by Winter
+and Perlman ([11]; see also [12]), who used RFs to show that there is a systematic sound-sym-
+bolic relationship between size and phonemes in English words.
+In the following, we construct and test RFs using the fictional names of characters known
+as Poke´mon. Initially released in 1996 as a video game, Poke´mon is an incredibly popular
+mixed-media franchise, particularly in its country of origin, Japan [13]. The present study
+measures the classification accuracy of RFs against that of Japanese university students. The
+RFs are trained to classify Poke´mon into pre-evolution and post-evolution categories using
+only the sounds that make up their names. In Experiment 1, three RFs are constructed using
+the sounds that make up the names of Japanese, Mandarin Chinese (hereafter: Chinese), and
+South Korean (hereafter: Korean) Poke´mon. These RFs are trained using a subset of each data-
+set and then tested on the remaining data. While all RFs classify Poke´mon at a rate better than
+chance, the Japanese RF was found to perform the best, hence the remaining experiments are
+conducted on Japanese participants and Japanese Poke´mon names only. The Japanese RF is
+then tested using the results of an elicitation experiment where Japanese participants were
+asked to name previously unseen Poke´mon presented next to a pre/post-evolution parallel. A
+further RF is constructed using the responses from the elicitation experiment and tested both
+on the elicitation responses and the official Japanese names. In Experiment 2, we retrain the
+RFs presented in Experiment 1 to include name length. These retrained RFs uncover an issue
+of overfitting caused by a lack of variability in decision trees. We resolve this issue through
+cross-validation by constructing multiple random forests (MRFs) with different starting values
+for the randomization of splitting the data into training and testing subsets. The mean accu-
+racy of the RFs in the Japanese MRF is then compared to the results of a classification experi-
+ment where Japanese participants were asked to classify the elicited responses from
+Experiment 1 into pre- and post-evolution categories. The results of the human participants in
+the categorization experiment are then measured against the results of the MRFs. To summa-
+rize, Experiment 1 tests whether RFs can learn to make classification decisions using the
+sounds that make up names and whether this learning is applicable to elicited samples, and
+Experiment 2 measures the performance of MRFs against humans.
+Sound symbolism
+One of the standard assumptions of modern linguistic theory is that the relationship between
+sound and meaning is arbitrary [1,14]. While language is undoubtedly capable of associating
+sounds and meanings in arbitrary ways, the last few decades have seen a growing number of
+studies that reveal systematic relationships between sounds and meanings [15–17]. One well-
+known example is the takete-maluma effect [18] which is the observation that voiceless obstru-
+ents are typically associated with jagged-shaped objects, while names with sonorant sounds are
+more often associated with round-shaped objects. This effect has been shown to hold cross-lin-
+guistically [19–23]. While relationships between sound and meaning can be systematic, they
+are typically stochastic in nature [24]; that is, sound-meaning relationships manifest them-
+selves as a probability distribution that show statistical skews but may not be hold in all lexical
+items. For example, English adjectives like tiny, mini, and itsy bitsy adhere to the high front
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+2 / 27
+Funding: AK - Grant obtained from Japan Society
+for the Promotion of Science (Tokyo, JP)
+GRANT_NUMBER: 20K13055 https://www.jsps.go.
+jp/english/index.html The funders had no role in
+study design, data collection and analysis, decision
+to publish, or preparation of the manuscript.
+Competing interests: The authors have declared
+that no competing interests exist.
+
+vowel equates to smallness pattern discussed above, while the English adjective small is a clear
+exception to this generalization [11]. Sound symbolism is demonstrably important for lan-
+guage acquisition processes [21,25]; symbolic words are more common in both child-directed
+speech and early infant speech [26,27], and indeed, research has shown that infants are sensi-
+tive to sound symbolism [6–9].
+Poke´monastics is a relatively new subfield of sound symbolism that examines sound sym-
+bolic relationships between the names of video game characters known as Poke´mon and their
+attributes. In the video games, the player character collects Poke´mon, which they use to battle
+other players. As Poke´mon earn experience, many have the option to evolve. Poke´mon evolu-
+tion permanently changes the Poke´mon, they typically grow larger and stronger, and their
+names change. Poke´monastic studies have shown that Poke´mon evolution status can be sig-
+naled via some sound symbolic means in English and Japanese by an increase in name length,
+increased use of voiced obstruents, and in vowel use where the high front vowel [i] is typically
+associated with pre-evolution Poke´mon [28–31]. Based on these established relationships and
+the likelihood that the participants would be familiar with the subject, Poke��mon evolution was
+determined to be a suitable test case for measuring the ability of RFs against humans in under-
+standing sound symbolism (see also [11]).
+Random forests
+RFs, first introduced by Breiman [10], are ensemble method machine learning algorithms that
+are typically applied to classification and regression tasks. Since their inception, RFs have been
+a popular tool in machine learning, and several recent review articles attest to their efficacy
+[32–34]. Typically, RFs work by constructing many decision trees using a two-thirds subset of
+the data, they are then tested on the remaining data. Decision trees themselves are non-
+parametric supervised machine learning algorithms that resemble flow charts where each
+internal node represents a test of features. The decision tree splits at each node based on how
+important each feature is in the task. Splits eventually lead to a terminal node in the decision
+tree, which depicts the outcome of the decision-making process. Decision trees can be
+extremely useful; they are scale-invariant, robust to irrelevant features and inherently inter-
+pretable. However, decision trees are sensitive to noise and outliers, and are thus prone to
+overfitting data which limits their ability to generalize to unobserved samples [35,36]. Overfit-
+ting is a modelling error that occurs when a function is too closely aligned to a limited set of
+data points. This results in a model that performs well for the trained dataset but may not gen-
+eralize well to other datasets. To address the issue of overfitting, RFs use bootstrap aggregating
+(bagging: [37]) and the random subspace method [38]. Bagging involves using many decision
+trees to improve the stability and accuracy of the algorithm by averaging voting (in classifica-
+tion) or the output (in regression). In bagging, samples are randomly allocated to trees, typi-
+cally with replacement, which raises the issue of duplication. The random subspace method
+resolves this issue by randomly selecting a subset of features at each internal node, which
+allows the model to better generalize by introducing variability into the decision trees. In other
+words, bagging randomly selects samples while the subspace method randomly selects fea-
+tures. By randomizing the decision trees across both dimensions, random forests resolve the
+issue of overfitting inherent in decision trees.
+Experiment 1: Elicitation
+Material and methods
+The data, an explanation of the data, and a detailed annotated script for the following algo-
+rithms are available under the OSF repository.
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+3 / 27
+
+Official Poke´mon name data. All data were obtained from Bulbapedia ([39], last accessed
+in June 2022). As of June 2022, Bulbapedia has completed (mainspaced in the parlance of the
+website) lists for Japanese, Chinese, Korean, English, German, and French Poke´mon. Japanese,
+Chinese, and Korean names were selected for this experiment on the basis that Japanese kata-
+kana, Chinese pinyin, and Korean McCune-Reischauer romanisation are reasonably phonetic
+scripts. An algorithm was created for each language to count the number of times each sound
+occurs in each name. The algorithms and a detailed explanation for their implementation are
+included in the above OSF repository. This resulted in an almost entirely phonemic analysis
+except in the case of tones in Chinese, which are counted as separate features, and voicing on
+plosives in Korean. In Korean [40] and Chinese [41], there is no phonological opposition
+between voiced and voiceless plosives. However, Korean plosives are systematically voiced
+when they occur intervocalically [40], and this is reflected in the McCune-Reischauer romani-
+sation of Korean. Given that voiced plosives have been shown carry information pertaining to
+Poke´mon evolution in other languages [29,31], intervocalic plosives were counted separately
+in Korean.
+As of June 2022, there are 905 Poke´mon that span eight generations. This study only exam-
+ines the names of pre-evolution and post-evolution Poke´mon. Some Poke´mon do not evolve
+and are therefore not included in the current study. The sixth generation of the core video
+game series saw the introduction of a mechanic known as Mega Evolution that temporarily
+transforms certain Poke´mon. Mega evolution is not considered by the present study because
+this is a temporary transformation that has little effect on Poke´mon names other than the addi-
+tion of prefixes like mega. Other Poke´mon that were excluded from the analysis are mid-stage
+evolutionary variants. An example of a mid-stage Poke´mon is Electabuzz which was intro-
+duced in the first generation of the video game series. Its pre-evolution variant, Elekid, was
+introduced in the second generation, and its post-evolution variant, Electivire, was introduced
+in the fourth generation. In the present study, we exclude Electabuzz from the analysis because
+it is considered the mid-stage variant, despite other Poke´mon being added to the evolutionary
+family retroactively. Kawahara and Kumagai [28] analysed the relationships between the
+sounds in the names of Poke´mon and Poke´mon evolution where they did not exclude mid-
+stage Poke´mon. To achieve this, they had four categories based on evolution level rather than
+binary pre- and post-evolution categories. RFs are capable of multiclass classification; however,
+we opted for binary classification for the current analysis because, while the data is technically
+count data, it is almost entirely binary (e.g., 96.7% of all data points in the Japanese dataset are
+either 0 or 1). Therefore, it made sense to use a binary classifier given that the sound symbolic
+patterns are likely scalar across mid- and final-stage categories. The removal of mid-stage
+Poke´mon and Poke´mon with no evolutionary family resulted in 628 unique Poke´mon names,
+303 of which are pre-evolution and 325 of which are post-evolution. The reason for the distri-
+bution skew is because certain pre-evolution Poke´mon may evolve into multiple post-evolu-
+tion variants.
+Elicitation experiment.
+This experiment received ethics approval from the Nagoya Uni-
+versity of Business and Commerce. ID number 21048.
+The elicitation experiment has two main goals. The first is to determine whether an RF con-
+structed using the official Poke´mon name data can be used to classify names elicited from par-
+ticipants and vice versa. In other words, is there enough overlap between the official names
+and names provided by participants for each model to be useful in classifying Poke´mon from
+the alternate dataset. The second goal is to provide stimuli for a categorization experiment
+(Experiment 2) designed to measure the performance of human participants against the
+machine learning algorithms. To get a fair measurement of classification accuracy, it was
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+4 / 27
+
+important to test both humans and the machine learning algorithms on data that they had not
+previously been exposed to, hence the need for elicited samples.
+The elicitation experiment was conducted using Google Forms. Each Google form con-
+sisted of a short instructional paragraph, followed by twenty Poke´mon-like images. Following
+the method outlined in Kawahara & Kumagai [28], these images were not of existing Poke´mon
+and had likely not been previously viewed by the participants. The instructions noted that only
+native Japanese speakers were to take the survey. Participants were informed that they were to
+name twenty new Poke´mon. It was made clear to participants that they would be shown
+images of pre- and post-evolution Poke´mon. Participants were asked to provide names for
+Poke´mon in katakana which is the script used for Poke´mon names and nonce words in Japa-
+nese. Participants were instructed not to use existing words (Japanese or otherwise) to name
+the Poke´mon. Participants were given no further instructions (such as length limitations)
+regarding naming the Poke´mon. Participants were not asked if they were familiar with the
+Poke´mon franchise prior to completing the survey. All instructions were written in Japanese.
+Participants were informed that their participation was entirely voluntary, that they may quit
+the survey at any time. Consent was obtained verbally and it was explained to participants that
+their participation also constituted consent. No personal data were collected other than stu-
+dent email addresses which were collected to ensure that students were not completing the sur-
+vey twice. These were discarded prior to the analysis.
+Each image contained a pre-evolution and a post-evolution Poke´mon presented side by
+side. The pre-evolution Poke´mon was always located to the left of the post-evolution Poke´mon
+and was always presented as substantially smaller (see Fig 1) than its post-evolution counter-
+part. In each image, there was an arrow pointing to the Poke´mon that was to be named. Images
+with arrows pointing to the pre-evolution Poke´mon were always followed by an identical
+image, except the arrow would be pointing to the post-evolution Poke´mon. Trials were not
+randomized, and the pre-evolution image was always followed by the post-evolution image.
+Pre-evolution Poke´mon were always presented on the left and post-evolution Pokemon were
+always presented on the right. The images were created by a semi-professional artist (Devian-
+tArt user: Involuntary-Twitch), and samples are presented in Fig 1. The images very closely
+resemble the pixelated images used to represent Poke´mon in the earlier generations of Poke´-
+mon games.
+Participants were recruited from the Nagoya University of Commerce and Business via a
+post on the student bulletin board. Students were not compensated for their time monetarily
+or otherwise. The human participants needed to be somewhat familiar with the subject matter
+because sound-symbolic relationships in fictional names may not adhere to those found in nat-
+ural languages. Given the popularity of Poke´mon in Japan and that the participants were Japa-
+nese university students, Poke´mon was determined to be a good test case for assessing the
+accuracy of RFs against that of humans. Forty-nine students responded to the survey. In total,
+980 responses were recorded; however, some responses were blank and other responses con-
+tained duplicate names, the distribution of which suggested that participants had possibly con-
+ferred while taking the survey. These were discarded, resulting in 967 unique names (482 pre-
+evolution; 485 post-evolution). Elicited names were transcribed using the same algorithm used
+for the official Japanese Poke´mon names. None of the names collected in the elicitation experi-
+ment were names of existing Poke´mon.
+Random forests.
+Random forests were constructed and tested using the ranger package
+0.13.1 [42]. The number of trees included in each RF was manually tuned by constructing nine
+RFs at different tree number values with different starting points for randomization (set.seed).
+Optimal values were determined by examining mean out of bag (OOB) accuracy and its stan-
+dard deviation. OOB error refers to incorrectly classified samples. For all RFs, 20,000 trees
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+5 / 27
+
+were determined to be a suitable size because we observed no reduction in OOB error with
+increased trees and because calculating feature importance using the Altmann method [43] at
+20,000 trees approached the processing capability of the computer the RFs were constructed
+upon. Hyperparameters pertaining to the number of features examined at each node, the sam-
+ple fraction, and node size were tuned using the tuneRanger package 0.5 [44]. Essentially, the
+tuning process determines how much variability there is between trees. Highly variable trees
+Fig 1. Sample stimulus pairs of pre- and post-evolution Poke´mon characters used in Experiment 1. These images
+are reproduced with the permission of the artist.
+https://doi.org/10.1371/journal.pone.0279350.g001
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+6 / 27
+
+↑will produce highly variable results but might encounter issues with datasets that contain
+many unimportant features or null values. Low variability in decision trees results in more sta-
+ble algorithms but may mask the importance of weaker features because they will often be
+paired with strong features. The accuracy of the RF is determined by feeding the testing data
+into the model and assessing the OOB error. The OOB error gives an overall representation of
+the accuracy of the algorithm but does not communicate which features are important in clas-
+sification, which is instead determined by feature importance. There are several ways to calcu-
+late feature importance, the present study uses permutation. In permutation, each feature is
+randomized individually, and then the algorithm is reconstructed with all other features
+remaining the same. Feature importance is calculated on the increase of OOB error due to ran-
+domization. One issue with the interpretability of RFs is that feature importance does not com-
+municate directionality. For example, those sounds that are important to classification may be
+considered as “pulling” each sample into one category or the other, while feature importance
+communicates the strength of the “pull”, it does not communicate whether that “pull” is in the
+direction of the pre- or post-evolution category. In the present study, we report on the distri-
+bution of speech sounds to pre- and post-evolution categories.to indicate directionality,
+though it should be noted that they are not necessarily the same measure.
+In total, there were six RFs constructed for Experiment 1. The first three RFs presented in
+the results section were trained using a randomly sampled subset consisting of two-thirds of
+the Japanese, Chinese and Korean Poke´mon names. The fourth RF is trained using two-thirds
+of the results of the elicitation experiment. All four RFs are then tested using the remaining
+one-third subset of each dataset. We then calculate feature importance for each RF to examine
+potential cross-linguistic patterns, and patterns between the Japanese Poke´mon data and the
+elicited data. The remaining two RFs are constructed using the entirety of the official Japanese
+Poke´mon names and the entirety of the samples collected in the Elicitation experiment. These
+two RFs are then tested using the alternate dataset. In other words, one RF is constructed
+using all the official names and tested on the elicited responses, while the other is constructed
+using all the elicited samples and tested on the official names. This is done to determine
+whether there is enough overlap in the two datasets for the algorithms to be useful in classify-
+ing the opposite dataset.
+Results
+The three RFs trained and tested on the official Poke´mon names all classified Poke´mon at a
+rate better than chance. Given that there is an uneven distribution of pre- and post-evolution
+Poke´mon, any model that naïvely classified to the majority category would achieve an accuracy
+of around 52% (OOB error 48%) depending on the split of the training and testing subsets.
+The Japanese RF was the most accurate (OOB error 29.05%), followed by the Chinese RF
+(OOB error 39.05%), and finally, the Korean RF (OOB error 40.95%). A confusion matrix for
+the Japanese RF is presented in Table 1 and feature importance for the Japanese RF is pre-
+sented in Table 2. Note here that in Experiment 2, we report on the results of MRFs with differ-
+ent starting values for the randomization of both splitting the data in the training and testing
+Table 1. Confusion matrix for the Japanese RF.
+Classification
+Pre-evolution
+Post-evolution
+Sample
+Pre-evolution
+69
+38
+Post-evolution
+23
+80
+https://doi.org/10.1371/journal.pone.0279350.t001
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+7 / 27
+
+subsets, and the RFs themselves. The results of the MRFs (OOB error: M = 34.07%,
+SD = 2.48%) suggest that this result was an outlier caused by a particularly advantageous split
+between training and testing subsets. This process was conducted for the Chinese (OOB error:
+M = 40.85%, SD = 3.35%) and Korean (OOB error: M = 43.28%, SD = 3.09%) datasets as well.
+The RF trained and tested on the elicited names (Elicited RF) classified samples at a rate better
+than chance. As with the official datasets, there was an uneven distribution to categories, a
+naïve model would accurately classify samples in the elicited data 50.16% (OOB error 49.84%)
+of the time. The Elicited RF achieved an OOB error of 30.96%. Feature importance was calcu-
+lated for each model to determine which sounds contributed to classification. Feature impor-
+tance and significance is calculated using the Altmann [43] permutation method on the
+training subsets. Permutation involves randomizing features individually; the random forest is
+then reconstructed for each feature. Feature importance is the increase in OOB error for the
+feature being randomized. The Altmann permutation method involves running multiple per-
+mutations to estimate more precise p values. Feature importance significance is calculated by
+normalizing the biased measure based on a permutation test. This returns a significance result
+for each feature, not for the random forest itself [43]. All RFs in the present study use the Alt-
+mann permutation method with the number of iterations set at 100. Directionality was deter-
+mined by the distribution of features in the training subsets of the data. The distribution of
+features in the Japanese training subset is presented in Fig 2. In the Japanese RF, the most
+important features were the bilabial nasal (/m/), the coda nasal (/ɴ/), long vowels (/:/), and the
+voiced velar plosive (/g/). Of these features, only /m/ occurs more frequently in the pre-evolu-
+tion samples.
+As with the Japanese RF, the distribution of most features that were important in the Chi-
+nese RF skewed towards the post-evolution category. A confusion matrix for the Chinese RF is
+presented in Table 3 and feature importance scores for its features are presented in Table 4,
+and distribution is presented in Fig 3. Tones are an important feature in the RF; where the fall-
+ing tone occurs more frequently in the post-evolution samples, the neutral tone occurs more
+frequently in the pre-evolution samples. The velar nasal (/η/) was also found to be an impor-
+tant feature in the Chinese RF.
+Table 2. Feature importance (Importance) and p values for features that achieved a feature importance greater
+than 0.1% in the Japanese RF.
+Feature
+Importance
+p value
+/m/
+0.78%
+0.030
+/ɴ/a
+0.63%
+0.020
+/:/b
+0.45%
+0.049
+/g/
+0.40%
+0.049
+/a/
+0.39%
+0.139
+/ɾ/
+0.35%
+0.089
+/Q/c
+0.24%
+0.059
+/ɸ/
+0.20%
+0.079
+/ʈ ͡ɕ/
+0.19%
+0.069
+/d ͡ʒ/
+0.19%
+0.129
+/d/
+0.19%
+0.228
+a /ɴ/ represents the coda nasal.
+b /:/ represents the second portion of long vowels.
+c /Q/ represents the initial portion of geminate consonants.
+https://doi.org/10.1371/journal.pone.0279350.t002
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+8 / 27
+
+In the Korean RF, vowels /ɯ/, /a/, /ʌ/, and /u/ were important, as was the voiced labial-
+velar approximant /w/. Interestingly, while the close back unrounded vowel, /ɯ/ was present
+more often in post-evolution samples, the close back rounded vowel /u/ was present more
+often in pre-evolution samples. Table 5 presents a confusion matrix for the Korean RF, Table 6
+presents the feature importance and p values, and Fig 4 presents the distribution.
+Fig 2. Distribution of features to pre- and post-evolution categories in the Japanese training subset. Features
+appear in order of importance from left to right. Asterisks denote significant features.
+https://doi.org/10.1371/journal.pone.0279350.g002
+Table 3. Feature importance (Importance) and p values for features that achieved a feature importance greater
+than 0.1% in the Chinese RF.
+Classification
+Pre-evolution
+Post-evolution
+Sample
+Pre-evolution
+53
+50
+Post-evolution
+29
+78
+https://doi.org/10.1371/journal.pone.0279350.t003
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+9 / 27
+
+1.00 -
+0.75 -
+Distribution
+Condition
+0.50 -
+Post-Evolution
+Pre-Evolution
+0.25 -
+0.00
+Im/ IN*
+la/
+r
+FeatureTable 4. Feature importance (Importance) and p values for features that achieved a feature importance greater
+than 0.1% in the Chinese RF.
+Feature
+Importance
+p value
+Falling tone
+0.88%
+0.020
+/η/
+0.87%
+0.010
+/ʈ ͡ɕ/
+0.20%
+0.089
+/ɕ/
+0.14%
+0.109
+/e/
+0.13%
+0.238
+/o/
+0.13%
+0.257
+Neutral tone
+0.13%
+0.188
+https://doi.org/10.1371/journal.pone.0279350.t004
+Fig 3. Distribution of features to pre- and post-evolution categories in the Chinese training subset. Features
+appear in order of importance from left to right. Asterisks denote significant features.
+https://doi.org/10.1371/journal.pone.0279350.g003
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+10 / 27
+
+1.00-
+0.75-
+uopnqusia
+Condition
+0.50 -
+Post-Evolution
+Pre-Evolution
+0.25 -
+0.00
+Falling tone
+lel
+Neutral tone
+Inr
+Ic/
+FeatureTable 5. Confusion matrix for the Korean RF.
+Classification
+Pre-evolution
+Post-evolution
+Sample
+Pre-evolution
+57
+50
+Post-evolution
+36
+67
+https://doi.org/10.1371/journal.pone.0279350.t005
+Table 6. Feature importance (Imp.) and p values (p) for features that achieved a feature importance greater than
+0.1% in the Korean RF.
+Feature
+Importance
+p value
+/ɯ/
+2.558%
+<0.001
+/a/
+1.326%
+0.0297
+/w/
+0.226%
+0.0693
+/ʌ/
+0.207%
+0.1287
+/u/
+0.113%
+0.1782
+https://doi.org/10.1371/journal.pone.0279350.t006
+Fig 4. Distribution of features to pre- and post-evolution categories in the Korean training subset. Features appear
+in order of importance from left to right. Asterisks denote significant features.
+https://doi.org/10.1371/journal.pone.0279350.g004
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+11 / 27
+
+1.00 -
+0.75 -
+Distribution
+Condition
+0.50 -
+Post-Evolution
+Pre-Evolution
+0.25 -
+0.00 -
+Iur
+/a
+IN
+Iu/
+FeatureMost of the features that were important in the Japanese RF were also important in the Elic-
+ited RF. These include voiced plosives (/g/ & /d/), the open front unrounded vowel (/a/), the
+coda nasal (/ɴ/), and long vowels (/:/). Interestingly, all the features that achieved a feature
+importance greater than 0.1% in the Elicited RF occurred more frequently in post-evolution
+Poke´mon. The confusion matrix for the RF constructed and tested on the data from the elicita-
+tion experiment are presented in Tables 7 and 8 presents the feature importance scores, and
+Fig 5 presents the distribution chart.
+Given that the Japanese RF and the Elicited RF feature importance patterns are reasonably
+similar, we wanted to test whether these RFs would be able to accurately classify samples from
+their opposite dataset. Important features that are shared between the two models are non-
+labial voiced obstruents such as /d/ and /g/, coda nasals, long vowels, and the low front vowel
+/a/. Interestingly, the distributional skew for all of these features is towards the post-evolution
+category. We tested each existing RF on the entirety of their opposite dataset (not just the test
+subsets). The Japanese RF was able to accurately classify the elicited samples 61.43% of the
+time (OOB error 38.57%), and the Elicited RF was able to accurately classify the official Japa-
+nese Poke´mon name samples 66.72% of the time (OOB error 33.28%) where naïve models
+would be expected to achieve an accuracy of 52% and 50.16% respectively. The confusion
+matrix for the RF trained using the official Japanese Poke´mon names and tested using the elic-
+ited samples is shown in Table 9. Table 10 shows the confusion matrix for the for the RF
+trained using the elicited samples and tested using the official Japanese Poke´mon names.
+Discussion
+All the RFs presented above performed better than a naïve algorithm would. For the Japanese,
+Chinese, and Korean RFs, a naïve algorithm would be expected to achieve an OOB error of
+48%. While the Japanese RF was shown to be the most accurate (OOB error 29.05%), the Chi-
+nese (OOB error 39.05%) and Korean (OOB error 40.95%) error rates were well below 48%.
+The elicited RF, for which a naïve algorithm would be expected to achieve an OOB error of
+50%, achieved an OOB error of 30.96%. Important to remember here is that the RFs were
+Table 7. Confusion matrix for the Elicited RF.
+Classification
+Pre-evolution
+Post-evolution
+Sample
+Pre-evolution
+103
+55
+Post-evolution
+45
+120
+https://doi.org/10.1371/journal.pone.0279350.t007
+Table 8. Feature importance (Imp.) and p values (p) for features that achieved a feature importance greater than
+0.1% in the Elicited RF.
+Feature
+Importance
+p value
+/d/
+1.263%
+<0.001
+/ɯ/
+1.059%
+<0.001
+/a/
+0.918%
+<0.001
+/g/
+0.838%
+<0.001
+/d ͡ʒ/
+0.448%
+<0.001
+/o/
+0.219%
+0.2277
+/ɴ/
+0.180%
+0.2376
+/:/
+0.140%
+0.2772
+/z/
+0.138%
+0.1287
+https://doi.org/10.1371/journal.pone.0279350.t008
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+12 / 27
+
+trained on only two-thirds of the data, so the RFs were efficient learners, given that they only
+had 419 samples to learn from. The RF trained on the official Japanese data and tested on the
+elicited data was trained on all 628 official Japanese samples and tested on all 967 elicited
+responses. Despite having more samples from which to learn, the RF trained on the official
+names and tested on the elicited responses (OOB error 38.57%) was less accurate than the RF
+trained and tested on the official names (OOB error 29.05%). Similarly, the RF trained on the
+Fig 5. Distribution of features to pre- and post-evolution categories in the elicited training subset. Features appear
+in order of importance from left to right. Asterisks denote significant features.
+https://doi.org/10.1371/journal.pone.0279350.g005
+Table 9. Confusion matrix for the RF trained using the official Japanese Poke´mon names and tested using the
+elicited samples.
+Classification
+Pre-evolution
+Post-evolution
+Sample
+Pre-evolution
+245
+237
+Post-evolution
+136
+349
+https://doi.org/10.1371/journal.pone.0279350.t009
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+13 / 27
+
+1.00
+0.75-
+Distribution
+Condition
+0.50 
+Post-Evolution
+Pre-Evolution
+0.25
+0.00
+Ia/t
+la/ 
+Ig/* Ids/
+1o/
+IN
+Featureentirety of the elicited responses and tested on the official names (OOB error 33.28%) was less
+accurate than the RF trained and tested on the elicited responses (OOB error 30.96%). Despite
+these differences, the official names and the elicited responses are similar enough to perform
+better than naïve algorithms.
+The feature importance scores of the RFs reveal interesting relationships between Poke´mon
+evolution status and the sounds that make up their names, some of which hold across lan-
+guages. While high front vowels did not achieve a feature importance greater than 0.1% in any
+of the RFs, the low front vowel /a/ and the high back unrounded vowel /ɯ/ were important in
+the Japanese, Korean, and Elicited RFs and were distributionally skewed towards post-evolu-
+tion in all cases. The result for the phoneme /a/ as representing post-evolution Poke´mon is in
+line with the well-known observation that nonce words containing [a] are larger than those
+containing [i] [2,45,46] given that post-evolution Poke´mon are typically larger than their pre-
+evolution counterparts. Interestingly, the high back rounded vowel /u/ was important in the
+Chinese model, but it skewed towards the pre-evolution category. Vowels were found to be
+important in the Korean model, particularly /ɯ/, /a/, /ʌ/, and /u/. Korean vowels have been
+found to hold sound symbolic correspondences between “light” and “dark” vowels [47]. These
+correspondences run counter to cross-linguistic patterns. For example, light vowels are
+defined as being low vowels and are said to reflect small, fast-moving entities, while dark (or
+high) vowels are said to reflect larger, slow-moving entities [48]. Our findings do not support
+this observation. Although the distribution of dark vowels /ɯ/ and /ʌ/ skew towards the post-
+evolution category, the distribution of the light vowel /a/ skews towards the post-evolution cat-
+egory, while the dark vowel /u/ skews towards the pre-evolution category. The finding that /a/
+is important to the Korean model and skews towards the post-evolution category is in line
+with [5] who found that Korean listeners judge nonce words to be larger when the contain [a].
+Long vowels were important in both the Japanese and the Elicited RFs, and they skewed
+towards post-evolution in both cases. This finding is reflected in previous Pokemon studies
+[29,30], which also show that long vowels are associated with increased size. These studies
+tend to suggest this can be explained by the iconicity of quantity which is the finding that larger
+objects are typically associated with longer names [49]. This is explored further in Experiment
+2. Lastly, tones in the Chinese RF were important to the model. The falling tone had the high-
+est feature importance in the Chinese RF and it skewed toward the post-evolution category.
+The neutral tone, on the other hand, skewed toward the pre-evolution category. In a similar
+Poke´monastic study, Shih et al., [50] found that the falling tone seems to be associated with
+increased power, evolution stage, and increased distribution to the male gender. This is seem-
+ingly more complex than what Ohala’s Frequency Code hypothesis [51] would predict as it
+simply states that low tones should reflect largeness while high tones should predict smallness;
+but makes no prediction regarding tone pitch contour. Shih et al., [50] propose that the falling
+tone has the steepest pitch of all Chinese tones, and that this may explain why this tone is icon-
+ically linked to largeness in Chinese.
+The Japanese nasal /ɴ/ and the Chinese nasal /η/ were important in the Japanese, Chinese,
+and Elicited RFs and skewed towards post-evolution in all cases. This is an interesting finding
+Table 10. Confusion matrix for the for the RF trained using the elicited samples and tested using the official Japa-
+nese Poke´mon names.
+Classification
+Pre-evolution
+Post-evolution
+Sample
+Pre-evolution
+180
+123
+Post-evolution
+83
+242
+https://doi.org/10.1371/journal.pone.0279350.t010
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+14 / 27
+
+given that both consonants can only occur in the coda position, although the coda nasal /η/ in
+Korean did not achieve a feature importance greater than 0.1%. Cross-linguistically, nasal con-
+sonants are generally associated with large entities [2,52], likely due to their low frequency [2].
+In Japanese, however, bilabial consonants have been found to be associated with images of
+cuteness and softness [53], which may explain why /m/ was both important in the Japanese
+model and was skewed towards the pre-evolution category. High back vowels in the Korean
+model present an interesting case study when examined through the lens of the relationship
+between cuteness and labiality in Japanese. In the Korean model, both high back vowels were
+found to be important. While the high back rounded vowel /u/ skewed towards the pre-evolu-
+tion category, the high back unrounded vowel /ɯ/ skewed towards the post-evolution cate-
+gory. This result suggests that the association between cuteness and labiality may be a cross-
+linguistic one; however, this suggestion is tentative given that the Korean labial-velar approxi-
+mant both skewed towards post-evolution and was important in the model. Berlin [2] suggests
+that nasal consonants can imply largeness given their low frequency energy; however, the bila-
+bial nasal /m/ skewed towards the pre-evolution category and was found to be important in
+the Japanese RF. In line with Shih et al. [31], who found that voiced plosives were reflective of
+size in Japanese and English Poke´mon names, voiced plosives /d/ and /g/ were important in
+the Japanese and Elicited RFs. Intervocalic plosives in Korean were counted separately due to
+maintaining systematic voicing in these positions; however, these did not achieve a feature
+importance greater than 0.1%.
+Experiment 2: Categorization
+Random forests
+The RFs presented in Experiment 1 were constructed using only the sounds that make up the
+names of Poke´mon. In Experiment 2, we reconstruct those RFs with name length as an addi-
+tional feature. Length was not included in the previous RFs because previous studies suggest
+that it is likely a highly important feature [29,31], the inclusion of which would likely mask the
+importance of other features. In all other aspects, the RFs presented in Experiment 2 follow the
+same method as those in Experiment 1, except in the case where multiple random forests
+(MRFs) are constructed independently of each other. In MRFs, the starting value for the ran-
+domization for splitting data into training and testing subsets, as well as the starting value for
+the randomization for each RF was set as the number where the RF fell in the RF sequence. So
+the first RF in each MRF had a set.seed value of 1 while the ninth RF had a set.seed value of 9.
+For MRFs, tuning was conducted on the first RF only and those hyperparameter settings were
+applied to all nine RFs in each MRF. This is because, as far as we can tell, there is no way to
+make the tuning process replicable and the data for individual RFs come from the same source.
+We test each MRF nine times using the testing subset for each random split. In those instances
+where the entirety of a dataset is tested against the entirety of a different dataset, as in the case
+of testing the accuracy of the official Japanese MRF against the elicited Japanese MRF, nine
+iterations of each test were run with different starting values for the randomization of the
+MRFs.
+Categorization experiment
+This experiment received ethics approval from the Nagoya University of Business and Com-
+merce. ID number 21057.
+In the categorization experiment, we took the elicited responses from Experiment 1 and
+asked Japanese participants to classify them as either pre-evolution or post-evolution Poke´-
+mon. One hundred samples were selected randomly from the 967 elicited responses. There
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+15 / 27
+
+was no control for distribution in the random sampling process because there is no distribu-
+tion control in the splitting of subsets for RFs. These samples were used to populate five Google
+Forms that held twenty elicited names each. The surveys were designed in this manner, rather
+than randomly sampling twenty names from the entire dataset, to simulate the voting process
+that decision trees undertake in RFs. This is discussed further in the results section below. The
+forms explained (in Japanese) to the participants that they were to assign new Poke´mon to either
+pre- or post-evolution categories. These choices were presented as buttons labelled 進化前 [pre-
+evolution] and 進化後 [post-evolution]. Participants were not asked if they were familiar with the
+Poke´mon franchise prior to completing the survey. Five QR codes were generated for each of the
+five Google Forms. The codes were printed on handouts and distributed to Japanese university
+students at the Aichi Prefectural University and the Nagoya University of Business and Com-
+merce. Other than the QR code, there was no other information on the handout except for the
+heading ポケモンクイズ [Poke´mon Quiz]. Handouts were distributed to students prior to club
+activities and scheduled classes. Students were not given any time in class to complete the survey.
+In total, 119 participants responded to the survey, and there were 10 instances where participants
+had failed to designate a category, resulting in 2,370 responses. As with Experiment 1, participants
+were informed that their participation was entirely voluntary, that they may quit the survey at any
+time. Consent was obtained verbally. No personal data were collected other than student email
+addresses which were collected to ensure that students were not completing the survey twice.
+These were discarded prior to the analysis. It was requested that students who had undertaken
+Experiment 1 were to refrain from taking Experiment 2.
+Results
+The aim of Experiment 2 is to compare the performance of RFs against that of humans in clas-
+sifying Poke´mon into pre- and post-evolution categories. In the categorization experiment,
+human participants had access to name length, so length was included in the algorithms to
+give the RFs access to this information. In the following, the distribution of length is examined
+across pre-and post-evolution in all four datasets. Then, all previous RFs are reconstructed to
+include length to ascertain its effects on OOB error. We also calculate the feature importance
+of length to determine how much it is contributing to OOB error. Finally, we report on the
+results of the categorization experiment and compare the accuracy of the human participants
+against that of the machine learning algorithms. Length was calculated on the sum of all
+sounds in each dataset except for Chinese tones. In an exploration of sound symbolic relation-
+ships in Poke´mon names, Kawahara and Kumagai [28] calculated name length on the number
+of moras in Japanese names because the mora is the most psycholinguistically salient prosodic
+unit [54]. Although decision trees are scale-invariant, we calculated length on the number of
+features to bring the Japanese length parameter in line with the Chinese and Korean parame-
+ters. Chinese tones were excluded from the length calculation because tones are a measure of
+pitch contour and do not contribute to the overall length of a name the same way that other
+speech sounds do. Despite this, Chinese names were longer than those in all other data sets,
+with both pre-evolution (M = 8.76, SD = 2.09) and post-evolution (M = 9.31, SD = 2.09) Poke´-
+mon names consisting of a median of nine sounds. Length in the Japanese, Korean, and elic-
+ited datasets were similar, with all pre-evolution names consisting of a median of seven
+sounds, and all post-evolution names consisting of a median of eight sounds. The difference
+between mean pre- and post-evolution length was greatest in the Elicited dataset (1.52), fol-
+lowed by the Japanese dataset (0.9), Korean (0.73), and finally Chinese (0.55). Mean, median
+and standard deviation for length across datasets are presented in Table 11. Fig 6 presents a
+boxplot of length in Chinese, Japanese, and Korean by evolution status.
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+16 / 27
+
+Table 11. Feature importance (Imp.) and p values (p) for features that achieved a feature importance greater than
+0.1% in the Korean RF.
+Language
+Measure
+Pre-evolution
+Post-evolution
+Chinese
+Median
+9
+9
+Mean
+8.76
+9.31
+Standard Deviation
+2.09
+2.09
+Japanese
+Median
+7
+8
+Mean
+7.28
+8.18
+Standard Deviation
+1.38
+1.26
+Korean
+Median
+7
+8
+Mean
+7.33
+8.06
+Standard Deviation
+1.88
+1.81
+Elicited
+Median
+7
+8
+Mean
+6.79
+8.31
+Standard Deviation
+2.07
+2.38
+https://doi.org/10.1371/journal.pone.0279350.t011
+Fig 6. Boxplot of length for pre- and post-evolution Poke´mon across the three languages.
+https://doi.org/10.1371/journal.pone.0279350.g006
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+17 / 27
+
+16 -
+12 -
+Evolution
+Length
+Pre-evolution
+Post-evolution
+8
+4
+Chinese
+Japanese
+Korean
+LanguageThe distribution of length in the official Japanese Poke´mon dataset and the elicited dataset
+were extremely similar. While there were more outliers in the elicited dataset, the median,
+upper and lower quartiles, and minimum and maximum scores (excluding outliers) were
+almost identical. Fig 7 presents a boxplot for the official Japanese Poke´mon name length, and
+the elicited name length presented side by side to illustrate these similarities.
+Length was excluded from the RFs in the previous section because it is clear from previous
+studies that length would be an important feature [29,31] and would likely mask the impor-
+tance of speech sounds. Its inclusion should therefore increase the accuracy of the models (or
+reduce OOB error). However, this was not the case. All previous RFs were reconstructed to
+include the length feature. These RFs underwent the same tuning process outlined in Experi-
+ment 1. The feature importance of length is presented in Table 12. Table 12 also presents the
+OOB error rates for the RFs constructed with (+L) and without (-L) length. The inclusion of
+length increased the OOB error of all but the Chinese RF, which should be the RF least affected
+by length because the difference in average length between pre- and post-evolution Poke´mon
+Fig 7. Official Japanese Poke´mon name length (Official) and elicited name length (Elicited) presented side by side
+in a boxplot.
+https://doi.org/10.1371/journal.pone.0279350.g007
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+18 / 27
+
+20 -
+15 -
+Designation
+Length
+Elicited
+10
+Official
+5-
+Pre-evolution
+Post-evolution
+Evolutionwas the smallest in the Chinese dataset. Confoundingly, length was shown to be an important
+feature in the RFs, yet its effects were not being exhibited by the difference in OOB error
+between +L and -L RFs.
+To explore a potential explanation for this, we examined the randomization processes used
+in the construction of RFs. For all RFs until this point, we used the same set.seed value, except
+for those used to tune the number of trees in each forest (num.trees). This value was used as
+the starting number to generate randomization for both the splits between training and testing
+data and the RFs themselves. In the method section of Experiment 1, we tuned num.trees by
+running nine iterations of each num.trees value with different set.seed values for the randomi-
+zation of RFs. We applied this method to the randomization of the splits between training and
+testing subsets and found a substantial amount of variation in OOB error. We ran nine itera-
+tions of each of the RFs presented in this study. Here, however, we adjusted the set.seed values
+for both the subset splits and the RFs. The set.seed values ranged from 1–9 for both +L and -L
+RFs, resulting in the same nine subset splits. The results of these are displayed in Table 13.
+There is no way to control for randomization in the tuning process, each time the tuning pro-
+cess is conducted, it returns different hyperparameter values even when conducted on the
+same data. Given that the nature of the data remained the same, the hyperparameter values
+used for the MRFs were taken from the previous RFs.
+To understand the reason why the Japanese RF in Experiment 1 achieved such a low OOB
+error, we examined the mean feature importance values for features in the -L Japanese MRFs
+and compared them to the feature importance of features in the -L Japanese RF. Table 14
+shows the confusion matrix for the Japanese MRF. Table 15 presents the feature importance
+values in the Japanese RF and the mean feature importance values in the Japanese MRF. Here
+we see that the Japanese RF outlined in Experiment 1 was over-emphasising the importance of
+features /m/, /ɸ/, and /Q/, and under-emphasising the importance of /ɴ/, /:/, /ɾ/, /d/, /ɯ/, /o/,
+and /s/. The latter three were not included in earlier tables and charts because they did not
+achieve a feature importance greater than 0.1% in the Japanese RF.
+Given that the randomization of subsets has such a large impact on OOB error, we recon-
+ducted the tests of the Japanese RF using the elicited data and the Elicited RF using the official
+Japanese Poke´mon data using both -L and +L datasets. We tested the entirety of the Japanese
+Table 12. OOB error rates for the RFs constructed in Experiment 1 (-L OOB), the OOB error for RFs constructed
+using length (+L OOB), and the feature importance of length in those RFs (L Imp).
+RF
+-L OOBa
++L OOBb
+L Impc
+Chinese
+41.90%
+41.43%
+0.25%
+Japanese
+29.05%
+30.95%
+4.74%
+Korean
+40.95%
+41.43%
+1.33%
+Elicited
+30.34%
+30.96%
+6.66%
+https://doi.org/10.1371/journal.pone.0279350.t012
+Table 13. Results of multiple random forests. The mean OOB error for RFs constructed without length (-L OOBM) and their standard deviation (-L OOBSD), the
+mean OOB error for the RFS constructed with length (+L OOBM) and their standard deviation (+L OOBSD), and the mean feature importance of length (L ImpM) in the
++L MRFs.
+MRF
+-L OOBM
+-L OOBSD
++L OOBM
++L OOBSD
+L ImpM
+Chinese
+40.85%
+3.35%
+39.36%
+4.52%
+0.51%
+Japanese
+34.07%
+2.40%
+31.69%
+3.01%
+5.47%
+Korean
+43.28%
+3.09%
+40.85%
+2.85%
+2.44%
+Elicited
+36.29%
+1.98%
+32.47%
+2.44%
+6.99%
+https://doi.org/10.1371/journal.pone.0279350.t013
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+19 / 27
+
+and elicited datasets on the Elicited and Japanese MRFs, respectively. Table 16 presents the RF
+trained on the official Japanese Poke´mon names and tested on the elicited samples. Table 17
+shows the confusion matrix for the RF trained using the elicited samples and tested on the offi-
+cial names.
+Table 14. Confusion matrix for the Japanese MRF trained and tested on multiple subsets from the official Poke´-
+mon names that include length as a feature.
+Classification
+Pre-evolution
+Post-evolution
+Sample
+Pre-evolution
+609
+314
+Post-evolution
+285
+682
+https://doi.org/10.1371/journal.pone.0279350.t014
+Table 15. Feature importance of sounds for Japanese RF trained on official Poke´mon names (RF Imp), mean fea-
+ture importance of sounds for Japanese MRF trained on official names (MRF ImpM), and mean standard devia-
+tion for the Japanese MRF (MRF ImpSD). Asterisks reflect a mean p value of less than 0.05.
+Feature
+RF Imp
+MRF ImpM
+MRF ImpSD
+/m/
+0.78%�
+0.39%�
+0.22%
+/ɴ/
+0.63%�
+1.29%�
+0.38%
+/:/
+0.45%�
+0.83%�
+0.21%
+/g/
+0.40%�
+0.45%
+0.20%
+/a/
+0.39%
+0.37%
+0.23%
+/ɾ/
+0.35%
+0.67%�
+0.26%
+/Q/
+0.25%
+0.01%
+0.05%
+/ɸ/
+0.21%
+0.09%�
+0.11%
+/t ͡ɕ/
+0.19%
+0.10%
+0.07%
+/d ͡ʒ/
+0.19%
+0.11%
+0.14%
+/d/
+0.19%
+0.50%
+0.26%
+/ɯ/
+0.51%�
+0.29%
+/o/
+0.27%
+0.18%
+/s/
+0.10%
+0.07%
+https://doi.org/10.1371/journal.pone.0279350.t015
+Table 16. Confusion matrix for the MRF trained on all official Japanese Poke´mon names and tested on all elicited
+samples.
+Classification
+Pre-evolution
+Post-evolution
+Sample
+Pre-evolution
+3055
+1510
+Post-evolution
+1283
+2855
+https://doi.org/10.1371/journal.pone.0279350.t016
+Table 17. Confusion matrix for the MRF trained on all elicited samples and tested on all official Japanese Poke´-
+mon names.
+Classification
+Pre-evolution
+Post-evolution
+Sample
+Pre-evolution
+1436
+1291
+Post-evolution
+585
+2340
+https://doi.org/10.1371/journal.pone.0279350.t017
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+20 / 27
+
+To explore the issue of overfitting further, we reconstructed the -L Japanese MRF; this time,
+however, we skipped the tuning process and used the default hyperparameter settings in the
+Ranger package. This was done because we considered that the most likely reason for overfit-
+ting was low variability in decision trees due to the hyperparameter settings suggested by the
+tuning process. The untuned -L Japanese MRF (OOB error M = 35.94%, SD = 1.75%) was less
+accurate than the tuned MRF (OOB error M = 34.07%, SD = 2.4%), but the standard deviation
+was lower, suggesting that overfitting was less prevalent in individual RFs. We then recreated
+the untuned MRF using the entirety of the official Japanese names and tested it on the entirety
+of the elicited names and found the same pattern whereby the untuned MRF (OOB error
+M = 38.24%, SD = 0.42%) was less accurate, but more stable than the tuned MRF (OOB error
+M = 37.31%, SD = 1.26%) presented in Table 18.
+We considered a potential alternative explanation for the high standard deviation in tuned
+MRFs; that variability caused by the randomization of subset splits may be explained by an
+over/under-representation of pre-/post-evolution Poke´mon in the testing/training subsets. A
+simple regression model was constructed to predict the effect of increased post-evolution
+Poke´mon in the testing subset on OOB error for all the Japanese, Chinese and Korean RFs
+taken from the MRFs. Elicitation data was not included because the distribution of samples to
+pre-/post-evolution categories is different in the elicited responses. No correlation between
+distribution in subsets and OOB error was observed, F(1,25) = 0.31, p = 0.581, R2 = 0.01. We
+must therefore consider that the variability in accuracy when randomizing the subsets is most
+likely due to overfitting resulting from low variability in decision trees.
+In the classification experiment, 119 Japanese participants each classified twenty names
+into either pre- or post-evolution categories. The twenty names were taken from 100 randomly
+selected samples from the results of Experiment 1. The participants were reasonably accurate
+(M = 61.58%, SD = 17.84%) at assigning the elicited Poke´mon names to pre- and post-evolu-
+tion categories. This assessment was based on the individual responses taken from their mean
+accuracy. This is arguably an unfair assessment of human ability, given that sound symbolic
+associations are decided upon by speech communities, not individual speakers. In RFs con-
+structed for classification tasks, each decision tree votes for the classification of samples. The
+RF chooses the classification based on majority voting. To apply this method to the results of
+the classification experiment, we treated each response as a vote and examined the results of a
+majority vote analysis. Put simply, we examined the mode rather than the mean for each sam-
+ple. Using majority voting, the participants in the classification experiment were able to accu-
+rately classify 71% of the samples. The same 100 samples were then tested using each RF in the
+MRF constructed with the official Japanese names. The MRF was able to accurately classify the
+samples far more accurately than the humans, correctly classifying samples 75.88% of the time
+(OOB error M = 24.12%, SD = 1.61%).
+Discussion
+Experiment 2 was designed to test whether machine learning algorithms perform on par with
+humans, though it may not be immediately clear which of the MRFs presented in Table 18
+Table 18. Results of testing the Japanese MRF on the elicited data from Experiment 1 and the Elicited MRF on the Japanese data. This includes both MRFs that do
+not contain length as a feature (-L OOBM) and those that do (+L OOBM) as well as their standard deviation (-L OOBSD, +L OOBSD).
+Train data
+Test data
+-L OOBM
+-L OOBSD
++L OOBM
++L OOBSD
+Japanese
+Elicited
+37.31%
+1.26%
+32.09%
+0.75%
+Elicited
+Japanese
+35.86%
+2.01%
+33.19%
+2.03%
+https://doi.org/10.1371/journal.pone.0279350.t018
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+21 / 27
+
+should be used as a fair yardstick for the accuracy of the algorithms. We consider the results of
+the MRF trained using the length and sounds of all of the official Japanese Poke´mon names
+and tested on the 100 samples used in Experiment 2 (OOB error M = 24.12%) as the fairest
+measure for the performance of the algorithms, because they were trained on the maximum
+amount of information available that was also available to the human participants and tested
+on the same samples used in Experiment 2. Converting the responses of the human partici-
+pants to OOB error shows us that the human participants (OOB error M = 38.42%,
+SD = 17.84%) were far less accurate than the algorithm (OOB error M = 20.12%, SD = 1.61%),
+even when using the majority vote method (OOB error = 29%).
+The finding that the algorithms were more accurate than individual participants at classify-
+ing Poke´mon is unintuitive, particularly given the limited data upon which the MRFs were
+trained. One interpretation of this finding is that human participants do not give their best
+effort all the time, while machine learning algorithms do. This lack of effort may come down
+to a lack of motivation, not taking the survey seriously, or any number of other factors that are
+simply impossible to take into account. However, we contend that this does not account for
+the entirety of the difference in classification accuracy for the following reasons. Firstly, the
+categorisation experiment was voluntary; participants were not rewarded monetarily or other-
+wise for their participation. While the printed handouts were distributed prior to classes, the
+students were not given any time in class to complete the experiment. It was done entirely in
+their own time. Additionally, the task was brief, taking around 2–3 minutes to complete.
+Lastly, the subject matter was specifically chosen because it was appealing and familiar to the
+population sample. Based on these factors, we expect that participant interest would have been
+high and that many participants would have been invested in the experiment.
+Therefore, we believe that another interpretation may better explain the difference between
+participant and algorithm accuracy. That humans are susceptible to cognitive biases while
+machine learning algorithms are not. For example, humans will often apply oversimplified
+images or ideas to types of people or things, this is known as stereotyping. Through the lens of
+RFs, stereotyping is the overapplication of a feature to a category. Other cognitive biases sug-
+gest that humans do not intuitively understand probabilities, this is important given that
+sound symbolism is stochastic, not deterministic [24]. These biases include the recency bias
+(also known as the availability bias) which is the expectation that events that have occurred
+recently will reoccur regardless of their probability and the conjunction fallacy which is the
+assumption that a specific condition is more probable than a general one even when said spe-
+cific condition includes the general condition [55]. Indeed, other studies have shown that
+machine learning algorithms can outperform humans (see [56] for a recent review). For exam-
+ple, McKinney et al. [57] presented a machine learning algorithm that outperformed six expert
+readers of mammographs in breast cancer prediction performance. Compared to the expert
+radiologists, the algorithm showed an absolute reduction in both false positives and false nega-
+tives. Given the nature of the task and the human participants, we can reasonably safely assume
+that the difference in performance was not based on disinterest or lack of motivation on the
+part of the radiologists. We must therefore consider that the OOB error difference between the
+human participants and the algorithm in this study is potentially due to a difference in learning
+efficiency and the application of that learning.
+Length was omitted from the RFs presented in Experiment 1 because we wanted to isolate
+the feature importance of speech sounds, and the descriptive statistics suggested that Length
+was going to be an important feature that may mask the importance of weaker features.
+Indeed, Length was found to be important in all the MRFs. It was most important in the Elic-
+ited (6.99%) and Japanese (5.47%) MRFs which suggest that word length carries a considerable
+amount of sound symbolic information in Japanese. It was less important in the Korean
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+22 / 27
+
+(2.44%) and Chinese (0.51%) MRFs. Isolating Length from the other features in Experiment 1
+and introducing it into the RFs in Experiment 2 uncovered the issue of overfitting that led to
+the use of MRFs. The -L Japanese RF in Experiment 1 (OOB 29.05%) performed better than
+the +L Japanese RF in Experiment 2 (OOB 30.95%), despite Length being a highly important
+feature (4.54%) in the +L Japanese RF. The most likely explanation for overfitting is that there
+was little variability in decision trees. This hypothesis was tested by recreating the Japanese
+RFs using the default hyperparameter settings in the Ranger package. Running the untuned
+MRFs resulted in more stable RFs that were only slightly less accurate than their tuned coun-
+terparts. This finding supports our hypothesis that overfitting in Experiment 1 was the result
+of a lack of variability in decision trees. The lack of decision tree variability is likely due to a
+high number of features being examined at each node (mtry) which was suggested by the tun-
+ing process due to the large percentage of null values in the dataset (82.26%).
+A potential solution to this issue was explored, which involved constructing each RF using
+the default hyperparameter settings; however, this resulted in an increased OOB error in all
+cases. Another potential solution would be to reduce the number of null values by reporting
+on phonological features rather than the sounds themselves. This would reduce both the num-
+ber of null values and the number of features resulting in a less fine-grained data resolution.
+Instead, we constructed MRFs made up of independent RFs with different starting values for
+the randomisation of both the splitting of data into subsets and the RFs themselves. At first
+glance, MRFs may appear to be stacked RFs (SRFs: [58]), but this is not the case. Stacking [37]
+is a method of improving algorithm accuracy by combining weaker models into a super
+learner [59]. For example, Ha¨nsch [58] sequentially adds RFs to SRFs using the estimates of
+earlier RFs to improve the accuracy of the final model. Our method is more like k-fold cross-
+validation which involves randomly dividing the data into k groups, or folds, and then recom-
+bining the data by way of a partial Latin square to create multiple training/testing subsets
+which are then used for constructing and testing multiple iterations of the algorithm [60]. K-
+fold cross-validation was not used in the present study because if the user adheres to the two-
+thirds subset rule, they are limited in choice for the number of iterations.
+Conclusion
+The present study builds and tests machine learning algorithms using the names of Poke´mon.
+Those algorithms are constructed to classify Poke´mon into pre- and post-evolution categories.
+In Experiment 1, the algorithms are constructed using the speech sounds that make up Japa-
+nese, Chinese, and Korean Poke´mon names. The feature importance calculations of these algo-
+rithms show that while some sound-symbolic patterns hold across languages, many important
+features are unique to each language. Experiment 1 also includes an elicitation experiment
+whereby Japanese participants named previously unseen Poke´mon. We then construct RFs
+using the entirety of the official Japanese Poke´mon name data and the elicited responses and
+test them on their opposite dataset. The OOB error of these tests shows that the sound sym-
+bolic patterns in these datasets are reasonably similar, suggesting that either those sound sym-
+bolic patterns already exist in the Japanese language, or the participants are familiar with
+Poke´mon naming conventions. Previous studies have shown no correlation between Pokemon
+familiarity and sound symbolism effect size in nonce-word Poke´monastic experiments [61,62],
+suggesting that their results were not driven by existing knowledge of Poke´mon names. In
+Experiment 2, all algorithms are reconstructed to include name length as a feature. This
+uncovers an issue of overfitting, which we resolve using MRFs. The performance of the MRFs
+is then measured against the performance of Japanese participants. The MRFs are shown to
+perform more accurately than humans.
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+23 / 27
+
+RFs are said to be appropriate for “small N, high p” datasets [63], such as those found in the
+present study. However, Experiment 2 uncovers a clear case of overfitting in Experiment 1.
+The RFs constructed with length as a feature showed that length was important, yet this impor-
+tance was not always reflected in OOB error. For example, the Japanese +L MRF
+(OOB = 31.69%) performed worse than the -L RF in Experiment 1 (OOB = 29.05%). Given
+that length was found to be important in the MRFs, this suggests that the individual RFs were
+overfitting because of the lack of variability in decision trees. Further evidence for this can be
+found in the difference between the accuracy of the RF trained on official Japanese Poke´mon
+names to classify Elicited names (OOB = 38.57%) and the -L MRF trained on official Japanese
+Poke´mon names to classify Elicited names (OOBM = 37.31%). In other words, the Japanese RF
+in Experiment 1 was more accurate than the Japanese MRF at classifying its own testing subset
+but less accurate at classifying the elicited samples because its function was too closely aligned
+to the initial dataset, resulting in a reduced capacity to classify external samples.
+Sound symbolism is the study of systematic relationships between sounds and meanings.
+These relationships are not deterministic but rather stochastic, so they need to be observed
+through a statistical analysis. This paper details random forest algorithms that learn from these
+stochastic relationships and apply that learning to a classification task. Said task is the classifi-
+cation of Poke´mon into pre- and post-evolution categories. This finding has important impli-
+cations for the Natural Language Processing field of research, adding to the findings of Winter
+and Perlman [11] and showing that machine learning algorithms can make classification deci-
+sions driven (at least mostly) by sound symbolic principles, and should do so if the goal of an
+algorithm is to understand and use language the same way that humans do. The algorithms
+show how they make their classification decisions using feature importance, which is a useful
+metric for measuring the sound symbolic qualities of specific linguistic features. This is partic-
+ularly useful when assessing universal sound-symbolic patterns. The present paper also expo-
+ses an issue of overfitting inherent in random forests constructed using decision trees with low
+variability. This issue is resolved by randomizing training and testing subset splits across mul-
+tiple random forests. The machine learning algorithms are shown to be efficient learners in
+this task, achieving a higher classification accuracy than the human participants, despite hav-
+ing access to a limited number of samples from which to learn.
+Author Contributions
+Conceptualization: Alexander James Kilpatrick, Shigeto Kawahara.
+Data curation: Alexander James Kilpatrick.
+Formal analysis: Alexander James Kilpatrick.
+Funding acquisition: Alexander James Kilpatrick.
+Investigation: Alexander James Kilpatrick.
+Methodology: Alexander James Kilpatrick.
+Project administration: Alexander James Kilpatrick.
+Resources: Alexander James Kilpatrick.
+Software: Alexander James Kilpatrick.
+Supervision: Alexander James Kilpatrick.
+Validation: Alexander James Kilpatrick.
+Visualization: Alexander James Kilpatrick.
+PLOS ONE
+Random forests, sound symbolism and Poke´mon evolution
+PLOS ONE | https://doi.org/10.1371/journal.pone.0279350
+January 4, 2023
+24 / 27
+
+Writing – original draft: Alexander James Kilpatrick, Aleksandra C´wiek.
+Writing – review & editing: Alexander James Kilpatrick, Aleksandra C´wiek, Shigeto
+Kawahara.
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+

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+Astronomy & Astrophysics manuscript no. main
+©ESO 2023
+January 9, 2023
+What does a typical full-disc around a post-AGB binary look like?⋆
+Radiative transfer models reproducing PIONIER, GRAVITY, and MATISSE data
+A. Corporaal1, J. Kluska1, H. Van Winckel1, D. Kamath2, 3, 4, and M. Min5
+1 Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
+e-mail: [email protected]
+2 School of Mathematical and Physical Sciences, Macquarie University, Sydney, NSW, Australia
+3 Astronomy, Astrophysics and Astrophotonics Research Centre, Macquarie University, Sydney, NSW, Australia
+4 INAF, Observatory of Rome, Via Frascati 33, 00077 Monte Porzio Catone (RM), Italy
+5 SRON Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA, Leiden, The Netherlands
+Received ; accepted
+ABSTRACT
+Context. Stable circumbinary discs around evolved post-Asymptotic Giant branch (post-AGB) binary systems composed of gas and
+dust show many similarities with protoplanetary discs around young stellar objects. These discs can provide constraints on both
+binary evolution and the formation of macrostructures within circumstellar discs. Here we focus on one post-AGB binary system:
+IRAS 08544-4431.
+Aims. We aim to refine the physical model of IRAS 08544-4431 with a radiative transfer treatment and continue the near-infrared
+and mid-infrared interferometric analysis covering the H-, K-, L-, and N-bands. Results from geometric modelling of this data in our
+previous study constrain the shape of the inner rim of the disc and its radial dust structure. We aim to capture the previously detected
+amount of over-resolved flux and the radial intensity profile at and beyond the inner dust disc rim to put constraints on the physical
+processes in the inner disc regions.
+Methods. We used a three-dimensional Monte Carlo radiative transfer code to investigate the physical structure of the disc by re-
+producing both the photometry and the multi-wavelength infrared interferometric data set. We first performed a parametric study
+to explore the effect of the individual parameters and selected the most important parameters which were then used in a thorough
+grid search to fit the structural characteristics. We developed a strategy to identify the models which perform best to reproduce our
+extensive multi-wavelength data set.
+Results. We found a family of models that successfully fit the infrared photometric and interferometric data in all bands. These models
+show a flaring geometry with efficient settling. Larger grains are present in the inner disc as probed by our infrared interferometric
+observations. Some over-resolved flux component was recovered in all bands but the optimised models still fall short to explain all the
+over-resolved flux. This suggests that another dusty structure within the system plays a role, which is not comprised in our models.
+The structure of this over-resolved component is unclear but it has a colour temperature between 1400 and 3600 K.
+Conclusions. Multi-wavelength infrared interferometric observations of circumstellar discs allow to study the inner disc regions
+in unprecedented detail. The refined physical models can reproduce most of the investigated features, including the photometric
+characteristics, the radial extent, and the overall shape of the visibility curves. Our multi-wavelength interferometric observations
+combined with photometry show that the disc around IRAS 08544-4431 is similar to protoplanetary discs around young stars with
+similar dust masses and efficient dust growth. The resulting disc geometry is capable of reproducing part of the over-resolved flux but
+to fully reproduce the over-resolved flux component, an additional component is needed. Multi-scale high-angular resolution analysis
+combining VLTI, VLT/SPHERE, and ALMA data is needed to fully define the structure of the system.
+Key words. Stars: AGB and post-AGB - techniques: interferometric - binaries: general - protoplanetary discs - circumstellar matter
+1. Introduction
+Circumstellar discs composed of gas and dust are found at vari-
+ous stellar evolutionary stages. One class of evolved stellar sys-
+tems that show such circumstellar discs are post-asymptotic gi-
+ant branch (post-AGB) binaries. Their spectral energy distribu-
+tions (SEDs) show infrared excesses pointing toward the pres-
+ence of hot dust. It has now been well established that this prop-
+erty of the SEDs provides observational evidence for the pres-
+ence of stable, massive circumbinary discs around these systems
+(e.g. de Ruyter et al. 2006; Van Winckel 2003, 2017; Kamath
+⋆ Based on observations collected at the European Southern Obser-
+vatory under ESO programmes 094.D-0865, 0102.D-0760, 60.A-9275,
+and 0104.D-0739
+et al. 2014, 2015; Kluska et al. 2022). Observational properties
+of these systems are recently reviewed by Van Winckel (2018).
+These discs show evidence of stability and a Keplerian veloc-
+ity field has been spatially resolved at millimetre wavelengths in
+CO in several objects (Bujarrabal et al. 2013b, 2015, 2017, 2018;
+Gallardo Cava et al. 2021). More objects were detected in single
+dish and also their narrow CO profile is indicative of rotation
+(Bujarrabal et al. 2013a). Moreover, the dust grains show evi-
+dence of strong processing in a stable environment resulting in
+grain growth and a high degree of crystallinity as revealed by
+infrared spectroscopic observations and millimetre photometry
+(e.g. Gielen et al. 2011; Sahai et al. 2011).
+While the presence of these circumbinary discs around post-
+AGB binaries is well established, their formation, structure, and
+Article number, page 1 of 18
+arXiv:2301.02622v1  [astro-ph.SR]  6 Jan 2023
+
+A&A proofs: manuscript no. main
+evolution are still insufficiently understood. There is observa-
+tional evidence for interactions between the system’s compo-
+nents. Indirect observational evidence for disc-binary interac-
+tions is provided by analyses of time-series of spectroscopic ob-
+servations and photospheric abundance determinations. The for-
+mer observations show fast outflows or jets originating from an
+accretion disc around the companion (Gorlova et al. 2012, 2015;
+Bollen et al. 2017, 2019, 2022). To launch the jets, the mass-
+accretion rates onto the companion are found to be on the order
+of 10−6 to 10−4 M⊙/yr (Bollen et al. 2020). Such a mass-accretion
+rate cannot be due to the mass-loss rate of the post-AGB primary
+and is pointing towards accretion from the circumbinary disc as
+the main feeding mechanism of the circum-companion disc. Ad-
+ditional evidence for accretion from the circumbinary disc comes
+from the depletion of refractory elements observed on the pho-
+tosphere of the post-AGB primary itself (e.g. Van Winckel et al.
+1995; Maas et al. 2005; Giridhar et al. 2005; Oomen et al. 2018).
+Such depletion is caused by the re-accretion of gas from the cir-
+cumbinary disc (Oomen et al. 2019) while the refractory ele-
+ments remain on the dust grains in the disc. However, the cause
+of this dust-gas separation is still debated.
+Kluska et al. (2022) showed a link between the strength of
+the photospheric depletion and the lack of near-infrared (near-
+IR) excess in the SEDs of the systems (labelled transition discs),
+showing that the most depleted targets are surrounded by discs
+with a large dust free cavity. In young stellar objects (YSO) such
+a depletion pattern is also observed in targets hosting a transition
+disc and are often linked to planet-disc interactions.
+Constraining the physical processes in the inner disc regions
+and the disc-binary interactions of circumbinary discs around
+post-AGB binaries will improve our understanding of the late
+evolution of binary systems. Interferometric techniques in the
+infrared are needed to spatially resolve the inner disc regions as
+well as the inner binary.
+Two-dimensional radiative transfer modelling efforts of
+high-angular resolution interferometric data of such discs (Deroo
+et al. 2006, 2007; Hillen et al. 2014, 2015, 2017; Kluska et al.
+2018) have shown that passively irradiated disc models devel-
+oped for protoplanetary discs are able to reproduce the data.
+This points toward a similar structure of discs around post-AGB
+binary systems and protoplanetary discs around YSOs. An im-
+portant difference is, however, that estimated lifetime of discs
+around post-AGB binaries is only in the order of (104 − 105 yr),
+while protoplanetary discs live up to a few Myr.
+While there is growing evidence that initial phases of planet
+formation around YSOs can be short and grain growth is very
+efficient on short timescales (∼ 105 yr) (e.g. Sheehan & Eisner
+2018; Segura-Cox et al. 2020; Cridland et al. 2022; Lau et al.
+2022), it is as yet unclear what the formation timescale is of full-
+grown planets. Discs around post-AGB binaries, thus, represent
+an interesting laboratory to test processes for planet formation
+and this in a different parameter space than around YSOs.
+Here
+we
+focus
+on
+one
+post-AGB
+binary
+system,
+IRAS 08544-4431, hereafter IRAS 08544. IRAS 08544 is a
+luminous (∼
+10500 L⊙) post-AGB star with a confirmed
+companion (Maas et al. 2003). The binary is surrounded by
+an optically thick circumbinary disc with the near-IR excess
+providing evidence for a full disc with the inner rim located
+at the dust sublimation radius (de Ruyter et al. 2006; Hillen
+et al. 2016; Kluska et al. 2018; Kluska et al. 2022). It is indeed
+classified as a Category 1 disc in Kluska et al. (2022) and
+is the prototypical full disc surrounding a post-AGB binary.
+Interferometric observations in the near-IR have revealed the
+structure of the inner rim of the disc by geometric models, im-
+age reconstruction techniques, and radiative transfer modelling
+(Hillen et al. 2016; Kluska et al. 2018). With these techniques,
+a small resolved flux excess at the location of the companion
+was also identified. As this excess cannot be explained by
+photospheric emission from the main sequence companion star,
+this excess likely originates from a circum-secondary accretion
+disc.
+The radiative transfer model of IRAS 08544 from Kluska
+et al. (2018) provides a good fit to the SED and the H-band
+interferometric measurements. These authors find that the disc
+inner rim coincides with the theoretical dust sublimation radius.
+Their disc model requires, however, an ad-hoc over-resolved flux
+component of unknown origin on top of the disc model (see also
+Sects. 4.2 and 6.2). By performing 2D geometric modelling of
+both near-IR and mid-IR interferometric observations, Corpo-
+raal et al. (2021) reproduce the visibility data of the H-, K-,
+L- and N-bands and find that the inner rim of the circumbi-
+nary disc is rounded and puffed-up. In the near-IR, the inner rim,
+the stars, and a spatially extended component are detected. The
+over-resolved flux contribution to the total flux is rather constant
+throughout the H-, K-, and L-bands. In the mid-IR, however, the
+stars contribute only a few per cent to the total flux, and thus
+the visibilities are mainly dominated by thermal emission and
+scattering from the circumbinary disc. While the 2D geometric
+models (i.e. parameterised rings on the image plane) are able to
+reproduce most features in the visibility data, we now want to de-
+velop a 3D radiative transfer model of a disc with self-consistent
+handling of dust settling to infer the physical properties of the
+circumbinary disc.
+Here we present such a physical model of the circumbinary
+disc around IRAS 08544. We aim to reproduce the main charac-
+teristics of both the observed photometry and the interferometric
+visibilities in four infrared bands. We focus on investigating the
+over-resolved flux component and retrieve the radial profile of
+the inner disc regions using data of the current three four-beam
+combiners at Very Large Telescope Interferometer (VLTI): PIO-
+NIER, GRAVITY, and MATISSE. We summarise the observa-
+tions in Sect. 2 and the physical setup of the radiative transfer
+model in Sect. 3. We investigate our parameter space in Sect. 4
+and use the results of the most impacting parameters to refine the
+disc model in Sect. 5. We discuss the results and implications in
+Sect. 6 and summarise the conclusions in Sect. 7.
+2. Observations
+2.1. Photometry
+The energetics of the target are taken from the catalogue of
+Galactic post-AGB binary systems of Kluska et al. (2022) and
+we refer to this paper for a full description of the data collection.
+In short, the full SED is assembled by collecting public broad-
+band photometric data from a broad range of wavelengths (from
+0.3 µm to 0.8 mm). The parameters of the photospheric model of
+the post-AGB star were derived from Kurucz stellar atmosphere
+models (Castelli & Kurucz 2003).
+2.2. Interferometry
+We used the infrared interferometric data set presented in Cor-
+poraal et al. (2021). Here, we summarise the main characteris-
+tics of the data. The data set consists of observations obtained
+on the following three current four-telescope beam combiner in-
+struments the VLTI at Mount Paranal in Chile: PIONIER (Le
+Bouquin et al. 2011), GRAVITY (Gravity Collaboration et al.
+Article number, page 2 of 18
+
+A. Corporaal et al.: What does a typical full-disc around a post-AGB binary look like?
+2017), and MATISSE (Lopez et al. 2014, 2022). These instru-
+ments provide simultaneous observations on six baselines and
+three independent closure phases per pointing.
+The PIONIER instrument operates in the H-band (between
+1.5 µm and 1.85 µm). The data set was taken in 2015 (prog. ID:
+094.D-0865, PI: Hillen), using the three configurations on the
+1.8 m Auxiliary Telescopes (ATs). A log of these observations is
+reported in Hillen et al. (2016).
+GRAVITY is operating in the K-band (2.0 - 2.4 µm). The
+data are taken in 2018 and 2019 (prog. ID: 0102.D-0760, PI:
+Bollen), using the three configurations of the ATs at high resolu-
+tion (R ∼ 4000) in single field mode.
+The data set taken with the MATISSE instrument covers the
+L-band (2.9-4.2 µm) and N-band (8-13 µm). Observations were
+taken during 2019 (prog. ID 60.A-9275, PI: Kluska) and 2020
+(prog. ID 0104.D-0739, PI: Kluska) with the three configura-
+tions of the ATs. All observations are taken in the low spectral
+resolution mode (R ∼ 30). N-band photometry was not taken
+during the 2019 observations such that the coherent flux mea-
+surements could not be normalised and the visibility amplitudes
+could not be determined. As a result, the reported visibilities in
+the N-band are correlated fluxes.
+3. Physical setup
+To infer the physical structure of the dusty circumbinary disc
+of IRAS 08544 from the observations we used the Monte Carlo
+radiative transfer code MCMax3D (Min et al. 2009). In such a
+code, the photon packages emitted from the central stellar source
+are scattered, absorbed, or re-emitted by the dust particles. The
+user specifies the dust distribution by setting the disc structure
+and its density distribution as well as the dust grain properties
+and composition. In MCMax3D, the disc can be made of several
+zones with different parameters for the disc structure, which is
+defined by the dust disc inner and outer radii, the vertical dust
+settling, and the scale height. This flexibility allows the explo-
+ration of complex disc geometries.
+Vertical dust settling is handled self-consistently with a sin-
+gle parameter, the turbulent mixing strength, α (Shakura & Sun-
+yaev 1973), following the prescription of Mulders & Dominik
+(2012). Stronger turbulence mixing strengths imply more effi-
+cient settling as larger grains decouple from the gas.
+The disc vertical scale height is defined by a power law:
+h(r) = h0
+� r
+Rin
+�β
+,
+(1)
+where h0 is the scale height at the radius we set to coincide with
+the disc inner radius, Rin, and β is the flaring exponent describing
+the disc curvature. Grain sizes are distributed by a power law
+with index q:
+n(a) ∝ a−q
+for
+amin < a < amax
+(2)
+where amin and amax are the minimum and maximum grain sizes,
+respectively.
+The surface density profile is prescribed by a radial power
+law with index p and radius r:
+Σ(r) ∝
+� r
+rc
+�−p
+(3)
+The surface density is scaled to the dust mass in the disc. We ap-
+ply a two-zone model to adopt a double power law of the surface
+density to smooth the disc inner rim (Hillen et al. 2015; Kluska
+et al. 2018). In our two-zone model, there is an additional pa-
+rameter, the turn-over radius, rmid, at which the surface density
+profile (Eq. 3) changes as we apply different values of p in the
+zones. To ensure continuity in the full disc model, rmid corre-
+sponds to the outer radius of the inner zone and the inner radius
+of the outer zone. Besides the inner and outer radii of these discs
+and the surface density distribution, we do not consider differ-
+ences in the inner and the outer disc. The outcome of the Monte
+Carlo run is a three-dimensional temperature distribution within
+the disc. This thermal structure is used to calculate ray-traced
+synthetic spectra and images in a subsequent step.
+4. Parameter study
+We first aim to understand the impact of the individual param-
+eters on the SED and the visibilities. This will allow us to 1)
+select the parameters that impact the observables the most, such
+that we can constrain the disc parameters in a subsequent step
+to provide a good fit to the data and 2) investigate the parame-
+ter space of the individual parameters. This section is organised
+as follows: we describe our reference model in Sect. 4.1, discuss
+our strategy for our parametric study in Sect. 4.3, present the re-
+sults in Sect. 4.4, and select the most promising parameters to
+meet the shortcomings of the reference model in Sect. 4.5.
+4.1. The reference model
+Both the SED and the squared visibility measurements of the
+PIONIER data set for IRAS 08544 were reproduced by Kluska
+et al. (2018) using the 2D version of MCMax. Here, we build
+upon this model. We translated this model to MCMax3D to be
+able to incorporate azimuthally asymmetric features in the fu-
+ture. We assumed a distance of 1.22+0.01
+−0.003 kpc from the recent
+Gaia data release 3 (Gaia Collaboration et al. 2016, 2022) and
+rescaled the stellar and disc parameters accordingly. Gaia did not
+flag this object as an astrometric binary and hence this distance is
+determined using a single star fit to the astrometric data. The stel-
+lar photosphere of the post-AGB star is modelled from the results
+of the SED fitting by Kluska et al. (2022). Stellar masses from
+the updated distance estimate are calculated in the same way as
+the upper limit estimates of Kluska et al. (2018). The mass of
+the post-AGB star is estimated using the luminosity-core mass
+relation for post-AGB stars (Vassiliadis & Wood 1994), as it is
+expected to have lost most of its envelope. The stellar parameters
+are reported in Table 1.
+As we are interested in the radial and vertical disc structure,
+we neglect the binary nature of the system in our modelling as
+the post-AGB primary is much more luminous than the main se-
+quence companion and the binary separation is negligible com-
+pared to the size of the disc (Oomen et al. 2018). We also neglect
+the effect of the (asymmetric) irradiation of on the disc due to the
+non-central energy source (see Sect. 6.5 for future implications).
+In the post-processing phase, the contribution of the accretion
+disc around the secondary is taken into account in the photom-
+etry by adding a blackbody with a temperature of 4000 K and a
+flux contribution of 3.9% at 1.65 µm, which are taken from a fit
+to the PIONIER data by Hillen et al. (2016) and Kluska et al.
+(2018). Likewise, this contribution is added to the synthetic in-
+terferometric images by assuming for simplicity that the emis-
+sion coincides with the position of the primary.
+We used the refined model for dust opacity of the DIANA
+project (Woitke et al. 2016). Since the circumbinary discs around
+post-AGB binaries are mostly found to be oxygen-rich (e.g. Gie-
+len et al. 2011), the carbon fraction is set to zero, leaving the dust
+Article number, page 3 of 18
+
+A&A proofs: manuscript no. main
+Fig. 1: Spectral energy distribution and visibility curves of the four interferometric bands of the reference model. From left to right:
+the reddened spectral energy distribution, the H-band, K-band, and L-band squared visibility curves, and N-band correlated fluxes.
+The wavelength regimes of the H-, K-, L-, and N-bands and the reddened stellar photosphere are depicted in the SED plot by the
+grey vertical areas and by the brown curve, respectively.
+to consist of amorphous pyroxene silicates (Mg0.7Fe0.3SiO3).
+The ISM-like opacities that were assumed in the 2D model were
+found to best correspond to irregular shaped particles with a
+porosity, p, of 0% and a distribution of hollow spheres (Min et al.
+2005) with a maximum hollow volume ratio, fmax, of 0.7. We
+note that these are different from the standard DIANA opacity,
+as it takes a porosity of 25% and fmax = 0.8. The opacities are
+calculated using a MRN-like distribution following Mathis et al.
+(1977), who showed that the grain size distribution with q = 3.5
+reproduced the extinction curve in the Milky Way.
+We call this adapted version of the previous best-fit model
+our reference model. Parameters of this reference model can be
+found in Table 1. The performance of this model on the photom-
+etry and the infrared interferometric data sets are shown in Fig.
+1.
+4.2. The extended component
+The model of Kluska et al. (2018) needed an additional ad-hoc
+extended component contributing 8.1% in the H-band to fit the
+SED and reproduce the H-band interferometric measurements.
+This component points towards an over-resolved emission that
+is not reproduced by the reference model. Indeed, Fig. 1 shows
+that by excluding this extended component, both the photometric
+and the interferometric data are not well reproduced. The model
+visibility curve in the H-band lies above the PIONIER data as
+a result of this missing flux. This offset indicates that the stellar
+flux relative to the total is over-estimated.The model lacks pho-
+tometric fluxes in the near-IR and mid-IR, while the millime-
+tre fluxes are fitted well. For these reasons, we aim at finding
+models that have more photometric fluxes in all bands and more
+over-resolved flux in the H-, K-, and L-bands.
+One possibility to explain this over-resolved emission is that
+it is a scattering component coming from the outer disc and
+hence originating from a flared disc. For a more flared disc, we
+expect that the outer regions of the disc intercept more starlight
+such that there is more scattering.
+Besides the over-resolved flux component, the reference
+model fails to fit the visibilities at short baselines before the first
+zero, which provides a measure for the wavelength-dependent
+radial profile. This leads to a significant underestimation of the
+size of the emission in these bands.
+Table 1: Parameter space for the parametric study.
+Stellar propertiesa
+Parameter
+Value
+Teff (K)
+7250
+log g (dex)
+1.01
+RPost−AGB star (R⊙)
+65
+LPost−AGB star (L⊙)
+10500
+d (kpc)
+1.22
+Parameter
+reference model
+grid range
+α
+0.01
+[0.001, 0.01, 0.1]
+β
+1.2
+[1.2, 1.4, 1.6]
+amin(µm)
+0.1
+[0.01, 0.1, 1.0]
+gas/dust
+100
+[50, 100, 200]
+h0 (au)
+0.93
+[0.93, 1.40, 1.87]
+md (×10−3 M⊙)
+2.0
+[1.0, 2.0, 4.0]
+pin
+-1.5
+[-1.0, -1.5, -2.0]
+q
+2.75
+[2.50, 2.75, 3.25]
+rmid (Rin)
+3.0
+[2.5, 3.0, 3.5]
+Rin (au)
+7.20
+fixed
+Rout (au)
+175
+fixed
+amax (mm)
+1.0
+fixed
+i (deg)
+19
+fixed
+PA (deg)
+6
+fixed
+pout
+1.0
+fixed
+Dust composition
+reference model
+grid range
+porosity
+0%
+[0%, 25%]
+fmax
+0.7
+[0.7, 0.8]
+crystallinity fraction
+0.0
+[0.0, 0.2, 0.4]
+metallic iron
+0%
+[0%, 15%]
+Notes.a Values for the displayed stellar parameters are taken from
+Kluska et al. (2022) and Kluska et al. (2018) and re-scaled with the
+distances from Gaia DR3 Gaia Collaboration et al. (2022).
+4.3. Strategy
+In this section we define how we set up the exploration of the
+parameters while showing the results in Sect. 4.4.
+4.3.1. Exploration of the parameter space
+Since the effect of the different parameters has not been exam-
+ined extensively for such post-AGB circumstellar discs yet, we
+start by exploring our parameter space by investigating 1) the
+Article number, page 4 of 18
+
+V2
+V2
+V2
+Fcorr (Jy)
+0.9
+1.0
+100
+10-6,
+Reference model
+0.8
+Data
+0.8
+10-1,
+102
+0.7
+10-8
+s
+cm-2 s
+0.6
+0.6
+10-2
+0.5
+0.4
+101
+0.4
+10-3
+10-12
+0.3
+0.2
+0.2
+10-14
+0.0
+10-4
+100
+100
+102
+20
+40
+60
+80
+20
+40
+09
+0
+10
+20
+30
+40
+0
+5
+10
+15
+Wavelength (μm)
+B (M^)
+B (M入)
+B (M入)
+B (M入)A. Corporaal et al.: What does a typical full-disc around a post-AGB binary look like?
+Fig. 2: From left to right: the reddened SEDs, H-band squared visibilities, K-band squared visibilities, L-band squared visibilities,
+and N-band correlated fluxes for variations in the most impacting parameters. The SED plots are zoomed in to the area of interest
+(i.e. in the wavelengths for which have infrared interferometric data) to highlight the changes. The reddened stellar photosphere
+is depicted in brown in the SED plots. Here, the wavelength regimes of the H-, K-, L-, and N-bands are depicted by the grey
+vertical areas. The reference model is always depicted in blue. The visibilities are calculated at synthetic baselines between 1-
+150 m. Interferometric data are only shown around the central wavelengths of the wavelength bands. From top to bottom: variations
+in α, β, h0, q, md, and rmid.
+Article number, page 5 of 18
+
+V2
+V2
+V2
+Fcorr (ly)
+10-7
+0.9
+1.0
+10°
+α = 0.01
+α = 0.1
+0.8
+α= 0.001
+0.8
+Data
+10-1
+0.7
+102
+10-8
+s
+(erg cm-² s
+0.6
+0.6
+10-2
+0.5
+101
+5
+10-9
+0.4
+10-3
+0.3
+0.2
+0.2
+10-10l
+0.0
+10-4
+1001
+100
+101
+0
+20
+40
+60
+80
+20
+40
+60
+0
+10
+20
+30
+0
+5
+10
+0
+Wavelength (μm)
+B (M^)
+B (M)
+B (M入)
+B (M入)V2
+V2
+V2
+Fcorr (ly)
+10-7
+0.9
+1.0
+100
+β = 1.2
+0.8
+β = 1.4
+β = 1.6
+0.8
+10-1
+102
+Data
+0.7
+10-8
+0.6
+0.6
+10-2
+0.5
+0.4
+101
+10-9
+0.4
+10-3
+0.3
+0.2
+0.2
+10-10/
+0.0-
+10-4
+1001
+100
+101
+0
+20
+40
+60
+80
+20
+40
+60
+0
+10
+20
+30
+0
+5
+10
+0
+Wavelength (μm)
+B (M^)
+B (M)
+B (M入)
+B (M入)V2
+V2
+V2
+Fcorr (Jy)
+10-7
+0.9
+1.0
+100
+ho = 0.93 au
+0.8
+ho = 1.87 au
+ho = 1.40 au
+0.8
+10-1
+102
+Data
+0.7
+10-8
+0.6
+0.6
+10-2
+0.5
+0.4
+101
+5
+10-9
+0.4
+10-3
+0.3
+0.2
+0.2
+:
+10-10/
+0.0
+10-4
+1001
+100
+101
+20
+40
+60
+80
+20
+40
+60
+0
+10
+20
+30
+0
+5
+10
+0
+0
+Wavelength (μm)
+B (MΛ)
+B (M)
+B (M入)
+B (M入)V2
+V2
+V2
+Fcorr (ly)
+10-7
+0.9
+1.0
+10°
+q = 2.75
+q = 3.25
+0.8
+q = 2.5
+0.8
+Data
+10-
+102
+0.7
+10-8
+s
+(erg cm-2 g
+0.6
+0.6
+10-2
+0.5
+101
+10-9
+0.4
+10-3
+0.3
+0.2
+0.2
+10-10/
+0.0
+10-4
+1001
+100
+101
+20
+40
+60
+80
+20
+40
+60 
+0
+10
+20
+30
+0
+5
+10
+0
+Wavelength (μm)
+B (M^)
+B (M)
+B (M入)
+B (M入)V2
+V2
+V2
+Fcorr (Jy)
+10-7
+0.9
+1.0
+10°
+md = 2.0e-03 Mo
+md = 4.0e-03 Mo
+0.8
+md = 1.0e-03 M。
+0.8
+102
+Data
+10-1
+0.7
+10-8
+s
+(erg cm-2 g
+0.6
+0.6
+10-2
+0.5
+101
+5
+10-9
+0.4
+10-3
+0.3
+0.2
+0.2
+10-10l
+0.0
+10-4
+1001
+100
+101
+20
+40
+60
+80
+20
+40
+60
+0
+10
+20
+30
+0
+5
+10
+0
+0
+Wavelength (μm)
+B (M^)
+B (M)
+B (M入)
+B (M入)A&A proofs: manuscript no. main
+Fig. 2: Cont. Variations in rmid.
+Fig. 3: Same as Fig. 2 but for the dust composition. Top. Effect of changing the porosity and the maximum hollow volume ratio
+assuming the DIANA project opacities. Middle. Effect of adding a crystalline component. Bottom. Effect of adding a metallic iron
+component.
+disc geometry with the primary aim to study the extended com-
+ponent and 2) the radial profile of the disc within our interfero-
+metric observations.
+We varied various disc parameters based on previous works
+in circumstellar disc modelling (e.g., Woitke et al. 2016). Table
+1 lists the disc parameters along with the grid range for the para-
+metric study. In general, we tested values that are both higher
+and lower than the reference model values except for parameters
+influencing the amount of the over-resolved emission such as β
+and h0 (Eq. 1) that control the flaring. The infrared interferomet-
+ric observations are not sensitive to the maximum grain size, and
+the outer disc radius.These parameters are therefore fixed to one
+value throughout this study. The inner disc radius is well con-
+strained by the PIONIER data (Kluska et al. 2018) and is kept
+fixed to their best-fit value, scaled to the updated distance from
+Gaia DR3, of 7.2 au. The inclination, i, and position angle, PA,
+are constrained by the geometric model fitting of the PIONIER
+data by Hillen et al. (2016). We kept these parameters fixed to
+i = 19◦ and PA = 6◦ (measured north to east), respectively.
+Article number, page 6 of 18
+
+V2
+V2
+V2
+Fcorr (ly)
+10-7
+0.9
+1.0
+10°
+Fe = 0%
+Fe = 15%
+0.8
+Data
+0.8
+10-
+0.7
+102
+10-8
+s
+(erg cm-² s
+0.6
+0.6
+10-2
+0.5
+0.4
+101
+5
+10-9
+0.4
+10-3
+0.3
+0.2
+0.2
+10-10/
+0.0
+10-4
+1001
+100
+101
+0
+20
+40
+60
+40
+60
+20
+10
+80
+20
+0
+10
+30
+0
+5
+Wavelength (μm)
+B (M^)
+B (M)
+B (M入)
+B (M入)V2
+V2
+V2
+Fcorr (ly)
+10-7
+0.9
+1.0
+100
+rmid = 3.0 Rin
+rmid = 2.5 Rin
+0.8
+rmid = 3.5 Rin
+0.8
+102
+Data
+10-
+0.7
+10-8
+s
+(erg cm-² s
+0.6
+0.6
+10-2
+0.5
+0.4
+101
+5
+10-9
+0.4
+10-3
+0.3
+0.2
+0.2
+10-101
+0.0
+10-4
+1001
+100
+101
+20
+40
+60
+80
+20
+40
+60
+0
+10
+20
+30
+0
+5
+10
+0
+0
+Wavelength (μm)
+B (M^)
+B (M)
+B (M入)
+B (M入)V2
+V2
+V2
+Fcorr (Jy)
+10-7
+0.9
+1.0
+10°
+p = 0.0, fmax = 0.7
+0.8
+p = 0.0, fmax = 0.8 
+p = 0.25, fmax = 0.7
+0.8
+10-1
+102
+p = 0.25, fmax = 0.8
+0.7
+Data
+10-8
+s
+(erg cm-2 g
+0.6
+0.6
+10-2
+0.5
+0.4
+101
+5
+10-9
+0.4
+10-3
+0.3
+0.2
+0.2
+10-10
+0.0
+10-4
+1001
+100
+101
+0
+20
+40
+60
+80
+20
+40
+60
+0
+10
+20
+30
+0
+5
+10
+0
+Wavelength (μm)
+B (M^)
+B (M)
+B (M^)
+B (M入)V2
+V2
+V2
+Fcorr (Jy)
+10-7
+0.9
+1.0
+10°
+crystallinity = 0.0
+crystallinity = 0.2
+0.8
+crystallinity = 0.4
+0.8
+Data
+10-1
+102
+0.7
+10-8
+s
+(erg cm-² s
+0.6
+0.6
+10-2
+0.5
+0.4
+101
+5
+10-9
+0.4
+10-3
+0.3
+0.2
+0.2
+10-10/
+0.0-
+10-4
+1001
+100
+101
+0
+20
+40
+60
+80
+20
+40
+60
+0
+10
+20
+30
+0
+5
+10
+0
+Wavelength (μm)
+B (M^)
+B (M)
+B (M入)
+B (M入)A. Corporaal et al.: What does a typical full-disc around a post-AGB binary look like?
+The mid-IR interferometric observations are also sensitive to
+the radial opacity profile. To investigate the effects of the opac-
+ity on the observables, we varied three dust properties. First, we
+varied the porosity and the fmax of the amorphous silicate dust
+mixture, as determined by the DIANA project, and adopted the
+values of the DIANA standard. Second, we added a crystalline
+component by taking a mixture of the amorphous silicates of the
+DIANA project and crystalline forsterite. For the latter, forsterite
+was chosen since it was found that it is the dominant source of
+crystalline dust in discs around post-AGB binaries (Gielen et al.
+2008). The opacities are computed from the optical constants
+of Servoin & Piriou (1973) assuming the grain properties are
+in agreement with our standard dust mixture (i.e. with a poros-
+ity of 0% and fmax = 0.7). For the crystalline component, we
+tested crystallinity fractions of 0.2 and 0.4 by volume. Third, we
+mixed the amorphous silicates with metallic iron. The formation
+of metallic iron in the disc environment has been debated in e.g.
+Hillen et al. (2014). We tested a mixture of 85% amorphous sil-
+icate with 15% metallic iron by mass. For metallic Fe, optical
+constants of Palik (1991) were used to compute the opacities. In
+all cases the dust composition is assumed to be homogeneous
+throughout the disc.
+To investigate the impact of the models parameters on the
+radial morphology of the emission, we used the half-light radii
+(hlr) of the disc in different spectral bands. This quantity speci-
+fies the radius at which half of the flux emitted at a given wave-
+length is captured. These hlr are constrained by geometric mod-
+els in Corporaal et al. (2021).
+4.3.2. Ray-traced spectra and images
+We fitted the interstellar reddening to the photosphere of the
+post-AGB star for the reference model, by using the Levenberg-
+Marquardt fitting routine from the Python package lmfit (for a
+more thorough explanation of the applied reddening see Ap-
+pendix B). The SED of each model is then reddened with this
+value.
+Ray-tracing images is computationally expensive. For this
+reason, we computed images at a single continuum wavelength
+coinciding with the centres of the wavelength bands of interest.
+We compared the image to one given spectral channel of the
+near-IR interferometric data to avoid any intra band chromatic
+effects.
+To visualise the effect of varying the parameters on the
+interferometric visibilities, we plot the squared visibilities for
+synthetic baselines between 1-150 meters, to match the base-
+line range of the VLTI. Since IRAS 08544 is almost pole-on,
+the direction to which the uv-coordinates are calculated, does
+not change the visibility signal significantly. The synthetic uv-
+coordinate space was calculated along the semi-major axis in
+the image plane. Output images were converted to complex vis-
+ibilities via a Fourier transform at this synthetic uv-coordinate
+space. These complex visibilities were subsequently converted
+to squared visibilities for the H-, K-, and L-bands and to corre-
+lated fluxes in the N-band, in agreement with the observables of
+the data.
+4.4. Results of parametric study
+The impact on the SED and on the visibilities for the most im-
+pacting structural parameters of our disc model is illustrated in
+Fig. 2. In Fig. 3 we show the same but for variations in the
+dust composition. The impact of these parameters on the hlr are
+Table 2: Explored set of parameters for the application to
+IRAS 08544 and the parameters of the family of best-fit mod-
+els (see also Table 3).
+Parameter
+Grid range
+Values best models
+α
+[0.01, 0.1]
+[0.1, 0.01]
+β
+[1.2, 1.3, 1.4]
+[1.2, 1.3, 1.4]
+h0 (au)
+[0.93, 1.40, 1.87]
+[1.40, 1.87]
+md (×10−3 M⊙)
+[1.0, 1.5, 2.0, 3.0]
+[1.0, 1.5, 2.0]
+q
+[2.75, 3.0, 3.25]
+[2.75, 3.0]
+rmid
+[2.5, 3.0, 3.5]
+[2.5, 3.0, 3.5]
+metallic iron
+[0%, 15%]
+[0%, 15%]
+shown in Fig. 4. The impact on the photometry and interferome-
+try and on the hlr for a change in the other parameters are shown
+in Fig. A.1 and Fig. A.2, respectively.
+4.4.1. The impact of the turbulence parameter
+The turbulence parameter controls the strength of the dust set-
+tling and, therefore, the scale height of the larger grains. Stronger
+dust settling leads to the removal of large grains from the sur-
+face layers. A decrease in α decreases the photometric fluxes,
+increases the star-to-disc flux ratio significantly and increases the
+over-resolved flux component. An increase of α shows, however,
+only slight variations with respect to the reference model, such
+as a slight increase in the photometric fluxes, suggesting that the
+number density of small grains where infrared interferometric
+observations are sensitive to are not changing from α = 0.01
+to α = 0.1. Neither an increase nor a decrease in α has notable
+effects on the hlr.
+4.4.2. The impact of the scale height
+The disc flaring as controlled by β and h0 strongly affects all ob-
+servables that we investigated. Larger β and h0 significantly in-
+crease the near-IR and mid-IR photometric fluxes, as well as the
+over-resolved emission and the hlr. For more flared discs or more
+vertically extended discs, the star-to-disc flux ratio is decreased
+with respect to the reference model, as the these parameters in-
+crease the captured infrared flux of the disc due to scattering and
+absorption. Both parameters impact similarly with two important
+exceptions: first, they have different effects on the amplitude of
+the visibility bump in the N-band. Indeed, for increased values
+of h0, the amplitude increases while this amplitude remains un-
+changed for varying β. Second, increased values of β affect the
+radial extent of the N-band more than it affect the radial extents
+of the H-, K-, and L-bands. The model with β = 1.6 is also the
+only model in which the N-band extent deduced from the data is
+reached, while it also performs well for the near-IR hlr.
+4.4.3. The impact of grain-size distribution
+Variations in the value of q changes the dust grain size distri-
+bution between the number of small and large grains, with sizes
+below and above 1µm respectively. Varying q leads to changes
+in the mean dust grain size. These alter the SED and visibili-
+ties significantly. If the number of small grains is increased with
+respect to the large grains, or equivalently, if higher values of
+q are taken, the radial extents in all bands decrease, the SED
+fluxes increase in the H-, K-, and L-bands, and the over-resolved
+emission increase in all bands. Moreover, the star-to-disc flux
+ratio is then significantly decreased. The small grains contribute
+Article number, page 7 of 18
+
+A&A proofs: manuscript no. main
+Fig. 4: Half-light-radius variations of the radial emission as a function of wavelength for changes in the parameters that have the
+most promising impact on the photometric and interferometric observables (top and middle) and for changes in the dust composition
+(bottom). The half-light radii were calculated at the central wavelengths of the H-, K-, L-, and N-bands. The geometric model data
+points are taken from Corporaal et al. (2021). Top. Effect of changing α (left), β (middle), and h0 (right). Middle. Effect of changing
+md (left) and q (middle), and rmid (right). Bottom. Variations of the DIANA parameters (left), the crystallinity fraction (middle), and
+the metallic iron content (right).
+considerably to the total flux of the disc such that the disc emis-
+sion becomes more dominant relative to the star. A decrease in
+q slightly decreases the fluxes in the H-, K-, and L-bands, in-
+creases the star-to-disc flux ratio, increases the hlr, and increases
+the over-resolved flux component. The increase/decrease in the
+near-IR photometric flux is stronger than in the mid-IR, which is
+different than what we find for other parameters. The grain-size
+distribution thus affects both the radial intensity profile and the
+emission morphology.
+4.4.4. The impact of the dust mass
+The dust mass impacts the whole SED, the visibilities, as well as
+the hlr. Increasing/decreasing the dust mass decreases/increases
+the stellar contribution significantly and thus shifts the visibility
+curve in the near-IR as the dust mass in the inner regions is dou-
+bled/halved as well. For this reason, higher dust masses decrease
+the IR excess of the disc and slightly increase the N-band over-
+resolved flux and visa versa for smaller dust masses. In the H-,
+K-, and L-bands, the over-resolved flux is not altered by varying
+the dust mass.
+4.4.5. The impact of the turn-over radius
+Variations in the turn-over radius, rmid, were compared by
+putting it closer to/farther from the inner rim while keeping the
+total dust mass constant. To ensure continuity in the surface den-
+sity profile, the ratio of dust mass in the inner zone and the outer
+zone is altered. For larger rmid, the surface density in the inner re-
+gions is higher, while for smaller rmid the turn-over is at smaller
+radii thus with a lower surface density. Variations in rmid signifi-
+cantly affects all observables. Putting rmid closer to/farther from
+the inner rim decreases/increases the SED fluxes in the H-, K-
+Article number, page 8 of 18
+
+24
+29.2
+α= 0.001
+α=0.01
+20
+α = 0.1
+24.4
+Data
+16
+19.5
+hlr (mas)
+hlr (AU)
+12
+14.6
+8
+9.8
+4
+4.9
+0
+.0.0
+0
+2
+3
+4
+6
+8
+10
+12
+Wavelength (μum)24
+29.2
+β= 1.2
+β = 1.4
+20
+24.4
+β = 1.6
+Data
+16
+19.5
+hlr (mas)
+hlr (AU)
+12
+14.6
+8
+9.8
+4
+4.9
+0
+0.0
+0
+2
+3
+4
+6
+8
+10
+12
+Wavelength (μum)24
+29.2
+ho = 0.66 au
+ho=0.99au
+20
+ho = 1.32 au
+24.4
+Data
+16
+19.5
+hlr (mas)
+hlr (AU)
+12
+14.6
+8
+9.8
+4
+4.9
+0
+.0.0
+0
+2
+3
+4
+6
+8
+10
+12
+Wavelength (μm)24
+29.2
+md = 1.0e-03 Mo
+md = 2.0e-03 M。
+20
+md = 4.0e-03 M。
+24.4
+Data
+16
+19.5
+hlr (mas)
+hlr (AU)
+12
+14.6
+8
+9.8
+4
+4.9
+0
+.0.0
+0
+2
+3
+4
+6
+8
+10
+12
+Wavelength (μm)24
+29.2
+q = 2.5
+q = 2.75
+20
+q = 3.25
+24.4
+Data
+16
+19.5
+hlr (mas)
+hlr (AU)
+12
+14.6
+8
+9.8
+4
+4.9
+0
+.0.0
+0
+2
+3
+4
+6
+8
+10
+12
+Wavelength (μum)24
+29.2
+rmid = 2.5 Rin
+rmid = 3.0 Rin
+20
+rmid = 3.5 Rin
+24.4
+Data
+16
+19.5
+hlr (mas)
+hlr (AU)
+12
+14.6
+8
+9.8
+4
+4.9
+0
+.0.0
+0
+2
+3
+4
+6
+8
+10
+12
+Wavelength (μm)24
+29.2
+p = 0.0, fmax = 0.7
+p = 0.0, fmax = 0.8
+20
+p = 0.25, fmax = 0.7
+24.4
+p = 0.25, fmax = 0.8
+Data
+16
+19.5
+hlr (mas)
+hlr (AU)
+12
+14.6
+8
+9.8
+4
+4.9
+0
+.0.0
+0
+2
+3
+4
+6
+8
+10
+12
+Wavelength (μm)24
+29.2
+crystallinity = 0.0
+crystallinity = 0.2
+20
+crystallinity = 0.4
+24.4
+Data
+16
+19.5
+hlr (mas)
+hlr (AU)
+12
+14.6
+8
+9.8
+4
+4.9
+0
+.0.0
+0
+2
+3
+4
+6
+8
+10
+12
+Wavelength (μum)24
+29.2
+Fe = 0%
+Fe = 15%
+20
+Data
+24.4
+16
+19.5
+hlr (mas)
+hlr (AU)
+12
+14.6
+8
+9.8
+4
+4.9
+0
+0.0
+0
+2
+3
+4
+6
+8
+10
+12
+Wavelength (μum)A. Corporaal et al.: What does a typical full-disc around a post-AGB binary look like?
+Fig. 5: Flowchart representing the different steps taken to find the models that represent best the photometric and interferometric
+data of IRAS 08544. The blue boxes describe different calculation steps and the orange boxes indicate the three selection steps.
+Table 3: Parameters of the family of best models and the score,
+sc, that these models got based on the number of characteristics
+that the model can reproduce.
+Model α
+β
+h0
+md
+q
+rmid
+Fe
+sc
+(×10−3)
+(au)
+(M⊙)
+(Rin)
+(%)
+1
+0.1
+1.3
+1.40
+1.0
+2.75
+2.5
+0
+12
+2
+0.01
+1.4
+1.40
+1.0
+2.75
+2.5
+0
+11
+3
+0.01
+1.2
+1.87
+1.0
+2.75
+2.5
+0
+11
+4
+0.01
+1.3
+1.40
+1.5
+2.75
+3.0
+0
+11
+5
+0.1
+1.2
+1.40
+1.5
+2.75
+3.5
+15
+11
+6
+0.1
+1.2
+1.40
+1.0
+3.0
+3.5
+0
+10
+7
+0.1
+1.2
+1.40
+2.0
+2.75
+3.5
+0
+10
+8
+0.1
+1.2
+1.40
+1.0
+3.0
+3.0
+0
+10
+9
+0.1
+1.2
+1.40
+1.5
+2.75
+3.0
+0
+10
+10
+0.01
+1.2
+1.40
+1.0
+3.0
+3.5
+15
+10
+, and L-bands and decreases/increases the hlr in all bands, and
+increases/decreases the star-to-disc flux ratio.
+4.4.6. The impact of the dust composition
+The dust composition determines the radial opacity profile of the
+disc. Variations of the DIANA opacities show a decrease in the
+star-to-disc flux ratio and a slight decrease in the hlr for mod-
+els with p = 0.25. Changes in fmax do not show notable effects
+on the interferometry. The crystallinity factor decreases the star-
+to-disc flux ratio and shows the emission in all bands is simi-
+larly radially extended as the reference model. The addition of
+metallic iron slightly increases the star-to-disc flux ratio, slightly
+decreases the radial extents in the near-IR and increases the ra-
+dial extents in the mid-IR. Different from the DIANA opacity
+changes and the addition of the crystallinity factor, the addition
+of metallic iron increases the photometric flux in the K- and L-
+bands significantly, while the H- and N-band fluxes remain not
+significantly affected. For all models with changing dust com-
+position, the over-resolved flux component remains unchanged
+with respect to the reference model and the hlr values are also
+barely affected.
+4.5. Selection of most impacting parameters
+Here, we outline our selection of the parameters which have
+most impact on the observables with a specific focus on the over-
+resolved flux component. We find the following six structural
+parameters that have the most impact on the SED and the in-
+terferometry in terms of the photometric flux, the over-resolved
+flux, and the radial extent: α, β, h0, md, q, and rmid. For α, β, h0,
+and q we found that increased values of these parameters with
+respect to the reference model show results improved the fit to
+the SED in the near-IR and mid-IR, the over-resolved flux com-
+ponent, and on the radial profile of the disc. Variations in the
+other parameters with significant impact, md and rmid, need to
+be explored in both ways with respect to the reference model.
+Therefore, we selected values for these six parameters that are
+either the values of the reference model or in the direction which
+improves the fit to the observables, i.e. the SED or the interfero-
+metric measurements.
+Changes in the dust compositions have smaller impacts on
+the explored observables as compared to the structural parame-
+ters. Effects of the opacity are therefore not considered to play
+a significant role in explaining the over-resolved component.
+Moreover, it remains difficult to break the degeneracies between
+the effect of the different dust compositions. However, the ad-
+dition of metallic iron as an additional opacity source signifi-
+cantly increases the K- and L-band photometric fluxes, while not
+changing the morphological characteristics of the emission. For
+this reason, we included the metallic iron as a parameter in our
+grid. We tested both models assuming a dust composition fixed
+to the composition of our reference model and models with a
+dust composition that is a mixture of silicates (85%) and metal-
+lic iron with content (15%) and kept this fixed throughout the
+disc. In what follows, we take these seven parameters for a more
+in-depth study while we keep all other parameters fixed to the
+values of the reference model.
+5. Application to IRAS 08544
+In this section, we perform a more in-depth study of the charac-
+teristics of the circumbinary disc around IRAS 08544. We out-
+line our strategy to find our family of best-fit models in Sect. 5.1
+and present the results in Sect. 5.2. We performed a systematic
+study within the parameter space derived from the results in pre-
+Article number, page 9 of 18
+
+Compute radiative
+transfer models
+and assess photometry
+and within our set limits in at
+1280 models
+least two bands.
+28%
+Ray-trace monochromatic images
+and compute visibilities at uv-
+(329 models)
+coordinates around central wavelengths
+Rank models based on the photometric
+fluxes, star-to-disc flux ratio, hlr, and V?
+Select models with the highest
+small
+Calculate chromatic images
+and compute visibilities at
+uv-coordinates of the data.
+15%
+(55 models)
+10 highest
+ranked
+Select best families of models.A&A proofs: manuscript no. main
+Fig. 6: The distribution of the parameter space resulting from each of the three selection steps. The parameter space is defined by
+(from left to right) α, β, h0, md, q, rmid, and Fe. The distribution for the total number of models (1380 models) is indicated by the
+circles and are equally distributed over each of the parameters. The final distribution of parameters of our family of best-fit models
+are indicated by the diamonds.
+vious section. Table 2 displays the range of the varied parame-
+ters.
+5.1. Strategy
+To limit computational time, our strategy consists of three selec-
+tion steps. A graphical representation of our strategy is depicted
+in Fig. 5. We first ran the radiative transfer models, computed
+the photometry, and subsequently assessed the performance of
+the SEDs with respect to the data before creating the images and
+comparing the model to the interferometric measurements.
+From the full investigation of the parameter space, we get
+1280 unique models. Photometric data points in the H-, K-, and
+L-bands are taken from de Ruyter et al. (2006). To avoid dis-
+crepancy between the model and the data due to the 11.3 µm
+crystalline forsterite feature, interpolation is performed between
+all photometric flux points in the N-band (from 8-13 µm) to find
+the photometric flux at 10 µm. This is then compared to the pho-
+tometric flux at 10 µm of the models. The amorphous pyroxene
+silicate feature at 9.8 µm is broad and extents throughout the N-
+band such that we cannot bypass this feature. We selected con-
+fidence intervals in which we considered the model fluxes to be
+good enough within our framework. These are within 20% of the
+photometric data in the H, K and L-bands and 25% of the inter-
+polated data point at 10 µm in the N-band. These levels take into
+account the uncertainties in the photometric data points.
+We fitted the interstellar reddening for each model spectrum
+individually (see also Appendix B). The SED of each model
+is then reddened accordingly. The reduced χ2 of this reddened
+SED, χ2
+red,SED, was calculated by taking into account only photo-
+metric data points up to 20 µm, since our interferometric observ-
+ables are sensitive to neither the geometry of the outer disc nor
+larger grain sizes.
+As a first selection step, we selected models based on the fol-
+lowing two criteria: 1) models that have χ2
+red,SED that are smaller
+than the χ2
+red,SED of the reference model (which has χ2
+red,SED = 90)
+and 2) models that comply with the observed photometry within
+our set limits in at least two out of the four bands. For models
+that satisfy these criteria, we computed the monochromatic im-
+ages at the central wavelengths of each band. To compute the
+synthetic visibilities, we used the uv-coordinates of the data and
+performed a Fourier transform at these uv-coordinates.
+We then ranked the models for which we computed the
+monochromatic images based on their reduced χ2 (χ2
+red) of both
+the photometry and the interferometry of the different bands. The
+model with the lowest χ2
+red in each category got the lowest rank.
+The resulting five ranks per model were then added to get a set
+of models that perform well.
+As a second selection step, we selected the 15% models with
+the lowest ranks. We subsequently ray-traced the images of these
+models at six wavelength channels per band, resulting in chro-
+matic images. For the H-band, we calculated the images at the
+wavelengths of the six spectral channels of PIONIER. For the
+other bands, the wavelengths of the different images are equally
+distributed over the wavelength range of the bands. The images
+were subsequently combined and saved as an image cube. Model
+visibilities were calculated at the (u,v)-coordinates of the data by
+linearly interpolating between the wavelengths.
+The models were then assessed based on their performances
+on the photometric fluxes, star-to-total flux ratios, hlr, and the
+over-resolved flux component to have metrics on the disc geom-
+etry and the radial structure. For the over-resolved component,
+we took the value of the visibility at the shortest baseline(s) of
+the data and compared it to the predicted value of the model at
+the same baseline. Reported values are at baselines of B = 4.31
+Mλ in H and B = 4.55 Mλ in K. In the L-band we took into
+account the set of short baselines B < 0.6 Mλ from the shortest
+baseline (B = 2.35 Mλ) to have a fair comparison between the
+model and the data by taking into account the uncertainties on
+the V2. We took this larger range since there are data points at
+V2 = 0.46 and at V2 = 0.57 with overlapping uncertainties at
+the short baselines with an average of V2 = 0.51. In the N-band,
+the correlated flux for the shortest baselines (< 0.8Mλ) are be-
+tween Fcorr ∼ 160 Jy and Fcorr ∼ 220 Jy. Reported values for
+are at B = 2.47 Mλ and B = 0.67 Mλ in L and N, respectively.
+The confidence levels for which we consider our models pre-
+dicting sufficient over-resolved fluxes are chosen to be 20% in
+the near-IR bands and 40% in the mid-IR bands, proportional to
+the measurement uncertainties.
+Similarly, we set our confidence levels for which we con-
+sider our models to predict a large enough radial extent to 15%
+for the near-IR bands and 20% in the mid-IR bands. As noted
+in Sect. 4.3.1, we took the radial extents as constrained by geo-
+metric models in Corporaal et al. (2021) which are reported at
+the central wavelength of the four bands. To compare the model
+Article number, page 10 of 18
+
+Total models
+Frequency of models
+First selection
+21.0
+Second selection
+Third selection
+0.8
+0.6
+0.4 
+8
+0.2 
+■
+0.0
+α= 0.01
+α= 0.1
+β= 1.2
+β= 1.3
+β= 1.4
+q = 2.75
+q = 3.0
+q = 3.25
+rmid = 3.0Rin
+%0=
+Fe = 15%
+= 0.93 
+=1.40
+=1.87
+md=2.0 ×10-3 NA. Corporaal et al.: What does a typical full-disc around a post-AGB binary look like?
+Table 4: The SED characteristics of the family of best-fit models.
+Model
+FH
+FK
+FL
+FN
+χ2
+red,SED
+(×10−8)
+(×10−8)
+(×10−8)
+(×10−8)
+(erg/s/cm2)
+(erg/s/cm2)
+(erg/s/cm2)
+(erg/s/cm2)
+data
+2.60
+3.46
+5.48
+5.00
+-
+1
+2.50
+2.96
+4.59
+5.49
+45
+2
+2.52
+2.86
+4.32
+5.77
+55
+3
+2.63
+3.16
+4.79
+5.27
+38
+4
+2.48
+2.76
+4.16
+5.73
+55
+5
+2.46
+3.05
+5.06
+5.73
+54
+6
+2.56
+2.70
+3.64
+5.66
+56
+7
+2.43
+2.77
+4.31
+5.67
+55
+8
+2.68
+3.04
+4.17
+5.32
+44
+9
+2.46
+2.85
+4.37
+5.13
+39
+10
+2.80
+3.50
+4.84
+6.07
+55
+hlr with these results, we calculated the hlr of the image at these
+central wavelengths.
+The stellar flux contribution with respect to the total,
+Fs/Frest, was calculated from the central wavelength image of
+the H-band, as the star is contributing most significantly in this
+band, with a contribution ∼ 62% to the total flux at 1.65 µm (Cor-
+poraal et al. 2021). The total contribution at this wavelength is
+coming from the star, the disc, and an over-resolved component.
+The total flux emerging from the star, Fs was thus compared
+with that emerging from the regions farther than the star, Frest.
+The confidence levels for Fs/Frest are set to 10%, proportional
+to the uncertainties of the geometric models.
+We checked whether our models are compliant with the four
+photometric bands, the stellar contribution as seen in the H-band
+interferometry, the four hlr, the four over-resolved flux compo-
+nents, and the stellar contribution with respect to the total flux in
+the confidence levels we outlined above. We summed the amount
+of times a model complies with these levels. Each model can get
+a maximum score of 13. As a third selection step, we selected the
+ten models with the highest score and ranked the models with the
+same score against each other based on their photometric and in-
+terferometric χ2
+red.
+5.2. Results
+5.2.1. The parameters after each selection step
+The results of the three selection steps per parameter are shown
+in Fig. 6. The total number of models are equally distributed be-
+tween all parameters. From the inspection of the SEDs, we found
+329 (28% of the total) models display SEDs satisfying the crite-
+ria defined in Sect. 5.1. From this first selection we can deduce
+two general parameter preferences. First, both a high scale height
+at the inner rim (h0 = 1.87 au), or a high degree of disc flaring
+(β = 1.4) show an overprediction of the N-band and to a lesser
+extent also an overprediction in the L-band flux. Instead, models
+with a h0 = 0.93 au and a β = 1.2 are favourable with 53% and
+62% of the models after the first selection, respectively. Second,
+grain size distributions with q = 2.75 or q = 3.0 are preferred
+over larger values of q, pointing towards the presence of larger
+grains.
+After the second selection, 55 models are left. The infrared
+interferometric observations show strong constraints in four of
+the parameters. The second selection step shows similar con-
+tributions as the first selection step in β with a strong prefer-
+ence for β = 1.2. Moreover, there is a clear preference for
+h0 = 1.40 au. There are strong differences within the wavelength
+bands. The near-IR interferometric data are sensitive to h0 and
+strongly prefer h0 = 1.40 au. The L-band data show a preference
+for h0 = 1.40 au but values of h0 = 0.93 au are not ruled out. The
+N-band data show, however, a clear preference for h0 = 0.93 au,
+resulting from the influence on the amplitude of the second lobe
+in the interferometry. Models with higher h0 show higher am-
+plitudes, which resides above the data. Third, the infrared inter-
+ferometric observations clearly prefer small dust masses. Dust
+masses of 1.0×10−3 M⊙ are highly preferred in 62% of the mod-
+els, with a decreasing slope with increasing dust mass. Fourth,
+grain size distributions with q = 2.75 are preferred while none
+of the models have q = 3.25.
+The ten models that remain after the third selection step
+shows a strong preference for β = 1.2. 90% of the models
+after this selection have h0 = 1.40 au while all models with
+h0 = 0.93 au are ruled out. A scale height at the inner rim larger
+than for our reference model is thus preferred. The distribution
+of dust masses and the grain size distribution power law remain
+similar to the second selection. While the first and second selec-
+tion did not prefer models with or models without Fe, the third
+selection shows a strong preferences for models without metal-
+lic iron. The distributions of α and rmid after the three selection
+steps do not show a strong preference for certain values of these
+parameters.
+5.2.2. The family of best-fit models
+The models resulting from the third selection step are our fam-
+ily of best-fit models. The parameters of these models are dis-
+played in Table 3. The performance of these models on photom-
+etry is displayed in Table 4. The stellar contribution, the visibility
+χ2
+red of the different bands resulting from the chromatic images,
+and the values of the visibility at the smallest baselines of these
+models are given in Table 5. The hlr of these models are given
+in Table 6. Values represented in bold in Tables 4-6 are within
+our set confidence levels and are considered to be good enough
+within our framework. The score, sc, in Table 3 corresponds to
+the addition of the number of such bold representations in the
+tables. The SED and infrared interferometric visibility curves of
+Model 1 are shown in Fig. 7. The performance of Model 1 on
+the wavelength-dependent radial extents is shown in Fig. 8. The
+Article number, page 11 of 18
+
+A&A proofs: manuscript no. main
+Table 5: The χ2
+red and small baselines of the family of best-fit models.
+Model
+Fs/Frest
+χ2
+red,H
+χ2
+red,K
+χ2
+red,L
+χ2
+red,N
+V2
+smallH
+V2
+smallK
+V2
+smallL
+Fcorr,smallN
+(Jy)
+data
+1.65
+-
+-
+-
+-
+0.65
+0.43
+0.51
+208
+1
+1.62
+20
+313
+251
+41
+0.77
+0.51
+0.67
+287
+2
+1.58
+12
+265
+149
+29
+0.75
+0.50
+0.70
+283
+3
+1.45
+38
+317
+189
+34
+0.74
+0.49
+0.68
+270
+4
+1.74
+21
+272
+81
+28
+0.76
+0.51
+0.63
+287
+5
+1.70
+21
+338
+187
+22
+0.81
+0.53
+0.66
+273
+6
+1.66
+6
+242
+122
+17
+0.73
+0.50
+0.62
+250
+7
+1.82
+15
+271
+120
+24
+0.79
+0.52
+0.64
+301
+8
+1.34
+30
+301
+144
+21
+0.74
+0.51
+0.67
+240
+9
+1.75
+17
+308
+269
+32
+0.78
+0.53
+0.67
+276
+10
+1.38
+57
+354
+117
+11
+0.77
+0.52
+0.65
+233
+Notes. Reported χ2
+red are calculated from the chromatic images considering the full interferometric data
+set.
+images of the H-, K-, L-, and N-bands of Model 1 are shown in
+Fig. 9.
+Overall, our resulting family of models is able to explain
+many features of our dataset. Model 1 performs best in terms of
+the photometry, the over-resolved fluxes and the radial extents
+and scores 12 out of a maximum of 13. It is able to reproduce
+well the photometry, the over-resolved flux component, the stel-
+lar contribution, and the radial extent in the H-, K-, and L-bands.
+It only cannot reproduce the N-band radial extent within our set
+limits. Models 4, 5, and 7 are able to reach these levels in the
+N-band radial extent. Models 2-5, have a total score of 11 and
+Models 6-10 have a score of 10.
+Models 1-4, 6, and 8 reproduce the over-resolved flux com-
+ponent at short baselines within our set limits. The models are
+thus able to explain at least part of the over-resolved flux. How-
+ever, there is a systematic underprediction of the over-resolved
+flux component compared to the data in all the models (see
+Sect. 6.2 for a discussion).
+The photometric flux is well reproduced in all bands for
+Models 1, 3, 5, and 10. This shows that the disc geometry is able
+to reproduce the fluxes in the SED. The models with metallic Fe,
+Model 5 and 10, come closer to the K- and L-band fluxes than
+all other models, as expected from the parameter study (Sect. 4).
+These models lack, however, over-resolved flux, most notable
+for Model 5 in the H-band.
+Model 6 performs best in the infrared interferometric χ2
+red. In
+the H-band, the model is the highest ranked in the second and
+third selection, in the K- and L-bands it is ranked in the top 10%
+but in the N-band it is in the bottom 50%. The latter is due to the
+larger h0, which is preferable for H, K, and L but not for N.
+6. Discussion
+In this work, we showed that we can find a family of physical
+models that reproduce the extensive data set covering both pho-
+tometric and multi-wavelength infrared interferometric observa-
+tions from the H- to the N-band with a parameterised disc model,
+originally designed for protoplanetary discs around young stars.
+Here we discuss the implications of the results of Sect. 5.2. We
+discuss the constrained disc parameters and its implications for
+the disc structure (Sect. 6.1), the origin of the missing over-
+resolved flux (Sect. 6.2), the difference in preferred structural
+Table 6: The half-light-radii of the family of best-fit models.
+Model
+hlrH
+hlrK
+hlrL
+hlrN
+(mas)
+(mas)
+(mas)
+(mas)
+geometric model
+7.12
+7.31
+8.85
+20.22
+1
+7.00
+7.33
+8.55
+15.50
+2
+7.09
+7.38
+8.65
+16.00
+3
+7.24
+7.48
+8.65
+14.50
+4
+7.00
+7.33
+8.65
+17.00
+5
+6.87
+7.24
+8.55
+17.00
+6
+7.47
+7.62
+9.09
+15.90
+7
+7.18
+7.48
+8.99
+16.40
+8
+7.00
+7.24
+8.42
+14.50
+9
+7.00
+7.33
+8.55
+15.00
+10
+6.78
+7.05
+8.32
+16.10
+Notes. Reported half-light radii are at the central wavelengths
+of the interferometric bands.
+disc parameters between the H-, K-, and L-band and the N-band
+interferometric data (Sect. 6.3), the comparison to protoplane-
+tary discs around young stars (Sect. 6.4), and finally some future
+prospects (Sect. 6.5).
+6.1. Constrained disc parameters and disc structure
+This disc structure constrains the physical processes in the disc.
+Our results show that five out of the seven disc parameters can
+be well constrained with the combination of photometric data
+and infrared interferometric data in four bands (see Fig. 6). The
+following five parameters are constrained: the two disc flaring
+parameters β and h0, the dust mass md, the index of the grain
+size distribution, q, and the presence or absence of metallic iron
+as an additional opacity source.
+The scale height of the disc is well constrained, with a strong
+preference for a flaring index β = 1.2 and a scale height at the
+inner rim of h0 = 1.40 au, indicating that the disc is more ver-
+tically extended than expected from the 2D hydrostatic equilib-
+rium model of Kluska et al. (2018) (i.e. our reference model,
+h0 = 0.99 au), starting at the disc inner rim. Moreover, a value
+of q = 2.75 is strongly preferred over larger values of q. This
+Article number, page 12 of 18
+
+A. Corporaal et al.: What does a typical full-disc around a post-AGB binary look like?
+Fig. 7: Spectral energy distribution and visibility curves as in Fig. 1 for Model 1 from our family of best-fit models.
+Fig. 8: Half-light radii of Model 1 compared to those from the
+geometric models of Corporaal et al. (2021) at the central wave-
+lengths of the H-, K-, L-, and N-bands.
+indicates that large grains are present in the inner disc regions,
+where the infrared interferometric observations are sensitive (see
+also Sect. 6.3). The infrared interferometry also prefers lower
+dust masses on the order of md = 1.0 × 10−3 M⊙. This translates
+to dust densities on the order of 2.1 − 4.8 × 10−13g/cm3 at the
+inner rim for for models with metallic iron and models without
+metallic iron, respectively. Dust masses above ∼ 2.0 × 10−3 M⊙
+are ruled out at the assumed distance of the system. The final
+selection also prefers discs without metallic Fe. The opacity of
+metallic iron affects strongly the dust density at the inner rim,
+which we can constrain with our infrared interferometric obser-
+vations. The factor of two difference between this density with
+the addition of metallic iron and without this addition points to-
+ward a constrained inner rim dust density of 4.8 × 10−13g/cm3.
+Finally, the degree of dust settling, α, and the turn-over-radius,
+rmid, are not well constrained with our data set.
+To illustrate the structure of the disc model, we present the
+dust grain distribution, the dust density, and dust temperature of
+Model 1 at two values of the vertical scale height, z/r, in Fig. 10.
+We consider the fraction between large and small dust grains
+within the disc a good measure for the dust grain distribution
+throughout a disc.
+As expected, the number of large grains is higher in the mid-
+plane than at larger scale heights in the disc, as larger grains are
+settled towards the midplane. The dust density decreases verti-
+cally, as expected for these discs, as the small grains populate the
+surface layers and the large grains settle toward the midplane.
+The dust temperature increases vertically as the grains populat-
+ing the midplane are more shielded from direct stellar radiation.
+The radial kink in the dust density distribution at the turn-over
+radius results from the changing surface density distribution in
+our two-zone model (Eq. 3).
+The radial and vertical dust temperature structure is well
+constrained. The dust density profile, however, depends on rmid,
+which is not strongly constrained (larger values are slightly pre-
+ferred) with our photometric and infrared interferometric ob-
+servations, and on the dust mass in the disc. These are inter-
+twined, as the surface density changes, depending on the turn-
+over-radius which is also linked to the dust mass in the two zone
+model as the two zones are well connected. Within our family of
+best models, the dust density at the inner regions can differ by a
+factor up to two for larger values of rmid.
+6.2. The missing over-resolved flux
+The excess of over-resolved flux in near-infrared VLTI data was
+noticed in previous modelling efforts of post-AGB discs (Hillen
+et al. 2014; Kluska et al. 2018; Kluska et al. 2019). None of
+the radiative transfer models in this work are able to reproduce
+the total over-resolved flux which makes that its origin is still
+unknown.
+We aim to investigate the origin of the over-resolved flux
+component by reproducing both the photometric fluxes up to
+20 µm, and the short baselines squared visibilities V2
+small in the H-
+, K-, and L-bands. The N-band data are displayed in correlated
+flux and are therefore not sensitive to the over-resolved emission.
+In Sect. 5.2 we tested whether the disc geometry (flaring, ver-
+tical dust settling) could be the origin of the over-resolved flux.
+Our family of best-fit models shows that in the H-, K- and L-
+bands, ∼50% of the over-resolved flux that was missing in our
+reference model has been captured. Therefore, the disc geometry
+can only account for a part of the over-resolved flux component.
+Moreover, while our confidence levels of 20% and 40% in the
+near-IR and mid-IR bands, respectively, are reached for all bands
+in six out of ten models, there is a systematic overestimation of
+the visibility at small baselines. Clearly, some over-resolved flux
+is still missing.
+Similarly to Kluska et al. (2018), we modelled the over-
+resolved emission with a single temperature blackbody nor-
+malised to 2.2 µm, which is about the central wavelength over the
+H-, K-, and L-bands. We added this component to the interfer-
+ometry of our family of best-fit models. We fitted the following
+two parameters: the relative flux of the over-resolved component
+with respect to the total flux and its temperature. We made a grid
+Article number, page 13 of 18
+
+V2
+V2
+V2
+Fcorr (ly)
+0.9
+1.0
+100
+Model
+10-6,
+ Data
+0.8
+0.8
+10-1,
+102
+0.7
+10-8
+0.6
+0.6
+10-2
+0.5
+0.4
+101
+0.4
+10-3
+10-12.
+0.3
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+100
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+60
+10
+20
+30
+40
+60
+0
+0
+5
+10
+15
+Wavelength (μm)
+B (M^)
+B (M入)
+B (M入)
+B (M入)24
+29.2
+Model 1
+Data
+20
+24.4
+16
+19.5
+hlr (mas)
+hlr (AU)
+12
+14.6
+8
+9.8
+4
+4.9
+0
+0.0
+0
+234
+6
+8
+10
+12
+Wavelength (μm)A&A proofs: manuscript no. main
+Fig. 9: Circumbinary disc images of Model 1 at the central wavelengths of the (from left to right) H-, K-, L- and N-bands. The
+fluxes of central stars are removed from the image to unveil the disc structures. The images are normalised to the total flux of each
+image.
+over these two parameters with temperatures in the range of Tback
+from 400 to 7000 K and flux contributions in the range of Fback
+1 to 10%. For the photometry, we used a Levenberg-Marquardt
+fitting routine from the Python package lmfit and evaluated the
+median and range of the best-fit values to all the different models
+in the family of best models. For the interferometry, we evalu-
+ated the over-resolved emission by computing the reduced χ2 on
+the shortest baselines of the H-, K-, and L-band interferometric
+data. We quantify which combination of parameters reproduces
+the short baselines visibilities in all bands the best. We subse-
+quently selected 5% of the models with the lowest total reduced
+χ2 values in the grid and computed the median temperature of
+their contributions.
+We found a better fit to the photometric fluxes if an addi-
+tional flux component with a median temperature of 1470 K and
+a median relative contribution of 7% at 2.2 µm was added. The
+ranges in best-fit temperatures and relative flux contributions we
+fitted to the family of best models are 1120-2100 K and 2-10%,
+respectively. This relative flux component is not very well con-
+strained and depends on the individual model. For Model 1, it is
+estimated to Fback = 7 ± 4% in K with Tback = 1610 ± 450 K.
+This would not significantly decrease its χ2
+red,SED from 45 to 44
+and increase the photometric fluxes FH, FK, FL, and FN by 4%,
+7%, 8%, and 3%, respectively, which brings them closer to the
+values of the data in H, K, and L.
+The short baselines are fitted better with an additional over-
+resolved component with a median temperature of 2100 K and
+a contribution of 10% to the relative flux in K, with a range in
+temperatures of 1400-3600 K and a relative contribution of 8-
+10% within the family of best-fit models. The impact of this ad-
+dition on the short baseline interferometric observables is shown
+in Fig. 11. The V2 of the family of best-fit models are clearly ly-
+ing above the data for all bands. With the added component, the
+V2 can go down to the V2 of the data within the uncertainties in
+the H- and K-bands. For the L-band it reaches within V2 ∼ 0.6
+while it does not reach within the lower-lying V2 at B < 3.0Mλ.
+While photometry does not bring strong constraints to the
+extended emission, the interferometric data is clearly better re-
+produced with the addition of such a component. Its temperature
+of 2100 K (with a range between 1400 and 3600 K) points to-
+wards a mixed origin of photons, that may come both from the
+star (Teff of 7250 K) and the hot inner disk rim (Teff of ∼1250 K).
+The exact location and geometry of the component are, however,
+not constrained by our data and it is, thus, difficult to claim what
+would be the disk geometry or the morphology of an additional
+component that could reproduce our data.
+Several other possibilities have been discussed in Kluska
+et al. (2018) and should be tested in future studies. One such
+possibility is that the emission originates from the jet launched
+from the accretion disc around the secondary star. These jets
+are commonly observed in post-AGB binaries and are detected
+in absorption during orbital phases when the companion is in
+front of the post-AGB primary (Bollen et al. 2022, and refer-
+ences therein). Such a jet is also detected in IRAS 08544 (Kluska
+et al. 2018). It is, however, as yet unknown if also dust grains are
+ejected in these jets.
+6.3. The N-band disagreement
+While we can obtain a good fit the multi-wavelength infrared
+interferometric data and the photometry at the same time, there
+seems to be a general disagreement with the N-band interfero-
+metric data. This is notable in both our second selection and the
+third selection of Sect. 5. The N-band data prefer models with 1)
+a scale height at the inner rim of h0 = 0.93 au to lower the ampli-
+tude of the second lobe, 2) the presence of metallic iron, and 3)
+a grain size distribution index of q = 3.0. Our global selection,
+however, rules out low h0 = 0.93 au and disfavours higher values
+for q and the presence of metallic iron. This is due to the finding
+that the H-, K-, and L-band interferometric data do not prefer
+these values.
+This N-band disagreement is best illustrated in Model 1.
+While it performs best in our total set of criteria, it has the worst
+value of the χ2 on the N-band interferometry. Moreover, while
+the wavelength dependent radial extent is always reached in the
+H-, K-, and L-bands for our ten best models, the N-band ex-
+tent is only reached in three out of ten models, in which there
+is still a systematic under prediction of the extent (see also Fig.
+8). Here we want to comprehend what conditions cause this dis-
+agreement between the H-, K-, and L-bands, which probe the
+very inner disc regions close to the inner rim, and the N-band,
+which probes deeper into the disc.
+The best model for the N-band interferometry, independent
+of the photometric or other interferometric data, has a small scale
+height at the inner rim, h0 = 0.93 au, an α = 0.01, a dust mass of
+md = 1.0 M⊙, a turn-over radius of rmid = 3.5 × Rin, and a dust
+size distribution index q = 3.0 with metallic iron mixed with the
+silicates throughout the disc. This difference in scale height at
+the inner rim with respect to the family of best-fit models sig-
+nificantly alters the dust density at different vertical heights, as
+the a disc with a higher h0 has more volume to distribute the par-
+ticles of the same mass than a disc with a smaller h0. For this
+reason, the dust density at the midplane is higher for smaller h0
+and the dust temperature is lower. The systematic higher q in
+the best ranked models within the N-band compared to the other
+Article number, page 14 of 18
+
+-40
+-40
+-40
+-40
+1.0
+-30
+-30
+-30
+-30
+0.8
+-20
+-20
+-20
+-20
+-10
+-10
+-10
+0.6
+01
+0
+0.
+0.4
+10
+10
+10
+20
+20
+20
+20
+0.2 
+30
+30
+30
+30 
+0.0
+40.
+10 0 -10 -20 -30 -40
+40 30 20 10 0 -10
+-20 -30 -40
+△α (mas)
+Aα (mas)
+△α (mas)
+Aα (mas)A. Corporaal et al.: What does a typical full-disc around a post-AGB binary look like?
+Fig. 10: Cuts of the dust distribution, dust density, and dust tem-
+perature of Model 1 at the midplane and at z/r = 0.15. The dust
+distribution is quantified as a ratio of the number of large grains
+(> 1µm) and small grains (< 1µm) and is calculated as a num-
+ber density per unit mass. The vertical dashed line indicates the
+location of rmid. The dashed yellow, green, and purple lines in
+the bottom plot indicates temperature profiles with power-law
+indexes of -0.67, -0.75, and -0.89, respectively.
+bands indicates that the outer regions contain more small grains
+compared to the inner disc regions.
+For future works, we suggest to consider distinctions be-
+tween the inner and outer disc regions within the dust distribu-
+tion, the dust species and/or structural parameters. This cannot
+be achieved by changing one parameter as there should be a bal-
+ance between decreasing the N-band photometric flux (as it is
+systematically over-estimated in the family of best-fit models)
+and increasing the radial extent in N while the extents in H, K,
+and L remain unchanged.
+6.4. Comparison with YSOs
+The family of best-fit models shows that a strong dust settling
+and a larger scale height starting at the inner rim are needed
+to explain part of the over-resolved flux component as well as
+the wavelength-dependent radial intensity profile. Flaring disc
+shapes in protoplanetary discs have been confirmed by various
+direct observations (e.g. Ginski et al. 2016; Pinte et al. 2018).
+Disc flaring in general is thought to be due to the radial internal
+temperature of the disc decreasing slower than r−1 (Kenyon &
+Hartmann 1987). 2D geometric modelling in the image plane of
+the near-IR and mid-IR interferometric data of IRAS 08544 con-
+firms that this is also the case for this circumbinary disc around
+an evolved binary system (Corporaal et al. 2021). In this work,
+we also show a temperature decrease with a power of −0.67 at
+the surface layers. The midplane temperature decreases with a
+power of −0.89 at the inner regions and with ∼ −0.75 at the
+outer regions (see Fig. 10).
+Strong turbulence mixing strengths (α = 0.1 and α = 0.01)
+are found in our disc modelling to explain the SED characteris-
+tics as well as the infrared interferometric visibilities. These are
+higher than found for most protoplanetary discs around YSOs
+for which values of α = 10−4 −10−2 are consistent with observa-
+tions (e.g. Mulders & Dominik 2012). These authors also show
+that this turbulent mixing strength depends on the adopted grain
+size distribution and gas-to-dust ratio. Higher turbulence param-
+eter values are found for a higher q or a lower gas-to-dust ratio.
+We note that α is not well constrained (see Fig. 6). While in H,
+K, and L lower values of q = 2.75 are preferred, higher values
+are preferred in N, indicating that the turbulence parameter could
+take different values in different disc zones and should be taken
+into consideration. In future studies, also differences in the gas-
+to-dust ratios in different zones of the disc could be taken into
+account. Observational these gas properties can be constrained
+using ALMA and JWST.
+The dust mass in the disc as constrained from the photometry
+and the infrared interferometry are on the order of ∼ 1×10−3 M⊙,
+which translates in a total mass of ∼ 0.1 M⊙ for the assumed gas-
+to-dust ratio. From an analysis on different star forming regions,
+Pascucci et al. (2016) found that the total protoplanetary disc
+masses range from about 0.1-1.0 M⊙ and that there is an increas-
+ing trend with stellar mass. From this, with our combined stellar
+mass of 2.4 M⊙, we could expect dust masses of ∼ 1 × 10−3 M⊙,
+which is aligned with our deduced dust masses. The combined
+stellar mass also makes them close to the average stellar mass of
+Herbig stars, which have an average dust mass of 4 × 10−4 up to
+0.1 M⊙ (Stapper et al. 2022), with a generally lower dust mass
+for non-structured full discs (van der Marel & Mulders 2021).
+The inferred dust masses for the disc around IRAS 08544 are
+therefore very similar to the higher mass end of the Herbig stars
+and thus similar to the of the ones found in protoplanetary discs
+around YSOs.
+6.5. Future prospects
+A limitation of our refined radiative transfer models presented
+here is that the system is assumed to be axisymmetric. Image
+reconstructions of the inner disc regions and the closure phase
+signal, however, reveal that the disc inner rim shows significant
+azimuthal variations (Hillen et al. 2016). In future modelling,
+the binary nature of the system needs to be taken into account to
+include the asymmetric illumination of the post-AGB star on the
+disc inner rim during orbital motion. The orbital phase of this
+asymmetric illumination will be important to constrain the inner
+rim structure. To fully constrain the physical processes in the
+disc, both the inner disc region and throughout the full disc, we
+need to combine high-angular resolution instruments at different
+spatial scales and at different orbital phases to obtain the full 3D
+orbit.
+Article number, page 15 of 18
+
+0.0040
+0.0038
+n(large)
+n(small)
+0.0036
+0.0034
+10-13
+m-
+cm
+Q 10-14
+midplane
+103
+z/r = 0.15
+T α r-0.67
+Tα r-0.75
+T α r-0.89
+T
+102
+0
+25
+50
+75
+100
+125
+150
+175
+r (au)A&A proofs: manuscript no. main
+Fig. 11: The range of squared visibilities at the shortest baselines of the family of ten best-fit models (shaded dark grey) and of those
+models with an addition of an over-resolved component with a relative contribution of 10% and a temperature of 2100 K (shaded
+light grey) for the (from left to right) H-, K-, and L-band.
+7. Conclusion
+We present radiative transfer models for the circumbinary disc
+around the post-AGB binary system IRAS 08544 that are able to
+reproduce an extensive data set of photometric data and infrared
+interferometric visibilities from the H- to the N-band. The pa-
+rameterised passive disc models characterise well the structure
+of the inner regions of the circumbinary disc.
+We first explored the impact of the individual parameters on
+the observables and selected the seven parameters (i.e. turbu-
+lence mixing strength, the two scale height structural parameters,
+the dust mass, the dust size distribution power law index, the
+turn-over radius of the surface density, and metallic iron) hav-
+ing substantial effects on the geometry and the radial structure
+of the disc. These seven parameters were used for a thorough
+grid search to isolate models that reproduce the photometric and
+interferometric dataset. With combinations of these disc parame-
+ters we aimed to reproduce the characteristics of the data set with
+a focus on the over-resolved flux component, the radial structure,
+and the photometric features.
+Our radiative transfer modelling can indeed reproduce the
+infrared visibility data of the disc as well as the photometric fea-
+tures. We find that the disc has a flared geometry with a vertical
+extension starting at the disc inner rim. The H-, K-, and L-band
+interferometric data, which are sensitive to the inner parts of the
+disc, prefer a low grain-size distribution power law index or, sim-
+ilarly, more large grains, while the N-band data prefer higher
+values. This points toward evidence for dust grain growth in the
+inner disc regions. The dust masses of our preferred models are
+very similar to dust masses inferred from observations of proto-
+planetary discs around young stars.
+The disc geometry can explain ∼ 50% of the over-resolved
+flux component, indicating that scattering is playing a role in
+explaining this component but there is still some over-resolved
+flux missing. This missing component has a temperature on the
+order of 1400−3600 K, pointing toward thermal photons emitted
+from the inner rim that are scattered further away in the system
+(from a complex disc morphology or the presence of a halo). The
+location and geometry of this component is not constrained with
+our data set.
+We conclude that the circumbinary disc around IRAS08544
+is in many ways very similar to the discs around YSOs. This
+bright object is therefore a very good candidate to search for disc
+asymmetries and eventual signs of macroscopic structure forma-
+tion around evolved systems. A full analysis of the asymmetries
+as well as the gas and dust distributions will be subject to fu-
+ture research. A complete model of the disc with an additional
+extended dusty component, including the effect of the asym-
+metric illumination of the binary stars and an adequate connec-
+tion between the inner and outer disc, awaits systematic studies
+in which high-angular resolution data at different spatial scales
+are combined using observational constraints from the VLTI,
+SPHERE, and ALMA. Moreover, JWST will allow us to look
+for the gaseous species inside the dust sublimation radius.
+Acknowledgements. We thank the referee for their fast and constructive com-
+ments and suggestions that substantially improved the clarity of the paper.
+A.C. and H.V.W. acknowledge support from FWO under contract G097619N.
+J.K. acknowledges support from FWO under the senior postdoctoral fellow-
+ship (1281121N). DK acknowledges the support of the Australian Research
+Council (ARC) Discovery Early Career Research Award (DECRA) grant
+(DE190100813). DK is supported in part by the Australian Research Council
+Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D),
+through project number CE170100013. This work has made use of data from the
+European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
+processed by the Gaia Data Processing and Analysis Consortium (DPAC,
+https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC
+has been provided by national institutions, in particular the institutions partici-
+pating in the Gaia Multilateral Agreement.
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+Appendix A: Parametric study
+Similarly to Figs. 2 and 4, Figs. A.1 and A.2 show variations of
+the parameters but for those that have no significant impact on
+the SED, visibilities, and half-light-radii.
+Appendix B: Applied reddening
+In our modelling, we fitted the interstellar reddening to the pho-
+tosphere of the post-AGB star. The total reddening of the star
+consists of a circumstellar and an interstellar part. The circum-
+stellar reddening can change from model to model, as the disc
+geometry changes with different combinations of parameters
+within our parameters ranges affecting the reddening properties.
+The circumstellar reddening is taken into account in MCMax3D
+using the extinction law of the interstellar medium as a function
+of wavelength and the total opacity of the dust grains used in the
+models. For each model, the interstellar reddening component is
+not subject to change substantially, as we look at the disc almost
+pole-on and no additional structures are placed within the line
+of sight. Literature values of this interstellar reddening compo-
+nent show a large range of values due to the galactic latitude of
+IRAS 08544 of 0.3◦ locating the target close to the galactic plane.
+Due to the proximity to the galactic plane and the relatively large
+distance > 1 kpc, interstellar extinction maps do not give reliable
+results for IRAS 08544. The fitted total line-of-sight reddening,
+E(B − V), is between 1.36 mag and 1.43 mag for models with an
+acceptable χ2
+red,SED (< 90). The E(B − V) reaches values of up to
+1.47 mag for a few models with a too flared geometry. Our fami-
+lies of best-fit models yield a E(B − V) = 1.366−1.38 mag. This
+is in agreement with the fit of Kluska et al. (2022) as expected
+for such a pole-on disc.
+Article number, page 17 of 18
+
+A&A proofs: manuscript no. main
+Fig. A.1: Variations of the parameter study for the parameters that do not impact the observables significantly. From top to bottom:
+variations in amin, the gas-to-dust ratio, and pin
+Fig. A.2: Half-light-radius variations of the radial emission as a function of wavelength for the parameters that do not have significant
+effects on the photometric and/or on the interferometric observables. The half-light radii were calculated at the central wavelengths
+of the H-, K-, L-, and N-bands. Effect of changing amin (left), the gas-to-dust ratio (middle), and pin (right).
+Article number, page 18 of 18
+
+V2
+V2
+V2
+Fcorr (ly)
+0.9
+1.0
+10°
+amin = 0.1 μm
+10-6.
+amin = 0.01 μm
+0.8
+amin = 1.0 μm
+0.8
+102
+Data
+0.7
+10-8
+s
+cm-2 s
+0.6
+0.6
+10-2
+0.5
+0.4
+101
+0.4
+10-3
+10-12
+0.3
+0.2
+0.2 
+10-14
+0.0
+10-4
+1001
+100
+102
+0
+20
+40
+60
+80
+0
+20
+40
+60
+0
+10
+20
+30
+0
+10
+Wavelength (μm)
+B (M入)
+B (M)
+B (M入)
+B (M入)V2
+V2
+V2
+Fcorr (ly)
+0.9
+1.0
+10°
+00T = #snp/se6
+10-6.
+0.8
+gas/dust = 50
+0.8
+gas/dust = 200
+102
+Data
+0.7
+10-8
+s
+cm-2 s
+0.6
+0.6
+10-2
+0.5
+0.4
+101
+0.4
+10-3
+10-12
+0.3
+0.2
+0.2 
+10-14
+0.0
+10-4
+1001
+10°
+102
+0
+20
+40
+60
+80
+0
+20
+40
+60
+0
+10
+20
+30
+0
+5
+10
+Wavelength (μm)
+B (M入)
+B (M)
+B (M入)
+B (M入)V2
+V2
+V2
+Fcorr (ly)
+0.9
+1.0
+100
+10-6,
+pin = -1.5
+0.8
+pin = -1
+0.8
+10-
+102
+0.7
+Data
+10-8
+s
+cm-2 s
+0.6
+0.6
+10-2
+0.5
+101
+0.4
+10-3
+10-12
+0.3
+0.2
+0.2 
+10-14
+0.0
+10-4
+1001
+10°
+102
+0
+20
+40
+60
+80
+0
+20
+40
+60 
+0
+10
+20
+30
+0
+10
+5
+Wavelength (μm)
+B (M^)
+B (M)
+B (M入)
+B (M入)24
+29.2
+amin = 0.01 μm
+amin =0.1 μm
+20
+amin = 1.0 μm
+24.4
+Data
+16
+19.5
+hlr (mas)
+hlr (AU)
+12
+14.6
+8
+9.8
+4
+4.9
+0
+.0.0
+0
+2
+3
+4
+6
+8
+10
+12
+Wavelength (μm)24
+29.2
+gas/dust = 100
+gas/dust = 200
+20
+gas/dust = 50
+24.4
+Data
+16
+19.5
+hlr (mas)
+hlr (AU)
+12
+14.6
+8
+9.8
+4
+4.9
+0
+.0.0
+0
+2
+3
+4
+6
+8
+10
+12
+Wavelength (μum)24
+29.2
+pin = -1.5
+Pin = -1
+20
+24.4
+pin = -2
+Data
+16
+19.5
+hlr (mas)
+hlr (AU)
+12
+14.6
+8
+9.8
+4
+4.9
+0
+.0.0
+0
+2
+3
+4
+6
+8
+10
+12
+Wavelength (μm)

7tE4T4oBgHgl3EQf2g0r/content/tmp_files/2301.05298v1.pdf.txt

@@ -0,0 +1,2262 @@
+Open Case Studies: Statistics and Data Science Education
+through Real-World Applications
+Carrie Wright1, Qier Meng1, Michael R. Breshock2, Lyla Atta2, Margaret A. Taub 1, Leah R.
+Jager 1, John Muschelli 1,3, and Stephanie C. Hicks1,*
+1Department of Biostatistics, Johns Hopkins Bloomberg School of Public Health
+2Department of Biomedical Engineering, Johns Hopkins University
+3Streamline Data Science
+*Correspondence to [email protected]
+Abstract
+With unprecedented and growing interest in data
+science education, there are limited educator materials
+that provide meaningful opportunities for learners
+to practice statistical thinking, as defined by Wild and
+Pfannkuch [1], with messy data addressing real-world
+challenges.
+As a solution, Nolan and Speed [2] ad-
+vocated for bringing applications to the forefront in
+undergraduate statistics curriculum with the use of
+in-depth case studies to encourage and develop statistical
+thinking in the classroom. Limitations to this approach
+include the significant time investment required to
+develop a case study – namely, to select a motivating
+question and to create an illustrative data analysis –
+and the domain expertise needed.
+As a result, case
+studies based on realistic challenges, not toy examples,
+are scarce.
+To address this, we developed the Open
+Case Studies (opencasestudies.org) project,
+which
+offers a new statistical and data science education
+case study model.
+This educational resource pro-
+vides self-contained, multimodal, peer-reviewed, and
+open-source guides (or case studies) from real-world
+examples for active experiences of complete data
+analyses. We developed an educator’s guide describing
+how to most effectively use the case studies, how to
+modify and adapt components of the case studies in
+the classroom, and how to contribute new case studies.
+(opencasestudies.org/OCS Guide).
+Keywords: applied statistics, data science, statistical
+thinking, case studies, education, computing
+1
+Introduction
+A major challenge in the practice of teaching data sci-
+ence and statistics is the limited availability of courses
+and course materials that provide meaningful opportu-
+nities for students to practice and apply statistical think-
+ing, as defined by Wild and Pfannkuch [1], with messy
+data addressing real-world challenges across diverse
+context domains. To address this problem, Nolan and
+Speed [2] presented a model for developing case studies
+(also known as ‘labs’) for use in undergraduate statistics
+courses with a specific goal to “encourage and develop
+statistical thinking”. Specifically, the model calls for each
+case study to be:
+“a substantial exercise with nontrivial solu-
+tions that leave room for different analyses,
+and for it to be a central part of the course. The
+lab should offer motivation and a framework
+for studying theoretical statistics, and it should
+give students experience with how statistics
+can be used to answer scientific questions. An
+important goal of this approach is to encourage
+and develop statistical thinking while impart-
+ing knowledge in mathematical statistics.” [2]
+In 2018, Hicks and Irizarry [3] stated that one of their
+five principles for teaching data science was to “organize
+the course around a set of diverse case studies” based
+on the model by Nolan and Speed [2], with a goal of
+practicing statistical thinking and bringing real-world
+applications into the classroom. Case studies are also
+being used in the classroom across a diverse set of fields,
+including statistics [4–8], evolutionary biology [9], engi-
+neering [10], and environmental science [11].
+However, there are several limiting factors to scal-
+ing up the use of case studies.
+First, the process of
+selecting motivating questions [12], finding real-world
+and motivating data [13, 14], describing the context
+around the data [15, 16], and preparing diverse didactic
+data analyses requires a large initial investment in time
+and effort [3]. Second, the individuals who are most
+primed to develop effective and insightful case studies
+are practitioner-instructors [17], or practicing applied
+statisticians and data scientists, who teach and practice
+in a field-specific context. For these individuals, success-
+Wright et al. | 2023 | arXiv |
+Page 1
+arXiv:2301.05298v1  [stat.AP]  12 Jan 2023
+
+fully constructing a diverse set of case studies across a
+wide range of contextual topics may require collabora-
+tion with individuals in other disciplines; this can be
+hard without protected time and effort from their aca-
+demic institutions [18]. Third, while there are rich repos-
+itories of data sets [7], there are few collections of as-
+sociated data analyses that show how the data can be
+used to demonstrate fundamental data science and sta-
+tistical concepts, potentially with unexpected outcomes
+[19]. This is especially true for complex and messy data,
+where analysis decisions must go beyond what can be
+summarized in a brief summary about the data, such as
+a README ��le [20, 21]. These challenges have resulted
+in a scarcity of case studies based on real-world chal-
+lenges instead of simple toy examples. Moreover, many
+data repositories have different recommended process-
+ing and analysis of subsets of data, which are commonly
+used as ”the” analysis, without proper discussion of al-
+ternative choices along the research pathway.
+To address these challenges, we developed an open-
+source educational resource, the Open Case Studies
+(OCS) project (opencasestudies.org). This resource con-
+tains in-depth, self-contained, multimodal, and peer-
+reviewed experiential guides (or case studies) that
+demonstrate illustrative data analyses covering a di-
+verse range of statistical and data science topics to teach
+learners how to effectively derive knowledge from data.
+These guides can be used by instructors to bring appli-
+cations to the forefront in the classroom or they can be
+used by independent learners outside of the classroom.
+Finally, we developed an educator’s guide describing
+how to most effectively use the case studies, how to
+modify and adapt components of the case studies in
+the classroom, and how to contribute new case studies.
+(opencasestudies.org/OCS Guide).
+The rest of the manuscript is as follows. First, we pro-
+vide an overview and discuss individual components of
+the Open Case Studies model (Section 2), a new model
+that extends the [2] case studies model. Second, we
+describe the Open Case Studies educational resource
+(Section 3). Third, we give guidance based on our expe-
+rience about how others can create their own case stud-
+ies (Section 4), including how to create interactive case
+studies. We conclude with a summary about the utility
+of such case studies inside and outside of the classroom
+(Section 5).
+2
+Putting OCS model into practice
+2.1
+An overview of the Open Case Studies model
+The case-studies model described by Nolan and Speed
+[2] divides each case study into five main components:
+(i) introduction, (ii) data description, (iii) background,
+(iv) investigations, and (v) theory, with an optional sec-
+tion for advanced analyses or related theoretical mate-
+rial. In our Open Case Studies (OCS) model, we expand
+upon these components to thirteen components. Table 1
+describes the components of the OCS model as well as
+the mapping between our model and the original model
+of Nolan and Speed [2].
+We highlight that while the structures of the two case-
+study models are similar, our OCS model has a different
+purpose than the one proposed by Nolan and Speed [2].
+Briefly, Nolan and Speed [2] designed case studies to be
+either (i) used in open-ended discussions in lecture or
+(ii) used as open-ended lab exercises where students do
+extensive analyses outside of class and write reports con-
+taining their observations and solutions. In both appli-
+cations, the case studies are designed to be open-ended;
+the background may be initially discussed in class or
+as part of an assignment, but students work indepen-
+dently or in a group to create their own solutions and
+summarize their own findings in a full-length report to
+answer the original question. In contrast, we made a
+design choice to build case studies that are full-length,
+in-depth experiential guides that walk learners through
+the entire process of data analysis, with an emphasis on
+computing [22], starting from a motivating question and
+ending with a summary of the results. Our goal is for
+educators either to directly use an entire case study in
+the classroom or to adapt a subset of the material for
+their use. For example, an educator can choose to show
+the solutions provided in the case study, show a differ-
+ent solution, or leave the discussion open-ended. Our
+reasoning for providing full-length guides is that it is
+typically easier for an educator to remove or modify ma-
+terial instead of creating it from scratch. In this way,
+we aim to reach a broader audience than just educators
+in a classroom, as any learner interested in a particular
+topic can walk through the case study to see an example
+of a complete data analysis. In addition, this method is
+particularly helpful for instructors who may not feel con-
+fident creating an analysis from scratch, especially if it is
+outside their main area of expertise, as our case studies
+built with domain experts and are peer-reviewed.
+2.2
+Components of the Open Case Studies model
+We will describe the thirteen individual components
+of our Open Case Studies model (Table 1) using one
+case study as an example.
+Currently all of our case
+studies showcase how to use the R statistical program-
+ming language [23] for data analyses, although other
+programming languages could be used with our model.
+Here, we use the “Exploring CO2 emissions across time”
+case study (opencasestudies.org/ocs-bp-co2-emissions),
+which explores global and country level carbon dioxide
+(CO2) emissions from the 1700s to 2014 (Figure 1). This
+case study also investigates how CO2 emission rates
+may relate to increasing temperatures and increasing
+rates of natural disasters in the United States (US). We
+also describe four other case studies (Table 2) and give
+Wright et al. | 2023 | arXiv |
+Page 2
+
+Mapping of components between two case study models
+Open Case Studies model
+Case-study model of Nolan and Speed [2]
+Component
+Description
+Component
+Description
+1. Motivation
+Motivating figure and text at the start of
+the case study
+2. Main questions
+Scientific question(s)
+Introduction
+Describes context of scientific
+question and motivation
+3. Learning objectives
+Both data science and statistics learning
+objectives
+4. Context
+Context of question(s) or data
+5. Limitations
+Any limitations in case study or with
+data used
+Background
+Information to put question in
+context using non-technical
+language
+6. What are the data?
+Summary of where the data came from
+and what the data contain
+Data description
+Documentation for data collected
+to address the question
+7. Data import
+Analyses for importing data
+8. Data wrangling and ex-
+ploration
+Analyses for wrangling and exploring
+the data
+9. Data visualization
+Analyses for data visualization
+Investigations
+Suggestions for answering the
+question (varies in difficulty)
+10. Data analysis
+Analyses containing statistical concepts
+and methods to answer question(s)
+Theory
+Describes relevant statistical con-
+cepts and methodologies to answer
+the question
+11. Summary
+Summary of results
+12. Suggested homework
+Question(s) to explore further
+13. Additional information
+Helpful links or packages used
+Extended material
+(optional)
+Describes advanced analyses or
+related theoretical material
+Table 1. Components of an Open Case Study Descriptions of the components of our Open Case Studies
+model (left) and their mapping to the components of the case studies model proposed by Nolan and Speed [2]
+(right). We note that the model from Nolan and Speed [2] orders ‘Data description’ before ‘Background’.
+However, Background is listed first here to more easily map to our Open Case Studies model.
+example topics covered in all case studies (Table S1).
+1. Motivation. Each case study begins with a motivating
+data visualization. This idea originated from Dr. Mine
+C¸ etinkaya-Rundel’s talk entitled ‘Let Them Eat Cake
+(First)!’, presented at the Harvard University Statistics
+Department’s 2018 David K. Pickard Memorial Lecture
+[24]. She argues that, similar to a recipe book about bak-
+ing cakes, showing a learner a visualization first can be
+motivating and give learners a sense of what they will
+be doing. This practice of showing a visualization at the
+start of a data analysis and then showing learners the
+code for how to produce the data visualization enables
+the learners to have a better sense of the final product
+and can be motivating to learn the more challenging con-
+cepts needed to make the visualization.
+The motivating figure from the CO2 emissions case
+study (Figure 1) is reproduced here. In the case studies,
+we also include text explaining the motivation for the
+case study.
+Our case studies are often motivated by
+a recent report or publication investigating a specific
+scientific question. In this section, we explain why the
+topic is of interest and define any terms that are needed
+to understand the main questions of interest (described
+in the next section).
+2. Main questions. In this section, we highlight and ex-
+plicitly state a precise set of scientific question(s) or prob-
+lem(s) before beginning the analysis [25]. When the case
+study is motivated by a previous publication, these ques-
+tions may not be exactly the same as what was originally
+investigated in the paper or report. For example, a case
+study may only investigate a small subset of the results
+presented in the report or publication. Alternatively, a
+case study may not investigate the same question(s) at
+all, but rather use the data from the report or publication
+to demonstrate a specific data science or statistics learn-
+ing objective. This framework also reiterates that many
+problems have a set of questions prior to analysis; find-
+ing an answer and engineering the question post-doc
+is not recommended. Data exploration is a large com-
+ponent of the analysis framework and is shown in case
+studies, but OCS impresses thoughtful questions should
+be determined prior to analysis.
+In the CO2 emissions case study, the scientific ques-
+tions are:
+1. How have global CO2 emission rates changed over
+time? In particular for the US, and how does the US
+compare to other countries?
+2. Are CO2 emissions in the US, global temperatures,
+and natural disaster rates in the US associated?
+3. Learning objectives. Each case study consists of a
+set of didactic learning objectives. We categorize each
+objective as related to either (i) data science or (ii) statis-
+tics where the latter are concepts traditionally taught in
+a statistics curriculum such as linear regression, multi-
+ple testing correction, significance and the former are
+Wright et al. | 2023 | arXiv |
+Page 3
+
+Figure 1. Example of a motivating figure in the “Exploring CO2 emissions across time” case study The complete case
+study can be found at (opencasestudies.org/ocs-bp-co2-emissions). Top row: Line plot showing the increase in CO2
+emissions over time (left). Longitudinal heatmap plot highlighting that the US has been one of the top emission
+producing countries historically and currently (right). Bottom row: Scatter plots showing the trends between CO2
+emissions and temperature across time.
+concepts often appearing outside of a traditional statis-
+tics course, such as re-coding data values, scraping data
+from a website, or creating a dashboard for a data set.
+Other categories could be considered depending on the
+purpose of the case study. This separation also allows
+for educators to adapt the material to other computa-
+tional frameworks and languages other than R, such as
+Python.
+We include these learning objectives for three reasons:
+(i) to help educators select a case study that meets the ob-
+jectives they want to teach and (ii) to help learners select
+a case study that demonstrates what they want to learn,
+and (iii) to provide both educators and learners with a
+clear understanding about the goals of a particular case
+study. For example, a study of the use of learning ob-
+jectives in an undergraduate science course found that
+students find learning objectives helpful for narrowing
+and organizing their studying [26].
+For the CO2 emissions case study, we designed the
+case study around the following learning objectives:
+(i) Data Science Learning Objectives:
+• Importing data from various types of Excel files and
+CSV files
+• Apply action verbs in dplyr [27] for data wran-
+gling
+• How to pivot between “long” and “wide” data sets
+• Joining together multiple datasets using dplyr
+• How to create effective longitudinal data visualiza-
+tions with ggplot2 [28]
+• How to add text, color, and labels to ggplot2 plots
+• How to create faceted ggplot2 plots
+(ii) Statistical Learning Objectives:
+• Correlation coefficient as a summary statistic
+Wright et al. | 2023 | arXiv |
+Page 4
+
+World CO2 Emissions per Year (1751-2014)
+Top 10 CO2 Emission-producing Countries
+Ordered by Emissions Produced in 2014
+Emissions (Metric Tonnes)
+China
+3e+07
+United States:
+India
+Russia
+2e+07
+Japan
+Germany
+1e+07
+Iran
+Saudi Arabia -
+South Korea'
+0e+00.
+Canada
+1800
+1900
+2000
+５0
+5
+５
+5
+05050
+7
+80
+9900
+7
+99-
+9
+9
+9
+gg~
+99
+0
+Year
+Limited to reporting countries
+Ln(CO2 Emissions (Metric Tonnes))
+8
+US CO2 Emissions and Temperature (1980-2014)
+CO2 Emissions (Metric Tons)
+5500000
+(Fahrenheit)
+2
+5000000
+4500000
+Temperature 
+Temperature (Fahrenheit)
+55
+0
+54 -
+led
+53
+S
+52 
+.
+2
+1980
+1985
+1990
+1995
+2000
+2005
+2010
+-2
+.1
+0
+Scaled Emissions (Metric Tonnes)Example case studies in the OCS resource
+Topic
+Question(s)
+Data source(s)
+Raw data
+Data science skills
+Statistical concepts
+Air Pollution
+[html]
+Can we predict annual fine
+particulate air pollution
+concentrations using
+predictors such as
+population density,
+urbanization, and satellite
+data?
+Gravimetric EPA air
+pollution data (from
+2008) and predictor data
+from NASA, the US
+Census, and NCHS
+Single
+curated CSV
+file
+tidymodels, correlation
+visualizations,
+geospatial visualizations
+machine learning, linear
+regression, random
+forest
+Vaping [html]
+How has tobacco / nicotine
+product use by American
+youths changed since 2015?
+Is there a relationship
+between e-cigarette /
+vaping use and other
+tobacco / nicotine product
+use?
+NYTS 2015-2019 survey
+data
+Excel files
+and
+codebooks
+for each year
+importing Excel files,
+importing multiple files
+efficiently, merging data,
+writing functions,
+functional
+programming,
+longitudinal
+visualizations
+survey weighting,
+logistic regression with
+survey weighting,
+longitudinal data,
+codebooks
+CO2
+Emissions
+[html]
+How have global CO2
+emission rates changed
+over time? In particular for
+the US, and how does the
+US compare to other
+countries? // Are CO2
+emissions in the US, global
+temperatures, and natural
+disaster rates in the US
+associated?
+CO2 emissions (from
+1751-2019, GDP and
+energy use data from
+gapminder. US
+temperature and disaster
+data form the NOAA
+XLSX and
+CSV files
+importing data from
+Excel files and CSV files,
+data joining,
+longitudinal data
+visualizations, plots
+with text and labels
+correlation coefficient,
+relationship between
+correlation and linear
+regression, correlation
+vs. causation
+US School
+Shootings
+[html]
+What has been the yearly
+rate of school shootings
+and where in the country
+have they occurred in the
+last 50 years (from January
+1970 to June 2020)?
+Open-source K-12 school
+shooting database
+(1970-2020)
+single CSV
+file, Google
+sheets
+importing Google sheets,
+date formats, geocoding,
+interactive tables, R
+Markdown, maps,
+interactive dashboards
+calculating percentages
+for data with missing
+values
+Table 2. Description of four example case studies in the OCS resource This table shows the topics covered
+in four individual case studies, as well as information about the raw data. EPA = the US Environmental
+Protection Agency, NASA = National Aeronautics and Space Administration, and NCHS = the National
+Center for Health Statistics, NYTS = the National Youth Tobacco Survey, NOAA = National Oceanic and
+Atmospheric Administration. CO2 emission data obtained from gapminder was originally from the World
+Bank. School Shooting data was obtained from the Center for Homeland Defense and Security at the at the
+Naval Postgraduate School (NPS).
+• Relationship between correlation, linear regression
+• Correlation is not causation
+In addition, by stating these objectives within the
+case studies, students may begin identify how they can
+apply these concepts for future analyses. Finally, we
+provide an interactive search table of learning objectives
+on the Open Case Studies website (opencasestudies.org)
+to make it easier to find a case study that would
+demonstrate a particular technique, method, or concept
+that an instructor or learner might be interested in.
+4. Context. The context section provides background
+information needed to understand the context of the
+question(s) of interest and the data that will be used to
+answer the questions [15, 16]. This may include infor-
+mation from the publication on which the case study is
+based, but also additional background literature. For an
+example from public health, the case study may describe
+what is currently known (or not known) about the health
+impact of the topic. This serves to demonstrate how the
+specific question(s) fit into a larger scientific context.
+For the CO2 case study, the context section includes
+a discussion of the potential impacts of climate change
+on human health, an overview of the likely progression
+of warming in the coming years, and potential impacts
+on other components of the environment such as ocean
+acidity and rainfall quantities.
+5. Limitations. In addition to the motivation and context
+for each case study, it is important to formally describe
+limitations of the analysis presented as it provides im-
+portant context for the educator or learner [7]. Examples
+of limitations include (i) limitations due to the available
+data, such as the use of surrogate variables or indica-
+tors, (ii) limitations in the methods used, such as annual
+average estimates for quantities that are likely to vary
+daily or monthly, and (iii) selection biases due to sam-
+pling of observed data. A key concept in data science
+is that the conclusions from an analysis can only be as
+good as the data that go into it and the methods used to
+analyze them, so presenting these limitations provides
+a valuable learning opportunity.
+In the CO2 case study, we describe limitations about
+how the data are incomplete because only certain
+countries reported CO2 emissions for certain years. We
+describe how additional emissions were also produced
+Wright et al. | 2023 | arXiv |
+Page 5
+
+by countries that are not included in the data.
+This
+helps the learners to understand that while the data will
+help us understand CO2 emissions, it will not provide
+the full picture.
+6. What are the data? To provide transparency about the
+data sources, we describe where and how the raw data
+were obtained and used in the case study. If the data are
+obtained from a website, survey or report, and where
+possible, we also describe how the data were originally
+collected. We typically describe what the variables are
+in each dataset later in the case study to better match
+the experience of the learners discovering the data after
+they import and explore it.
+The data sources for the CO2 case study are from
+Gapminder (gapminder.org) (originally from the World
+Bank) and the United States National Oceanic and
+Atmospheric Administration.
+In the case study, we
+present a table with the different data sources and a
+brief description of each one, including sources to cite.
+7. Data import. Next, we describe the steps and give
+the code required to read the raw data into the analy-
+sis environment. Currently, all of our case studies de-
+scribe analyses in the R programming language. In some
+cases, importing the raw data is fairly straightforward,
+and this section is quite short. Other case studies have
+longer and more involved data import sections that in-
+volve scraping data from a PDF, accessing data using
+an Application Programming Interface (API), or writing
+functions to efficiently access data from multiple files
+with the same format. Importantly, we describe all of
+our use of code in the case studies in a literate program-
+ming way [29], meaning that we describe each step in a
+way that can be understandable by learners.
+Since the data for the CO2 case study are stored in
+Excel and comma-separated-variable (CSV) files, we use
+standard data import functions read excel() from
+the readxl package and read csv() from the readr
+R package [30] to import our data.
+8. Data wrangling and exploration. Typically one of the
+longest sections for many of our case studies is the wran-
+gling section, which describes all of the steps required to
+take the imported raw data and get it into a state that is
+ready for analysis and creating visualizations. We also
+demonstrate how to perform exploratory data analysis
+[31].
+For example in the CO2 case study, the raw data needs
+to be converted from a “wide” to “long” format so that
+each country-year observation is in a single row. After
+wrangling the data from each source, we demonstrate
+how to join together data sets from different sources by
+matching on country-year combinations. Ultimately, we
+create one large data set containing all the variables we
+want to use for our analysis (in the columns) with one
+record for each country-year combination (in the rows).
+9. Data visualization. We show both simple and com-
+plex data visualizations to explore and demonstrate a
+variety of graphical design choices, including plot type
+and other aesthetic choices to best show the types of vari-
+ables of interest. In addition, most case studies describe
+how to facet or combine plots together so that all the ma-
+jor findings of the case study are illustrated in a single
+data visualization.
+In the CO2 case study, we create data visualizations
+for a subset of the variables. For example, we use line
+plots to visualize how CO2 emissions, in metric tons,
+have varied over time globally (Figure 1). We go into
+detail around coloring and labeling the lines, zooming
+in and out on the time-scale axis, as well as including
+informative plot titles and axis labels. We demonstrate
+that when looking at CO2 emissions from different
+countries across time, special consideration for labeling
+is required. We show that a heat map or tile plot does a
+great job of illustrating top country differences in a less
+overwhelming manner (Figure 1). We also demonstrate
+the utility of faceted plots to simultaneously visualize
+more variables over time. We also show how to start
+looking for associations or trends in the data through
+scatter plots with smoothed lines or linear regression
+lines added.
+10. Data analysis. Our case studies are intended to intro-
+duce how a particular statistical test or data science tech-
+nique might be implemented and interpreted to answer
+the scientific question(s) of interest. However, we walk
+the learner through an unexpected outcome and how we
+diagnosed it [19]. We provide background information
+about statistical concepts and how these concepts apply
+to our example analysis.
+The main topic of the analysis section for our CO2
+emissions Open Case Study is correlation and how
+correlation is related to linear regression.
+We dis-
+cuss background information such as a description
+about what summary statistics are, what the correlation
+coefficient is, and how the correlation coefficient is math-
+ematically calculated. We also describe the limitations
+of correlation analysis and how correlation does not
+imply causation. We demonstrate how to implement as-
+sessments of correlation and how to interpret the results.
+11. Summary. In this section we provide a summary fig-
+ure that visually indicates some of the major findings of
+the case study. The goal of this visualization is to demon-
+strate how to communicate the results of the analysis to
+a broader audience [6]. This often involves combining
+plots and adding annotations. This summary figure is
+the motivating figure used at the beginning of the case
+study. Along with this figure, we provide a synopsis
+of the case study in which the motivation, context, and
+Wright et al. | 2023 | arXiv |
+Page 6
+
+scientific questions are restated and summarized, while
+the major steps of wrangling, data exploration, and anal-
+ysis are described. The main findings of the analysis are
+discussed, with emphasis on what these findings might
+indicate for the larger context of the scientific question,
+in addition to what still remains unknown.
+In the CO2 emissions Open Case Study, the summary
+figure (Figure 1) combines several of the plots from the
+case study together to demonstrate the major findings.
+The synopsis recaps what data we worked with (CO2
+emissions for some countries from 1751- 2014) and what
+we have shown in the analysis, including touching on
+the learning objectives outlined at the beginning. We
+give a simpler explanation about the statistical concepts
+that were discussed in the analysis section, in this case
+about correlation and regression.
+We discuss more
+about what we were able to answer or not answer in
+terms of the questions of interest. We describe how we
+discovered a dramatic increase in global CO2 emissions
+over time and that some countries appear to be espe-
+cially responsible. We discuss that although the data
+suggests a relationship between temperature and CO2
+emissions in the US, there are many other important fac-
+tors to consider based on what we know about climate
+change. These include: the influence of CO2 emissions
+from other countries in the atmosphere, the influence of
+other greenhouse gases, the fact that the already existing
+CO2 in the atmosphere continues to trap heat for many
+years, and the fact that heat trapped in the ocean
+due to previous emissions causes delayed changes
+in surface temperatures. We also point out what the
+results of our analysis might mean for how we should
+consider mitigating climate change effects and how
+warming temperatures may impact society in the future.
+12. Suggested homework.
+Each case study suggests
+a homework activity for students to try on their own.
+These activities typically require the students to use the
+skills that they have learned on a new data set or to
+expand the analysis to evaluate another subset of the
+data. Students may also be asked to make visualizations
+based on these analyses.
+The suggested homework for the CO2 emissions Open
+Case Study are to:
+• Create a plot with labels showing the countries with
+the lowest CO2 emission levels.
+• Plot CO2 emissions and other variables (e.g. energy
+use) on a scatter plot, calculate the Pearson’s corre-
+lation coefficient, and discuss results.
+These suggestions would require learners to practice
+their visualization an analytic skills to further investi-
+gate the data with less guidance.
+13. Additional information. This section includes addi-
+tional information about the broader scientific topic of
+the case study, the methods used to analyze the data,
+and the specific data sets used in the analysis. Infor-
+mation is provided as links to external online resources
+such as blog posts, scientific articles, scientific reports,
+and educational websites. We also provide links to doc-
+umentation about the R packages used, as well as the
+specific package versions that were used. We also link
+to information about the specific subject-matter experts
+who contributed to the development of the case study.
+The CO2 emissions Open Case Study includes links
+to resources for learning more about the various R pack-
+ages used in the case study (such as here [32], readxl
+[33], readr [30], dplyr [27], magrittr [34], stringr
+[35], purrr [36], tidyr [37], tibble [38], forcats
+[39], ggplot2 [28], directlabels [40], ggrepel [41],
+broom [42], patchwork [43]) and how they were
+used, as well as information about the statistical topics
+touched on, including correlation, regression and time
+series analysis. These go beyond some of the material
+presented in the case study, to help point instructors or
+learners to additional resources for topics of interest.
+3
+The OCS educational resource
+The OCS resource can be found online (opencasestud-
+ies.org). In addition, we created an educator’s guide
+describing how to most effectively use the case studies,
+how to modify and adapt components of the case stud-
+ies in the classroom, and how to contribute new case
+studies. (opencasestudies.org/OCS Guide).
+3.1
+Open Case Study website and search tool
+Our case study resource is hosted on our Open Case
+Studies (OCS) website (Figure 2). To navigate the case
+studies, we provide an interactive search table, built us-
+ing the DT package [44], that allows those interested to
+search through our case studies by topic, statistical learn-
+ing objective, data science learning objective, and R pack-
+ages demonstrated. This table includes links to the code
+and data for each case study, as well as a links to web-
+sites that are rendered versions of each case study where
+the entire analysis can be read in full.
+3.2
+Open Case Studies on GitHub
+The code and data for each case study are hosted in
+a GitHub repository (Figure 2). Our case studies are
+built in R Markdown, allowing text, images, and gifs
+that describe the context and data analytic process to
+be interspersed with code chunks that show the actual
+code used in the analysis [29].
+We developed these
+prior to the release of the quarto publishing system
+(quarto.org/quarto). These case studies are then “knit”
+into rendered html-formatted files using GitHub actions
+[45] for continuous integration and deployment. By con-
+tinuous integration, we mean that changes are tracked
+Wright et al. | 2023 | arXiv |
+Page 7
+
+Figure 2. An overview of the OCS educational resource The Open Case Studies website contains a searchable database
+of all available case studies. Users can search by case study name, R packages used, learning objectives, and category.
+Each case study links to a website with a rendered version of the entire analysis and to the Github repository. The Github
+repository hosts the online lesson and all of the related code, data, image, plot, and document files needed to follow along
+or conduct new analyses. Some case studies now have interactive versions that include live quizzes and coding tutorials.
+Wright et al. | 2023 | arXiv |
+Page 8
+
+Open Case Studies: Mental Health of American
+Code-
+opencasestudies.org
+Youth
+OPEN
+CASE
+STUDIES
+Thepe
+e episodes (MDE) has
+OPEN
+CASE
+STUDIES
+Show10  entries
+Search:
+GitHub
+Case Study
+Packages
+Objectives
+Category
+Repository
+mental
+All
+All
+All
+All
+Scrape data directly from a website (rvest),
+Mental Health of
+Subset and filter data (dplyr), Write
+American Youth
+functions to wrangle data repetitively, Work
+Static Version:
+cowplot,
+with character strings (stringr), Reshape
+directlabels, dplyr,
+data into different formats (tidyr), Create
+Static Version
+forcats, ggplot2,
+data visualizations (ggplot2) with labels
+Repository
+ Bloomberg
+OPE
+ ggthemes, here,
+(directlabels)and facets for different groups,
+magick, purrr,
+Combine multiple plots (cowplot), Optional:
+American Health
+Interactive Version
+Initiative
+rstatix, rvest,
+Create an animated gif (magick), Discuss
+Repository
+Interactive Version:
+scales, stringr,
+the impact of self-reporting bias on survey
+tibble, tidyr
+responses, Define and create a contingency
+table, Implementation of a chi-squared test
+for independence, Interpretation of a chi-
+squared test for independence
+Showing 1 to 1 of 1 entries (filtered from 12 total entries)
+Previous
+Next
+Motivati
+anul
+innor_join()
+Try Again
+ GitHub
+· README.md
+: Code
+: Data
+=
+: Images
+</>
+()
+: Plots
+: Documents
+github.com/opencasestudies
+Case Study Repositoryand a history of the code from various authors is saved
+to a single main version [46] using Git and GitHub. By
+continuous deployment, we mean that the website ver-
+sions of the case studies are automatically rendered and
+available to the public once a new version is established
+on GitHub. These website versions of the case studies
+are also hosted on GitHub. Currently our case studies
+are all written using the R programming language, how-
+ever our current format could be extended to support
+tutorials using other programming languages as well.
+Our case studies have a table of contents that allows in-
+structors and learners to easily navigate from section to
+section, so that they can focus on the materials most use-
+ful for their needs. In addition, each case study starts
+with a graphic or plot that describes the basic findings
+of the case study. Each case study is organized with the
+same basic structure so that learners can navigate case
+studies more easily, and see patterns across case studies
+on how analysis is performed (Figure S1).
+3.3
+Open Case Study file structure
+Each case-study repository has a similar file structure,
+with a data directory containing both raw data and ver-
+sions of the data in various processed forms to allow
+instructors/learners to modularize the case studies for
+their own purposes (Figure 3). For example, an instruc-
+tor could skip the data import and wrangling sections
+of the case study and focus on the visualizations and
+analysis pieces using a fully cleaned data set. To sup-
+port this modular style of instruction, each case study in-
+cludes commands at the beginning of each section that
+imports the data in the final state of the previous sec-
+tion. These different stages of the data are organized in
+a data folder with five categories: raw, imported, wran-
+gled, simpler import, and extra. The raw data directory
+includes files in their original unaltered condition and
+in the original file format from the original data source
+(in some cases raw files are CSV files, Excel files, PDFs
+among other file formats). The imported data directory
+includes files containing the data in a format that is di-
+rectly compatible with R, such as RData files which are
+often abbreviated as Rda. The wrangled data directory
+also includes an RData file that contains a clean and tidy
+version of the data that has been pre-processed and is
+ready for analysis, as well as csv files for instructors that
+wish to demonstrate a simpler version of data import.
+The simpler import folder contains raw files that have
+been converted to CSV file format or other formats that
+can be more easily imported into R. The extra data folder
+contains data files that allow for individuals to conduct
+analyses beyond what was done in the case study (the
+file format for these extra files varies). Each repository
+also contains a README file [20, 21] that explains the
+modular aspect of the case study, as well as other infor-
+mation about how to use the case study for educational
+purposes (Table S2).
+3.4
+Interactive elements in Open Case Studies.
+To make our case studies more experiential, we have
+introduced interactive elements including quizzes and
+coding exercises using the learnr [47] and gradethis
+[48] packages.
+We include a mix of multiple choice questions and
+coding exercises in each case study. Coding exercises
+are embedded throughout the content of the case stud-
+ies and give students a chance to write code for a specific
+step in the analysis. The answers to these exercises (the
+code/output used in the case study to complete these
+steps) is then hidden in a click-to-expand section right
+after the exercise window. Students can compare their
+own code and output with these answers. We also create
+exercise subsections at the end of the main sections of
+the case study. These exercise subsections include both
+multiple choice questions and coding exercises.
+Stu-
+dents can use them to test their understanding of the
+content in each section. All multiple-choice questions
+provide real-time feedback, giving hints after wrong an-
+swers and allowing students to retry the questions if
+they submitted a wrong answer. For most of the coding
+exercises, hints and/or solutions are available. With the
+help of the gradethis package, some of these coding
+problems also provide real-time feedback after students
+submit their code.
+4
+Building your own case studies
+For educators interested in constructing their own case
+studies, in this next section, we describe our recommen-
+dations for the process based on our experiences and
+challenges throughout this project. We also describe
+these recommendations in our Educator’s guide (open-
+casestudies.org/OCS Guide).
+4.1
+Identifying questions and data for case studies
+The process of choosing data sources and questions of
+interest is arguably the most important part of construct-
+ing a case study. We can either identify an interesting
+and publicly available data set and then ask a timely and
+engaging question about a topic related to the data, or
+we can identify an interesting question and then work
+to find publicly available data to answer this question.
+This process of linking a question to publicly available
+data often involves a bit of trial and error and reshaping
+of the question while keeping in mind and potentially
+adjusting what the case study is meant to demonstrate.
+In our experience developing case studies, we found
+that identifying a data set first was often easier than re-
+lying on finding a data set to answer a particular ques-
+tion. While many of our case studies were specifically
+designed to address a public health challenge, we some-
+times struggled to find publicly available data that was
+appropriate for the question or set of questions of inter-
+Wright et al. | 2023 | arXiv |
+Page 9
+
+Figure 3. An overview of the data file structure on GitHub A tree illustrating the repository data directory structure.
+Each bubble describes the type of data files that can be found in the sub-folders.
+est. Collaboration with subject-matter experts can be
+especially helpful in addressing this challenge. For our
+case studies, we worked with public health experts in
+order to both identify interesting, timely, and testable
+questions and to find a public source of data to answer
+our questions.
+We found we could use the difficulty of obtaining data
+in a standard format (e.g., Excel, CSV) as a teaching op-
+portunity, and that being open-minded about the source
+of the data allowed us to demonstrate unconventional
+skills. For example, when we could not easily access
+the data stored in a table in a published report, we illus-
+trated the data science skill of pulling data directly from
+a PDF. As future data scientists, our students need the
+skills to be flexible to access data that cannot simply be
+read in or imported as-is into R.
+While we typically started developing each case study
+with a set of data science and statistical learning objec-
+tives in mind, there was sometimes a tension between
+finding a data set that would allow us to meet these spe-
+cific objectives and allowing the data to guide the direc-
+tion of the case study. We found that following opportu-
+nities presented by the data itself led us to give examples
+that were more authentic to a real-world data analysis
+situation. We recorded some of these challenges within
+the case studies themselves so students could better un-
+derstand the process of finding the right data to answer
+a question of interest (and the potential need to refocus a
+question). The limitations section in particular provides
+some of the most useful material for class discussions
+about the types of questions the data can and cannot an-
+swer and how sometimes we must simplify our analysis
+to reflect the limitations of the data available to us.
+As educators working during a time of reflection and
+social change around issues of gender and race in re-
+search, we also took care to point out some histori-
+cally overlooked aspects of our data sets.
+For exam-
+ple, collecting data with surveys that provide a lim-
+ited number of options about ethnicity or race or racial
+and gender intersections, limits our ability to accurately
+capture the diversity of the population being studied.
+As an example, we refer the reader to the case study
+about youth disconnection (opencasestudies.org/ocs-
+bp-youth-disconnection).
+Wright et al. | 2023 | arXiv |
+Page 10
+
+Case study
+Repository
+data
+Processed
+Unprocessed
+data: clean,
+original source
+tidy and ready
+data.
+Data
+for analysis.
+formatted into
+an R data
+frame object.
+raw
+wrangled
+Files that
+Raw datain
+imported
+allow for
+otherformats
+additional
+for simple
+analyses
+importing
+simpler
+extra
+importFor some case studies, we focused on finding mostly
+clean and complete data to allow us to demonstrate cer-
+tain concepts, like machine learning or how to create a
+dashboard. In these case studies where we knew that
+the analytical material was going to get quite intensive
+and lengthy, we specifically sought to find data sets that
+would allow us to jump right in with little difficulty in
+terms of gathering, cleaning, and importing the data.
+Our overall suggestions for starting a case study are:
+• Be open-minded and flexible about data sources:
+Unlike performing a real analysis where an analyst
+might choose to avoid complications in accessing
+the data (when the option is available to go with a
+data set that is easier to access), such complications
+can provide teaching opportunities to prepare stu-
+dents for cases where they will not have a simpler
+option available.
+• Determine the level of flexibility based on the
+goals of the case study: If the case study is intended
+to demonstrate a specific statistical method or data
+import method, more effort may be required to find
+the right data to meet this specific teaching expecta-
+tion. In our case we knew we were planning to
+make several case studies, thus we were able to
+let some of the case studies naturally flow in di-
+rections we didn’t initially intend. This ultimately
+led to some teaching opportunities we did not ex-
+pect. However, for some of our case studies we
+were more rigid about our data needs.
+• Think about the scope of the case study: Keep in
+mind the 1) type of learners that the case study is
+intended for and 2) data analysis method goals that
+the case study is intended to demonstrate. Try to
+avoid a case study that is both intensive for data
+import/wrangling and intensive for data analysis.
+At a later point reevaluation of the overall direction
+and scope of the case study may be needed. If the
+case study is too long, consider splitting it into mul-
+tiple case studies.
+• Keep it simple: Explaining a process at a beginner
+level often involved more space within a case study
+than anticipated. Keeping case study plans simple
+can help as unexpected teaching opportunities may
+arise that may require more instruction.
+4.2
+Do the analysis first but with a learner in mind
+To present an analysis narrative, it is necessary to first
+perform the analysis before working on the narrative
+description. However, the case study itself should not
+simply be a reproduction of the process used to analyze
+the data. Instead it should contain simplifications and
+modifications to create a clear and coherent presenta-
+tion for students. To do this, it is crucial to keep a good
+record of all the steps taken during this initial analysis,
+including explanations and comments to justify the anal-
+ysis choices made along the way. Special care should be
+taken to record exactly how the raw data is obtained.
+Often the way we would typically perform an anal-
+ysis ourselves might not always be the best for instruc-
+tion purposes. For example, an experienced data analyst
+might start by writing a function that wrangles multiple
+similar data files. However, this would not be the appro-
+priate way to start a case study for beginners. Instead
+one might choose to focus on wrangling a single specific
+file in great detail before trying to generalize the code
+as a function. Thus we try to determine an overall pro-
+cess of data import and wrangling for the intended level
+of audience before really generating the dialogue that
+describes this process.
+We also found that often the data exploration steps
+and the steps involved in the decision-making process
+of how to wrangle the data needed to be simplified for
+a case study. For example, we may ultimately decide
+to remove a data source from our analysis because we
+find errors in the data and dealing with these errors are
+beyond the scope for our intended audience. While it
+may be useful to tell students about these data errors
+[19] and how to address them, we also need to keep an
+appropriate level of detail so as not to overwhelm them.
+Another situation where we might modify the anal-
+ysis is if a process requires a considerable level of trial
+and error. Rather than showing the students all the itera-
+tions of the trial and error and all of the decisions around
+this process, we may only demonstrate a small portion
+so as not to make the case study too lengthy. In a case
+study about machine learning, for example, we aimed
+to achieve a certain level of performance so we spent a
+fair amount of time demonstrating how to optimize and
+tune parameters. While we briefly described our tuning
+strategy, we did not show all intermediate models, but
+ultimately showed two that were interesting and useful
+for describing parameter tuning.
+To conclude, we may have gone through a learning
+process in our own analysis, eventually arriving at a
+more refined approach. Instead of describing the entire
+process to get to this point, we would sometimes simply
+present the final approach, yet describe in the narrative
+that in practice more effort would be required. While we
+do want to present a realistic depiction of the data anal-
+ysis process, we also need to achieve clarity and focus.
+4.3
+Creating the case study narrative
+Once an analyst has performed the analysis to address
+the questions of interest, it is time to start writing the
+narrative. First, we introduce and motivate the main
+topic by presenting some research related to the particu-
+lar question evaluated in the case study.
+First, we describe the data import, wrangling, and
+analysis processes. As mentioned above, this will likely
+not be a faithful reproduction of our own analysis pro-
+cess, but will be recreated to best meet the pedagogical
+goals of the case study. In terms of added narrative, we
+Wright et al. | 2023 | arXiv |
+Page 11
+
+do our best to guide students through the new informa-
+tion we are presenting. The first time we use a function
+we describe what it does, its main arguments, and what
+packages it comes from. We describe the thoughts be-
+hind our decision making process from one step to the
+next, sometimes illustrating times where we try some-
+thing and it does not work to reflect a real-world data
+analysis.
+We also describe jargon and background information
+where possible with click-to-expand sections so as not
+to disrupt the general flow of the case study. For exam-
+ple, an expanded section would explain how ”piping”
+works, passing objects through a series of steps, to avoid
+slowing down students who are already familiar, while
+allowing us to not lose students that have never seen
+piping before. Other material for such expandable sec-
+tions includes describing the “grammar of graphics” for
+the ggplot2 package or providing background statis-
+tical information before performing a statistical test. In
+some cases we describe a concept at great length in an-
+other case study so we link to the description there, but
+in general we at least minimally describe most concepts
+and methods in each case study to keep them as self-
+contained as possible. Similarly, we found including
+portions of RStudio cheat sheets [49] to be very useful
+for certain topics, such as describing regular expressions
+or joining functions. In some cases we found it best to
+explain a concept or challenge with a simpler example
+first using a smaller data set imported into R or created
+in R ourselves. This material is also included in click-
+to-expand sections for students who might already be
+familiar with such concepts.
+While constructing the narrative, we think about
+where we can include question opportunities. These
+opportunities include places for an instructor to start a
+discussion about the analysis decision-making process,
+such as why a particular graph choice is not always ef-
+fective or why a wrangling method might not be repro-
+ducible. We may prompt students to try to remember
+how to perform a task that has already been shown in
+the case study previously. In our interactive case studies
+we also include quiz questions and coding exercises, as
+described in the Section 3.4.
+Finally, we end the narrative by summarizing how to
+communicate the major findings of the analysis [6]. We
+also describe how the results fit into the greater context
+of the field, what the implications are, the limitations of
+the study, and what is still unknown. We finish by going
+through the case study to create a list of all the resources
+shown throughout the case study.
+Through the process of creating this resource, we dis-
+covered a variety of challenges, as well as strategies that
+we used to overcome these challenges, as described in
+Table S3 and guidelines for creating new case studies
+(Supplemental Note S1).
+4.4
+Creating interactive case studies
+We have also included interactive elements in a subset
+of our case studies using packages (learnr [47] and
+gradethis [48]) that build on the shiny [50] package
+which allows R users to more easily create web appli-
+cations.
+The learnr package allows users to create
+multiple choice questions and coding exercises, while
+gradethis allows for customization of the feedback
+divided to learners as they answer questions or perform
+exercises.
+There are two methods to do this. One method is
+to host each exercise as an individual Shiny application
+and then embed these applications in the case study us-
+ing inline frames (HTML ‘iframe‘). The second method
+is to create one single application that incorporates exer-
+cises within the case study (Supplemental Note S2 and
+Supplemental Note S3).
+5
+Discussion and Conclusions
+In this paper, we introduce a model for creating fully
+open-source, peer-reviewed, and complete case stud-
+ies to create an archive of examples of best practices
+to guide students through data analyses involving real,
+complicated, messy, and interesting data. Our archive
+can be used in the classroom by instructors to guide stu-
+dents through any part of our case studies due to the
+easy navigation and common modularized architecture
+to structure the case studies. These can also be used
+by independent learners due to the thorough narrative,
+interactive elements, and complete analyses. Students
+and learners can learn about new topics or return to a
+case study to brush up on details of a particular method
+or technique. The data within our case studies and the
+narrated data analyses and data science methods can be
+used by instructors educating undergraduate and grad-
+uate students, as well as high school students in a vari-
+ety of topics including statistics, public health, program-
+ming, and data science. This provides an opportunity
+for instructors to use data that is relevant to current pub-
+lic health concerns and therefore of interest to a large
+variety of students without the work required to iden-
+tify such data or to determine what analyses are possible
+with such data. This will free instructors to focus on chal-
+lenging the students with more interactive discussions
+in class and allow students to learn more about the deci-
+sion processes required for analyzing data.
+In summary, OCS provides a consistent framework
+grounded by [2], is open, and additionally provides rec-
+ommendations on how to teach the material. With the
+OCS resources, educators can also make their own and
+expand OCS if they contribute back. We believe these ad-
+ditions try to bridge the gaps in the last mile of analysis
+education.
+Wright et al. | 2023 | arXiv |
+Page 12
+
+Back Matter
+Author Contributions
+Contributions listed according to the CRediT system.
+• Conceptualization: CW, MAT, LRJ, SCH
+• Software: all co-authors
+• Formal analysis: all co-authors
+• Investigation: CW, SCH
+• Data Curation CW, SCH
+• Writing - Original Draft: CW, SCH
+• Writing - Review & Editing: all co-authors
+• Visualization: CW, MB
+• Supervision: CW, MAT, LRJ, SCH
+• Funding acquisition: CW, MAT, LRJ, SCH
+Acknowledgements
+We would like to thank the Johns Hopkins Data Science
+lab (jhudatascience.org), in particular Roger Peng, Jeff
+Leek, Brian Caffo, and Jessica Crowell for their support
+and valuable feedback on the Open Case Studies project.
+We would like to thank Ira Gooding for his feedback
+on incorproating case studies into the Coursera plat-
+form. In addition we would like to thank all the data
+science and statistics reviewers of our case studies, in-
+cluding: Shannon Ellis, Nicholas Horton, Leslie Myint,
+Mine C¸ etinkaya-Rundel, Michael Love, and Christina
+P. Knudson, as well as the following student reviewers:
+Jensen Stanton, Tina Trinh, and Ruby Ho. We would
+also like to acknowledge the topic reviewers including:
+Roger Peng, Tamar Mendelson, Brendan Saloner, Renee
+Johnson, Jessica Fanzo, Daniel Webster, Elizabeth Stuart,
+Aboozar Hadavand, Megan Latshaw, Kirsten Koehler,
+and Alexander McCourt.
+We would also like to ac-
+knowledge Ashkan Afshin and Erin Mullany for giving
+us access to the data for the case study titled ”Explor-
+ing global patterns of dietary behaviors associated with
+health risk.” We would also like to thank the Johns Hop-
+kins Bloomberg School of Public Health Department of
+Biostatistics for initially funding this project.
+Funding
+The Open Case Study project reported in this publica-
+tion was supported by a High-Impact Project grant in
+2019-2020 by the Bloomberg American Health Initia-
+tive to create the majority of the case studies currently
+part of the project. A 2020 Digital Education & Learn-
+ing Technology Acceleration (DELTA) Grant from the
+Office of the Provost at the Johns Hopkins University
+supported the creation of interactive case studies and
+many of the tools that support their use, such as the
+search tool. The Open Case Studies guide was funded
+as an extension to funding for the Genomic Data Sci-
+ence Community Network (GDSCN). The GDSCN is
+supported through a contract to Johns Hopkins Uni-
+versity (75N92020P00235) NHGRI. JM was supported
+by Streamline Data Science, U24HG010263-01 (ANVIL),
+UL1TR003098 (NIH/NCATS): Institutional Clinical and
+Translational Science Award, and UE5CA254170.
+Conflict of Interest
+The co-authors Carrie Wright and Stephanie Hicks re-
+ceive royalties on a book available on Leanpub and a
+course on Coursera titled “Tidyverse Skills for Data Sci-
+ence”, both which incorporate three case studies from
+the Open Case Studies project.
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+Page 16
+
+Supplementary Materials
+Open Case Studies: Statistics and Data Science Education through Real-World
+Applications
+Carrie Wright, Qier Meng, Michael Breshock, Margaret A. Taub, Leah R. Jager, John Muschelli,
+Stephanie C. Hicks∗
+∗Correspondence to [email protected]
+Contents
+1. Supplemental Table S1-S3.
+2. Supplemental Figures S1-S5.
+3. Supplemental Notes S1-S3.
+Wright et al. | 2023 | arXχiv |
+Page S1
+
+Supplemental Tables
+Diversity of topics in the Open Case Study resource
+Question
+How does something change over time?
+How do groups compare?
+Types
+How do groups compare over time?
+How do paired groups compare?
+Are certain groups or subgroups more
+vulnerable?
+How does something compare across regions?
+How to predict outcomes for new data?
+Does this influence my data?
+Are variables related to one another?
+How to display this data?
+Data
+Multiple files
+PDF
+Types
+CSV
+Excel
+Website
+Image text
+API
+Google Sheets
+Survey data / Codebooks
+Wrangling
+Extracting data from a PDF
+Geocoding data
+Methods
+Recoding data
+Joining data
+Modifying data
+Working with text
+Reshaping data
+Repetitive process
+Data
+Formatted Table
+Scatter plot
+Visualizations
+Line plot
+Bar plot
+Box plots
+Pie chart / Waffle plot
+Heat map
+Correlation plots
+Missing data plots
+Maps
+Advanced
+Matching a plot style
+Faceted plots
+Visualizations
+Direct group labels
+Emphasizing a group
+Plot annotations
+Plot error bars
+Combining plots
+Interactive plots
+Interactive maps
+Interactive tables
+Adding images to plots
+Interactive dashboard
+Analysis
+t-tests
+ANOVA
+Concepts
+Linear Regression
+Logistic Regression
+and
+Mann-Kendall Trend Test
+Machine Learning
+Methods
+Chi-Squared Test of Independence
+Wilcoxon Signed-Rank Test
+Calculating percentages with missing data
+Wilcoxon Rank Sum Test
+Supplementary Table S1. Example topics in the OCS resource An example of how these are applied in an
+example case studies is provided in Table 2.
+Examples of modular case-study use
+User
+Example Use
+Data Folder
+Student
+Data science students looking for
+open source data for a class project
+raw
+Student
+Public health student practicing data
+wrangling and visualization
+imported
+Educator
+Course instructor assigns homework
+using related but new data that
+expands beyond the case study
+extra
+Educator
+Data analysis instructor who wants
+students to practice some simple
+data import, but has limited time
+simpler import
+Self-Learner
+Researcher looking for analysis
+examples
+wrangled
+Supplementary Table S2. Examples of modular case-study use Examples of potential uses for case studies,
+beyond demonstrating the full case study in a class or as a learner following along.
+Wright et al. | 2023 | arXχiv |
+Page S2
+
+Challenges of creating case studies in the OCS resource
+Challenge
+A Suggested Solution
+Finding the appropriate scope, data, and questions for the
+intended audience
+Be open minded about data sources, be flexible about revising or
+removing data
+Balancing the teaching goals with the teaching opportunities
+presented by the data
+Keeping the plan simple to allow room for unexpected teaching
+opportunities
+Showing the right amount of the data science process
+Performing the analysis first, then curating what will be included
+Balancing an assumption of some prior knowledge with making
+the case study self-contained
+Click-to-expand sections for additional information and links to
+other case studies and resources
+Catering to multiple audience types
+Modularizing case studies to allow different users to use only
+certain parts of the case studies and including click-to-expand
+sections for students that need more background
+Modularization of case studies
+Saving the data after each section and loading at the beginning
+of each section
+Supplementary Table S3. Challenges of creating case studies in the OCS resource A list of the major
+challenges that we experienced and suggested solutions.
+Wright et al. | 2023 | arXχiv |
+Page S3
+
+Supplemental Figures
+Supplementary Figure S1. Case Study Structure An image of the top of a case study showing the interactive table of
+contents to the left and a image summarizing the main findings of the case study.
+Supplementary Figure S2. Creating a Multiple Choice Quiz with learnr An example code chunk used to create a
+multiple choice quiz in the CO2 Emissions case study.
+Wright et al. | 2023 | arXχiv |
+Page S4
+
+Open Case Studies: Exploring global patterns of
+Code-
+IOPEN
+CASE
+Select Languagev
+STUDIES
+Powered by Google Translate
+A
+Mean National BMIl overtime
+B
+Change in BMI by region
+1985
+2017
+Men
+Women
+Motivation
+35
+6
+Main Questions
+0
+LearningObjectives
+0
+Obesity
+Context
+30
+4
+to 2017)
+Mean BMI
+USA)
+USA
+Limitations
+USA
+USA
+88
+SA
+USA
+What are the data?
+USA
+(1985 t
+Data Import
+25
+USA
+Data Wrangling
+Data Exploration
+0
+Data Analysis
+20
+Data Visualization
+Summary
+Suggested Homework
+Rur
+Jrban
+Additional Information*r dw_quiz, 
+echo = FALSE}
+quiz(caption = 
+question("which one of the following functions in the 'dplyr'
+ package allows us to see all of the variables (columns) at once.
+where several values of those columns are shown on the right of the variable names?"
+answer("'slice_head()
+"this function allows us to see the rows at the end of the data."),
+answer ("*glimpse()*
+, correct = TRUE),
+allow_retry = 
+TRUE,
+random_answer_order = TRUE
+question("which one of the pipe operators (from the 'magrittr' package) should be used right after a variable name if we want to
+perform a sequence of operations on that variable, and meanwhile, assign the final output to that variable (without redefining that
+variable using
+=?"
+answer("*%>%*"
+allow_retry = TRUE,
+random_answer_order = TRUE
+function in the ‘dplyr' package do? (more than one correct answers)"
+answer("select certain variables(columns) of the data.", message =
+"seiecting certain variables(columns) of the data is the
+answer("Rename a variable."
+ message = "Renaming a variable is the function of the
+‘renameO*
+function."),
+answer("Modify an existing variable.", correct = TRUE),
+allow_retry
+TRUE,
+random_answer_order = TRUESupplementary Figure S3. Multiple Choice Quiz Rendered in Case Study The multiple choice quiz in the CO2
+Emissions case study created using the code in Figure S2.
+Wright et al. | 2023 | arXχiv |
+Page S5
+
+Which one of the following functions in the
+dplyr package allows us to see al of the
+variables (columns) at once, where severa
+values of those columns are shown on the
+right of the variable names?
+O glimpse()
+O slice_sample()
+O
+slice_tail()
+slice_head()
+Submit Answer
+Which one of the pipe operators (from the
+magrittr package) should be used right
+after a variable name if we want to perform a
+sequence of operations on that variable, and
+meanwhile, assign the final output to that
+variable (without redefining that variable
+using <- or =)?
+%<1% 0
+%<>%
+%<%
+%%
+ Submit Answer
+Which of the following can the mutate()
+function in the dplyr package do? (more
+than one correct answers)
+O Create a new variable.
+O Rename a variable.
+O Select certain variablesicolumns) of the
+data.
+O Modify an existing variable
+Submit AnswerSupplementary Figure S4. Creating a Coding Exercise with learnr An example of the code chunks used to create a
+coding exercise in the Obesity case study.
+Wright et al. | 2023 | arXχiv |
+Page S6
+
+However,
+we can also do the same thing in a single step, where we replace the 'women
+dataset
+and then we split
+the data by
+Women"
+to create a new dataset all within the same chunk of code. write this code
+yourself!
+r,echo=FALSE7
+save(rural_urban, file = here::here("www", "exercise", "dw_code2.rda"))
+fr dw_code2-setup}
+.
+1ibrary(tidyverse)
+library(magrittr)
+load(here: :here("www"
+"exercise"
+"dw_code2.rda"))
+fr dw_code2. exercise=TRUE7
+country_split
+{r dw_code2-hint-1}
+country_split <
+str_subset(string = rural_urban
+pattern = "women") %>%
+r dw_code2-solutionf
+country_split <-
+str_subset(string = rural_urban,
+pattern = "women")
+%>%
+stringr::str_split(pattern=
+Women"） %>%
+unlist()
+# view the first rows of the data
+head(country_split)Supplementary Figure S5. Coding Exercise Rendered in Case Study The coding exercise rendered in the Obesity
+case study created using the code in Figure S4. Top: The exercise prompt. Bottom: Displays the hint function that can be
+used when stuck. The hints are made also with learnr.
+Wright et al. | 2023 | arXχiv |
+Page S7
+
+However, we can also do the same thing in a single step, where we replace the Women dataset with the code we used to create that
+dataset, so first, we select the "wiomen" and then we split the data by " women'" to create a new dataset all within the same chunk of
+code. Write this code yourself!
+R Code
+S start Over
+Q Hints
+ Run Code
+country_split <-
+2
+3
+Hints
+Next Hint >
+Li Copy to Clipboard
+country_split <
+2
+str_subset(string = rural urban,
+m
+pattern = "Women") %>%
+R Code
+S Start Over
+Q Hints
+Run Code
+country split <-
+2mSupplemental Notes
+Supplemental Note S1
+Guidelines for a creating case study in the OCS collection
+To preserve the integrity of the core ideas of our resource we suggest the following guidelines for case studies to
+be included in the Open Case Studies collection:
+• Case studies should be written in open source programming languages.
+• Case studies should use data that is publicly available or can be made publicly available. Please ensure that
+the data can be made public if it is not already.
+• Case studies should include disclaimers and appropriate license agreements.
+• Effort should be made to describe the original source of the data with transparency.
+• All included images (that are not original to the case study) should include a source.
+• Core sections of the case study are required: Motivation, Main Questions, Learning Objectives, Limitations
+(outlining the limitations of the data), What are the Data?, and Summary (including limitations for the analysis
+presented).
+• Case studies should aim to describe the decision making process involved in performing data-science related
+tasks.
+• Links to literature or other sources to motivate the scientific topic of the case study should be included where
+possible.
+• Despite often being motivated by articles, case studies are not intended to demonstrate the methods of a paper.
+They are intended as an educational resource where users are guided through the data science process.
+Supplemental Note S2
+Technical aspects of the two methods for creating interactive case
+studies
+In this note we describe how we implemented the two options for creating interactive case studies: 1) hosting each
+exercise as an individual Shiny application and then embedding these applications in the case study or 2) creating
+one single application that incorporates exercises within the case study by creating the case study as a learnr
+tutorial.
+Approach 1 - Embedding interactive elements.
+We did this by adapting the method of De Leon [51]. Building
+upon the iFrame Resizer JavaScript library, De Leon introduced the use of an HTML file that resizes the learnr
+tutorial windows in real time. This HTML file is included in the YAML header of the R Markdown file for the
+exercise, along with div tags in the last line to indicate the end of the content for the iFrame Resizer. We also
+added this feature by creating another HTML file containing the iFrame Resizer JavaScript and including it in
+the YAML header. Then, each exercise Shiny application is added to the case studies as an ‘iframe‘ with class
+“interactive”. This is done by creating the learnr exercises in separate R Markdown files. Each exercise is
+then published as an individual Shiny app online. Once published, the exercises now each have a unique URL
+address where they are hosted. The exercises can now be rendered within re-sizable windows inside the case study
+itself by nesting iFrame HTML code chunks within the case study R Markdown file using the exercise URLs as
+inputs. At the end of each case study R Markdown, an HTML tag specifying the application of the resizer only
+to class “interactive” is added. R knitr has an include app() function to embed Shiny applications, but De
+Leon’s method better accommodates the aesthetics of the website as well as the pop-ups in the tutorials like hints,
+solutions, and feedback.
+Approach 2 - Single interactive case study application.
+The second option is to make the entire case study a
+learnr tutorial, where exercises can be directly created in the case study R Markdown document. To do this, we
+add runtime: shiny prerendered to the YAML header of this document. In order to publish the single case
+study application, all of the files needed to render the case study (data, images, CSS style code, etc.) need to be put
+in a www/ sub-directory. This folder will be published along with the index.Rmd file to Shiny. The quizzes can
+be created inside a single R code chunk using the quiz(), question(), and answer() functions from learnr
+(Figures S2-S3). The coding exercises must be created using multiple R chunks. One chunk is written for each of
+the following: exercise setup, hints, solutions, and the exercise prompt (Figure S4). In the rendered case study,
+these individual chunks are combined into a single coding exercise element (Figure S5). Eventually, this document
+is published as a single Shiny application (opencasestudies.org/ocs-make-interactive-tutorial).
+Wright et al. | 2023 | arXχiv |
+Page S8
+
+Differences between the approaches.
+This first approach of creating interactive elements is beneficial as it is
+generally easier to maintain. If one exercise breaks, we can readily identify which exercise is broken. However,
+the method can be inefficient as multiple Shiny applications are needed for a single case study and since each
+application stands alone, the data in the case study cannot be directly passed into the exercise. Thus, when the
+exercise involves using the data from the case study at a specific point in the analysis, the setup can become more
+complicated.
+The second approach is more efficient because the entire case study is written in a single file. However, the
+startup of a Shiny application is required to be under 60 seconds. For case studies that take longer to render, some
+of the data analysis results need to be stored separately to reduce the startup time. Also, since a single file is more
+difficult to maintain, it is important to keep detailed maintenance notes for the case study.
+We decided to go with the second approach, because we created many exercises and this would have resulted in
+tens of Shiny applications. The first approach involves making each exercise question a learnr tutorial, which is
+then published as a Shiny application or with Posit Connect. With this approach, the actual rendered version of the
+interactive case study is still hosted on GitHub, but each exercise is embedded in the website as its own window
+using HTML ‘iframe‘. The second approach, that we we ultimately adopted, involves making the interactive case
+study a single learnr tutorial and in this case, instead of the rendered website version being hosted on GitHub, it
+is published as a Shiny application. In our case, we chose to use Posit Connect for this which does involve some
+payment. Currently, Shiny allows users to publish up to 5 applications for free for a single individual.
+Supplemental Note S3
+Examples of of interactive questions in case studies
+We included a couple of types of interactive elements. The first are coding exercises as part of the analysis in the
+case study. These exercises are placed where the analysis repeats what is already done with another data set. The
+students apply what they learned, while staying in the context of the case study. The second type are multiple
+choice quiz questions and coding exercises at the end of case study sections.
+As an example of the first type, if there are two similar data sets that need to have their variable names changed,
+we might demonstrate how to do it with one data set and ask the student to change the variable names of the
+other. Other circumstances for these exercises include, for instance, where an argument for a function is introduced.
+The exercise window can be used to explore the function of that argument by implementing the code with or
+without such argument. For example, when introducing the set.seed() function, the exercise window allows
+the students to experiment on how using a different seed would change the output.
+The second type of interactive elements are to assist with learners assessing their knowledge using quiz questions
+and coding exercises at the end of case study sections. We include these exercises in a separate subsection so that
+they are easy to navigate to. Instead of focusing on a specific step of the analysis, these problems integrate the
+content of the entire section. The multiple-choice questions test the students’ understanding of the concepts
+introduced. For example, they may be asked what the key assumptions of a linear regression model are, what an
+R function can do, etc. The coding questions check the students’ abilities to implement the R functions in data
+analyses. For example, they could be asked to convert a made-up data set from wide format to long format, to
+build a machine learning model using an R built-in data set, etc.
+Wright et al. | 2023 | arXχiv |
+Page S9
+

BNE5T4oBgHgl3EQfTA98/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf,len=218
+page_content='Investigation of radiation hardness of silicon semiconductor detectors under irradiation with fission products of 252Cf nuclide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' N V Bazlov1,2, A V Derbin1, I S Drachnev1, I M Kotina1, O I Konkov1,3, I S Lomskaya1, M S Mikulich1, V N Muratova1, D A Semenov1, M V Trushin1  and E V Unzhakov1 1 NRC "Kurchatov Institute" - PNPI, Gatchina, Russia 2 Saint-Petersburg State University, Universitetskaya nab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' 7/9, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Petersburg, Russia 3 Ioffe Physical-Technical Institute of the Russian Academy of Sciences, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Petersburg, Russia  e-mail: trushin_mv@pnpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='nrcki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='ru Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Influence of the prolonged irradiation by fission products of 252Cf radionuclide on the operational parameters of silicon-lithium Si(Li) p-i-n detectors, Si surface barrier detectors and Si planar p+n detector was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The obtained results revealed a linear shift of the fission fragment peaks positions towards the lower energies with increase of the irradiation dose for all investigated detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The rate of the peaks shift was found to depend strongly on the detector type and the strength of the electric field in the detector’s active region, but not on the temperature of irradiation (room or liquid nitrogen temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Based on the obtained results, the possibility of integration of the investigated types of Si semiconductor detectors in a radionuclide neutron calibration source is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Introduction Heavy nuclides subjected to spontaneous fission decay accompanied by emission of several fast neutrons can be utilized as a compact neutron calibration source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The most common spontaneous fission source is 252Cf which undergoes α-decay and spontaneous fission with a branching ratio of 97:3, whereas each spontaneous fission event liberates 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='8 neutrons and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='7 gamma-ray photons on average [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The timing of the moment of neutron production can be fixed by detecting the fission fragments signal with a semiconductor detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='  Semiconductor detectors possess sufficiently high energy resolution for detection of the high- energy heavy ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The main obstacle for the integration of such detectors in the neutron calibration source could be their limited lifetime under the influence of the nuclide radiation [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Degradation of the detector’s operational parameters effectively proceeds just in case of irradiation by alpha particles and fission fragments (FF), which are capable of transferring a significant fraction of their energy to the atoms of the detector lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Therefore, the degradation of the semiconductor detector will limit the maximum neutron source activity and/or the source expiration period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' This article is devoted to the investigations of degradation of the operational parameters of several types of silicon semiconductor detectors under prolonged irradiation with fission products of 252Cf (\uf061-  particles and fission fragments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The main issue was to study the rate of degradation of different detector types under irradiation by 252Cf fission products at various irradiation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Irradiation was performed at room and liquid nitrogen temperatures as well as with different detector’s operational biases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' with different electric field strength in the detectors active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Results of the preceding investigations were presented in previous articles [3-5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Detectors and experimental setup Three types of silicon semiconductor detectors were under investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Detectors of the first type are SiLi p-i-n detectors produced from p-type silicon ingot with resistivity of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='5 kΩ×cm and carrier lifetime of 1000 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Two similar detectors with a sensitive region of 20 mm in diameter and 4 mm thick were produced using standard Li drift technology [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The thickness of the undrifted p-type layer in these detectors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' the entrance window thickness) usually amounts to 300-500 nm [7], which is kept to suppress the excessive growth of the leakage current at high operation reverse voltage [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Detectors of the second type were two surface-barrier (SB) detectors fabricated from p-type boron- doped silicon wafer of (111) orientation and 10 mm in diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The resistivity and the carrier lifetime were 1 kΩ×cm and 1000 µs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The front side of the wafers was covered by a thin layer of amorphous silicon which served as a passivation coating [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='  The ohmic contact was made by sputtering of Pd layer on the whole rear side of the wafer, whereas the rectifying one – by evaporation of Al dot with diameter of 7 mm in the center of the wafer’s front side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Detector of the third type was p+n planar detector with the thickness of 300 \uf06dm produced in Ioffe Physical-Technical Institute (entrance window thickness was about 50 nm and the voltage of full depletion – nearly 150 V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Irradiation by a 252Cf source was performed in vacuum cryostat typically during 10-20 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The source representing a stainless steel substrate covered by an active layer under the thin protective coating was mounted 1 cm above the detector front surface that was collimated in order to exclude side surface effects of incomplete charge collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The spectra of the fission products of 252Cf were recorded continuously during the whole irradiation period in short 1-hour series, what allowed us to observe the spectra evolution directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Detector reverse current was also monitored during the whole irradiation period on 5-second basis with the following averaging on 1-hour measurement series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Details of the measurement setup were presented in [3-5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' (a) The first and the last spectra measured by SB2 detector in the beginning and at the end of the prolonged irradiation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The following peaks are marked: constant amplitude generator peak (g), peak of \uf061-particles at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='118 MeV (\uf061), peak at doubled energy of \uf061-particles (2\uf061) and the peaks due to FFs of light (LF) and heavy (HF) groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' (b) Dependence of the light and heavy FF peaks visible energies on exposure by FFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='  a)  106  α  first spectrum  last spectrum  g  105  Counts per hour  104  103  2α  HF  102  101  100  0  20  40  60  80  100  Energy, MeVb)  Heavy Fragments  80  Light Fragments  Peak Position, MeV  75  70  65  60  55  0  1  2  3  4  Exposure, FF 107  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Experimental results In order to study the influence of temperature of irradiation on the degradation of the detector’s parameters, the irradiation of identical SiLi detectors was performed at room (SiLi1 detector) and liquid nitrogen (SiLi2 detector) temperature, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' To study the influence of external electric field strength on the detector’s parameters degradation, two identical SB detectors were subjected to the irradiation with different applied reverse biases, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' with different electric field strengths in their active regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The operating biases applied to the respective detector during the irradiation period, the corresponding surface electric field strengths and the total exposures are collected in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' For all investigated detectors the similar signs of operational parameters degradation as a result of the prolonged irradiation by 252Cf fission products were revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' As an example, Figure 1a represents the spectra recorded by SB2 detector at the beginning and at the end of the prolonged irradiation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The peak at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='1 MeV corresponding to α-particles and another peak at doubled energy of the α- particles caused by their accidental coincidences were used as reference points for the calibration of the energy scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Two broad unresolved peaks appearing at higher energies correspond to fission fragments of light and heavy groups, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='  The main effect of the detector degradation is a gradual shift of fission fragments visible energy towards the lower values, see Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The positions of the peaks corresponding to heavy (HF) and light (LF) fission fragments were approximated using the Gaussian function for each 1-hour series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The dependences of the peaks positions with exposure by fission fragments can be well described by linear functions (Figure 1b) for any masses of fission fragments and for all investigated detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The obtained slope coefficients are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' It is interesting to note, that the obtained coefficients for the peaks of light and heavy fission fragments groups differ approximately by the factor of 2 – this holds for all types of investigated detectors and for all irradiation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' In more details this fact will be discussed separately in the next paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' A similar approximation of the positions of α-peaks didn’t reveal any measurable shift with the irradiation dose for all studied detectors [4-5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Another sign of the detector’s operational parameters degradation under irradiation is the rapid increase of the leakage current which proceeds linear with the number of absorbed fission products [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The obtained slope coefficients of the leakage current growth are also collected in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='  Discussion It could be noted in Figure 1 that the peak energies of light and heavy groups of fission fragments are below the predicted values of 104 MeV and 79 MeV [10], respectively, even on the spectrum measured by non-irradiated detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The same is true for all other investigated detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' This effect is known as pulse-height defect (PHD) in heavy charged particles spectroscopy by semiconductor detectors implying that the measured pulse height amplitude for heavy charged particles is somewhat lower than that for \uf061-particles of the same energy [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' It is generally considered that PHD is caused by a combination of energy losses (i) in the detector dead layer/entrance window, (ii) due to the atomic collisions and (iii) due to recombination of the electron-hole pairs created by the incident heavy particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Whereas energy losses by (i) and (ii) mechanisms are well understood, the full understanding of the charge losses due to recombination is still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Two models were suggested supposing that enhanced carrier recombination proceeds either in the bulk region on the radiation-induced defects created by incident FFs [11], or at the surface states of the semiconductor [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='  The later model is consistent with the TRIM [13] simulation results (Figure 2) showing that the density of electron-hole pairs generated by fission fragments reaches the maximum in the near-surface region of the detector and then gradually drops down towards the bulk, suggesting therefore that decisive influence on PHD would have the carrier recombination at the surface states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Previously, the PHD of about 7-10 MeV was reported for 252Cf fission fragments detection by semiconductor detectors not subjected to the prolonged irradiation [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' These PHD values are close to those ones obtained for the investigated planar and SB1 detectors operated at high reverse bias – see Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' We believe, that higher PHD values in non-irradiated SiLi are related with rather thick entrance window in these detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Whereas the increase of PHD for SB2 detector operated at lower  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Irradiation conditions and the degradation of the operational parameters of the investigated detectors: Ub – applied bias during irradiation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Fs – surface electric field strength;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' PHDLF/ PHDHF – pulse-height defects for light and heavy fragments peaks registered by non-irradiated detectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' NFF and N\uf061 – exposure by fission fragments and \uf061-particles, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' ∆EHF/∆NFF – slope coefficient describing the linear shift of heavy fission fragment maximum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' ∆ELF/∆NFF – slope coefficient describing the linear shift of light fission fragment maximum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' ∆I/∆N – rate of the reverse current increase relative to the total number of the registered fission products (wasn’t measured for SiLi2 detector);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' NFFmax – maximal permissible exposure by fission fragments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' t – expected active operation period of the detector in a neutron source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='  p+n planar  SB1  SB2  SiLi1  SiLi2  Ub, V  150  200  30  400  400  Fs, kV/cm  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='5  40  17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='5  PHDLF/PHDHF, MeV  8/10  9/11  18/19  28/29  35/37  NFF 108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='43  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='4  1  N\uf061\uf020\uf020 \uf031\uf030\uf031\uf030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='44  ∆EHF/∆NFF 10 5, keV/FF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='7  ∆ELF/∆NFF 10 5, keV/FF 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='9 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='2 12  ∆I/∆N 10 16, A/ion\uf020  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='9  14  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='4 NFFmax 108  22  12  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='2  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='7  t, years  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='5  electric field (Table 1) reflects the influence of the electric field strength on the charge carrier collection efficiency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' on the recombination of the generated electron-hole pairs (note that the active layer thickness in SB2 detector exceeds the projection range of incident FFs even at 30V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' As a result of the prolonged irradiation by 252Cf fission products, the linear shift of FF peaks positions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' the linear increase of PHD for fission fragments peaks, was revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Since the task of semiconductor detector operating as a part of neutron calibration source is the reliable detection of fission fragments signal, the irradiated detector could be considered to be "degraded" when the spectrum of the heavy fission fragment overlaps with much more intense signal at double energy of α- peak, what prevents us from discrimination between them [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The values of maximal “permissible” exposure by fission fragments NFFmax corresponding to the beginning of the peaks overlap at three standard deviations from their maxima were estimated for each detector using the corresponding slope coefficients derived for HF peak and the results are presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' TRIM simulated vacancies distribution profiles (solid lines) and linear densities of electron-hole pairs (dashed lines) generated by light and heavy FFs with mean energies and masses of 104 MeV and 79 MeV, 106 amu and 142 amu, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='  Heavy Fragments  300  2  Light Fragments  250  200  150  100  50  0  0  0  5  10  15  20  Depth, μm  Permissible exposure values NFFmax for the investigated detectors appeared to vary approximately by one order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The highest NFFmax values were found for planar and SB1 detector operated at 200V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Reduction of the operating bias and thus the electric field strength in case of SB2 detector has led to considerable decrease of the expected permissible exposure value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Therefore, the electric field strength affects not only the PHD on non-irradiated detector, but also the value of the expected maximal exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' However the NFFmax exposure values for SiLi detectors – which operated with lowest electric field as compared with other detectors – are significantly higher than that for SB2 detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Thus the expected maximal exposure appeared to be more sensitive to the electric field strength in the surface barrier detectors and less sensitive in SiLi and planar detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' It follows then that not only the electric field strength, but also a detector’s internal structure defines the PHD growth under irradiation and the maximal permissible exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' According to TRIM simulations, irradiation of Si detectors by fission fragments will lead to the creation of vacancy-interstitial pairs and therefore to the formation of high density of radiation- induced defects in the region from detector surface till the depth of 17 μm with the maxima at 14-16 μm (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Additionally TRIM indicates, that the energy of FFs is high enough to damage the detector surface by sputtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='  Therefore, prolonged irradiation with fission fragments will lead to an increase of the carrier recombination rate both in Si bulk and on the surface of the semiconductor, thus contributing to the PHD growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The transition region in the detectors produced by planar and by SiLi technology (p+n and p-i transition regions, respectively) is located inside the crystalline matrix at the typical depths of 50-500 nm from the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Apparently, the contribution of the surface recombination to the charge carrier losses will be more significant for surface-barrier detectors than for SiLi and planar ones, whereas the contribution of bulk defects – approximately similar in all detectors, what may be the reason for different sensitivity of NFFmax exposure to the electric field strength in these detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Additional investigations are needed to determine the dominant charge loss channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Suggested neutron calibration source should operate also at cryogenic temperatures (liquid nitrogen or slightly above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Performed irradiation of SiLi2 detector at liquid nitrogen temperature has shown, that in contrast to the electric field, temperature of irradiation seems to have no or only minor influence on the expected value of maximal exposure as it could be concluded from the comparable NFFmax values obtained for SiLi detectors irradiated at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Somewhat smaller NFFmax exposure obtained for SiLi2 detector is probably related with thicker entrance window in this detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='  Knowing the maximal expected exposure values NFFmax, it is possible to estimate the duration of active “lifetime” of neutron calibration source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' For the operation of neutron calibration source the reasonable neutron activity would be the around 20 neutrons/s and taking into account that each spontaneous fission releases in average 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='7 fast neutrons, the activity of 20 neutrons/s would correspond to ~6 spontaneous fissions per second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Therefore, considering the maximal exposure value from Table 1, the duration of active “lifetime” of such neutron calibration source will be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='2-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='6 years (without taking into account the decay of the radiation source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' During this operation period, a significant increase of leakage current up to ~100 μA can be expected at room temperature, as can be calculated from the obtained coefficients of leakage current growth (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Such high reverse current is unacceptable and therefore the detector cooling in order to reduce the reverse current during the neutron source operation will be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' The coefficients of current growth upon irradiation by fission products of 252Cf appeared to be an order of magnitude higher than the corresponding coefficients of 7-17×10–17 A/α determined by us earlier for the identical detectors subjected to long-term irradiation by \uf061-particles [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='  This fact confirms that prolonged irradiation by FFs leads to the creation of the effective recombination-generation defect centers participating in charge carrier recombination and the reverse current growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Conclusions Prolonged irradiation of three different types of Si semiconductor detectors by fission products of 252Cf nuclide has led to a gradual increase of pulse-height defect for the fission fragments peaks in all  investigated detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' This will eventually lead to the overlap with more intense \uf061-peak and therefore to the impossibility of further reliable detection of fission fragments by the semiconductor detector and thus to the limitation of the operation period of neutron calibration source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Obtained experimental results suggest, that in order to assure the longest operation period of the neutron calibration source it is worth to use the semiconductor detectors with lowest surface recombination rate and with highest possible electric field strength in their active region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Among the investigated detectors, the planar one most fully meets these requirements, whereas in relatively thick SiLi detectors it is difficult to achieve the high electric field strength and surface-barrier detectors may suffer from high surface recombination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' With properly chosen semiconductor detector the expected active operation period of 252Cf-based neutron calibration source may reach up to 12 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content='  Acknowledgements The reported study was funded by RFBR, project number 20-02-00571 References [1] Knoll G F 2000 Radiation Detection and Measurement, 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' (New York: John Wiley and Sons) ISBN 978-0-471-07338-3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
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+page_content=' Derbin A V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Drachnev I S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Konkov O I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Kotina I M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Kuzmichev A M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Lomskaya I S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Mikulich M S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Muratova N V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Niyazova N V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Semenov D A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Trushin M V and Unzhakov E V 2021 Journal of Physics: Conference Series 2103 012138 [4] Bakhlanov S V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Bazlov N V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
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+page_content=' Drachnev I S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Kotina I M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Konkov O I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Kuzmichev A M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Mikulich M S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Muratova N V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Trushin M V and Unzhakov E V 2021 Journal of Physics: Conference Series 2103 012139 [5] Bazlov N V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Bakhlanov S V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Derbin A V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Drachnev I S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Eremin V K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Kotina I M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Muratova V N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Pilipenko N V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Semenov D A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Unzhakov E V and Chmel E A 2018 Instruments and Experimental Techniques 61 323 [6] Bazlov N V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Bakhlanov S V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Derbin A V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Drachnev I S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Izegov G A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
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+page_content=' Muratova V N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Niyazova N V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
+page_content=' Semenov D A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNE5T4oBgHgl3EQfTA98/content/2301.05533v1.pdf'}
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+RobArch: Designing Robust Architectures against Adversarial Attacks
+ShengYun Peng1, Weilin Xu2, Cory Cornelius2, Kevin Li1, Rahul Duggal1,
+Duen Horng Chau1, and Jason Martin2
+1Georgia Institute of Technology, Atlanta, GA, USA
+{speng65,kevin.li,rahulduggal,polo}@gatech.edu
+2 Intel Corporation, Hillsboro, OR, USA
+{weilin.xu,cory.cornelius,jason.martin}@intel.com
+Abstract
+Adversarial Training is the most effective approach for
+improving the robustness of Deep Neural Networks (DNNs).
+However, compared to the large body of research in optimiz-
+ing the adversarial training process, there are few investi-
+gations into how architecture components affect robustness,
+and they rarely constrain model capacity. Thus, it is unclear
+where robustness precisely comes from. In this work, we
+present the first large-scale systematic study on the robust-
+ness of DNN architecture components under fixed parame-
+ter budgets. Through our investigation, we distill 18 action-
+able robust network design guidelines that empower model
+developers to gain deep insights.
+We demonstrate these
+guidelines’ effectiveness by introducing the novel Robust
+Architecture (RobArch) model that instantiates the guide-
+lines to build a family of top-performing models across
+parameter capacities against strong adversarial attacks.
+RobArch achieves the new state-of-the-art AutoAttack accu-
+racy on the RobustBench ImageNet leaderboard. The code
+is available at https://github.com/ShengYun-Peng/RobArch.
+1. Introduction
+Deep Neural Networks (DNNs) are vulnerable to adver-
+sarial attacks [6,17,29,32,47]. Many defense methods have
+been proposed to mitigate this pitfall [2,10,44,50,58,59,64],
+and among them, Adversarial Training (AT) [36] is the most
+effective way to defend against adversarial attacks. Com-
+pared to the large body of research devoted to improving
+the loss function [23, 31] and optimizing the AT proce-
+dure [14, 54, 64], few studies investigate how architectural
+components affect robustness despite its importance.
+Yet DNN architectures have been dominating general-
+ization improvements [16, 19, 34].
+Recent research has
+started to highlight the potential significant impact architec-
+ture choices could have on robustness [13,46], and showed
+20
+40
+60
+80
+100
+Number of Parameters (Millions)
+25
+30
+35
+40
+45
+50
+ResNet-18
+ResNet-50
+XCiT-S12
+XCiT-M12
+WideResNet50-2
+WideResNet50-2+DiffPure
+Swin-B
+XCiT-L12
+RobArch-S
+RobArch-M
+RobArch-L
+DeiT-S
++DiffPure
+ResNet-50+DiffPure
+ResNet-50+GELU
+PoolFormer
+-M12
+DeiT-S
+RobArch-L Achieves SOTA AutoAttack Accuracy
+RobArch family outperforms ConvNets and Transformers with similar #param
+AutoAttack
+Accuracy (%)
+Figure 1.
+Our RobArch model family outperforms the SOTA
+XCiT family on RobustBench ImageNet leaderboard [7]. Every
+RobArch model outperforms its XCiT counterparts at a similar
+capacity. RobArch-S outperforms ResNet-50 by 9.18 percentage
+points, and is even more robust than WideResNet50-2 despite hav-
+ing 2.6× fewer parameters. The robustness continues to increase
+as capacity increases. RobArch-L achieves the new SOTA AA ac-
+curacy on RobustBench. Table 3 presents accuracy details.
+that adjusting widths [55] or depths [26] could robustify a
+network.
+However, those studies did not constrain the model ca-
+pacity, making it hard to attribute the robustness gains to
+those adjustments, because increasing model capacity alone
+could already improve robustness [26, 36]. Thus, control-
+ling for model capacity while assessing robustness is im-
+portant, and recent research has provided supporting evi-
+dence. For example, despite the popular belief that trans-
+former models might be more robust than CNNs [5, 42],
+Bai et al. [3] demonstrated that Data-efficient image Trans-
+1
+arXiv:2301.03110v1  [cs.CV]  8 Jan 2023
+
+formers (DeiT) [49] and ResNet [19] with Gaussian Error
+Linear Unit (GELU) activations [21] attained comparable
+robustness if the model scales were balanced. Therefore, it
+remains unclear how these previously studied architectural
+components precisely affect robustness. Our research filled
+this critical research gap by making three key contributions:
+• The first large-scale systematic study on the robust-
+ness of DNN architecture components. To the best of
+our knowledge, our work is the first to comprehensively
+investigate and compare the robustness impacts of a wide
+range of architecture components on a large dataset such
+as ImageNet. Advancing over prior work, we carefully
+constrain the parameter budget to isolate and hone in on
+the benefit of each component. Such a systematic study
+enables us to discover a family of new architectures that
+outperform state-of-the-art (SOTA) networks. (Figure 1).
+• 18 actionable robust network design guidelines.
+Our systematic investigation for component robustness,
+through training over 150 models on ImageNet [12], en-
+ables us to distill 18 generalizable, actionable guidelines
+that empower model developers to gain deep insights and
+design networks with higher robustness. The guidelines
+present significant new knowledge and discoveries for our
+computer vision community. For example, we have dis-
+covered (1) deepening a network is more effective than
+widening it, and there is a sweet spot; (2) specific mod-
+ifications such as adding Squeeze and Excitation (SE)
+block, removing the first normalization layer in a block,
+and reducing the downsampling factor in the stem stage
+effectively boosts robustness; and (3) architecture designs
+that harm robustness include inverted bottleneck, large di-
+lation factor, Instance Normalization (IN), parametric ac-
+tivation functions [9], and reducing activation layers.
+• Top performance against strong adversarial attacks.
+We demonstrate our guidelines’ effectiveness by intro-
+ducing the novel Robust Architecture (RobArch) model
+that instantiates the guidelines to build a family of top-
+performing models across parameter capacities against
+strong adversarial attacks. In particular, we compare our
+RobArch family with the Cross-Covariance Image Trans-
+formers (XCiT) family [1] that is the SOTA on Robust-
+Bench [7]. Every RobArch model outperform its XCiT
+counterpart with a similar model capacity (Figure 1).
+RobArch-S surpasses ResNet-50’s AutoAttack (AA) ac-
+curacy by 9.18 percentage points, and is even more robust
+than WideResNet50-2 despite having 2.6× fewer param-
+eters. The robustness continues to increase as capacity
+increases. RobArch-L achieves the new SOTA AutoAt-
+tack (AA) [8] accuracy on the RobustBench ImageNet
+leaderboard. RobArch’s performance advantage extrap-
+olates to the Projected Gradient Descent (PGD) attack.
+Overall, the proposed RobArchs outperform both Con-
+vNets and Transformers with similar total parameters.
+2. Robust Architecture Design
+We carefully select architectural components from off-
+the-shelf DNNs (ResNet [19], RegNet [40], DenseNet [25],
+and ConvNeXt [34]) that improve generalization accuracy.
+Based on the commonalities in these network designs, we
+group the components into three modification categories:
+• Network-level: depth, width
+• Stage-level: stem stage, dense connection
+• Block-level: kernel size, dilation, activation, SE, nor-
+malization
+Since ResNet [19] is a milestone in the history of DNN
+architecture, we choose its most popular instantiation,
+ResNet-50 (∼26M parameters) as the base architecture,
+which consists of a stem stage, n = 4 body stages, and a
+classifier head, as our starting point. Each body stage con-
+tains multiple residual blocks with various depth and width
+configurations. Appendix A provides details of ResNet-50
+configurations.
+Notation and symbols used throughout this paper.
+• We denote D-d1-...-dn as the depth of each stage in an
+n-stage network (n ∈ {3, 4, 5, 6}).
+• For stage i, wi and wbi are the numbers of channels in the
+pointwise and non-pointwise convolutions, respectively.
+• Bottleneck multiplier bi is the ratio of channels in point-
+wise to non-pointwise convolution, bi = wi/wbi.
+• Assuming wgi is the group convolution width, gi is the
+total number of groups in the non-pointwise convolution
+layer: gi = ⌊wbi/wgi⌉ = ⌊wi/ (bi × wgi)⌉.
+• Width expansion ratio is e = wi+1/wi, i ≤ n − 1.
+• We use W-w1-...-wn, G-g1-...-gn, BM-b1-...-bn to rep-
+resent the number of channels, group convolution groups,
+and bottleneck multiplier in an n-stage network.
+Experimental settings.
+We train all models on Ima-
+geNet [12] with the recipes specified in Sec. 2.1. When
+studying a single architecture component (Sec. 2.2 - 2.4)
+and building cumulative networks (Sec.
+3.1 & 3.2), we
+use 10-step PGD (PGD10) with different attack budgets
+ϵ (ϵ ∈ {2, 4, 8}) for fast evaluations. After finalizing the
+model structures of the RobArch, we test all RobArchs
+against PGD100 and AA.
+All attacks are ℓ∞ bounded.
+To control for the effect of model capacity, we constrain
+the networks’ total parameters, i.e., similar to ResNet-50
+(∼26M), throughout the exploration.
+2.1. Training Techniques
+Standard-AT. Adversarial Training (AT) is the most reli-
+able defense to obtain robust DNNs [17, 36]. Standard-AT
+is formulated as a min-max optimization framework [36].
+Given a network fθ parameterized by θ, a dataset with sam-
+ples (xi, yi), and a loss function L, the robust optimization
+2
+
+3-Stage
+4-Stage
+5-Stage
+6-Stage
+5
+10
+15
+20
+GMAC
+Natural
+Accuracies
+PGD10-4
+Accuracies
+20
+30
+40
+50
+60
+4-Stage Networks Provides
+Top Accuracies at Moderate GMACs
+(a) 4-stage networks attain top accuracies at much
+lower Giga Multiply–Accumulates (GMACs) than 3-
+stage networks. 5-stage and 6-stage networks are sig-
+nificantly less robust.
+PGD10-8
+10
+20
+30
+40
+50
+60
+Not Followed
+Accuracy
+Natural
+PGD10-2
+PGD10-4
+Guideline 2 Followed
+Guideline 2 Followed
+For a 4-stage network,
+set d1 <d2 <d3
+dn
+(b) Higher accuracy when guideline 2 is fol-
+lowed: depth rule d1 < d2 < d3 > c × d4.
+We plot the mean accuracy (solid line) and
+95% confidence interval (the shading).
+Width
+Larger circle means higher PGD10-4 accuracy
+0
+10
+20
+30
+40
+50
+Depth
+200
+400
+600
+800
+1000
+1200
+1400
+1600
+D-5-8-13-1
+W-512-768-1152-1728
+D-8-12-20-2
+W-424-632-944-1416
+Stage 1
+Stage 2
+Stage 3
+Robustness Generally Improves
+As Depth Increases & Width Decreases
+(c) Robustness generally improves in all stages as
+depth increases and width decreases, until catas-
+trophic overfitting happens with significantly re-
+duced robustness.
+Figure 2. For network-level design, following guideline 2 to increase depth and decrease width in a 4-stage network provides optimal
+robustness. We study (a) how the number of stages affects accuracies, (b) stage depth settings, and (c) depth-width trade-off. We only
+plot the first three stages of a 4-stage network in (c) for better visualization since the last stage is much shallower as per the optimal depth
+configurations in guideline 2. These observations also apply to other PGD attack budgets, as shown in Appendix D.
+problem is formulated as:
+argmin
+θ
+E(xi,yi)∼D
+�
+max
+x′ L (fθ, x′, y)
+�
+,
+(1)
+The inner adversarial example x′ is generated on the fly dur-
+ing the training process, which aims to find an adversarial
+perturbation of a given data point x that achieves a high loss,
+x′
+k+1 =
+�
+x+∆
+(x′
+k + αsgn (∇xL(θ, x′
+k, y))) .
+(2)
+sgn(·) is the sign function, α is the step size, x′
+k is the ad-
+versarial example generated after k steps (1 ≤ k ≤ K),
+∆ = {δ : ∥δ∥∞ ≤ ϵ} is the threat mode, and �
+x+∆ is a
+projection operation that clips the perturbation back to the
+ϵ-ball centered on x if it goes beyond the attack budget.
+Fast-AT. Fast-AT speeds up the Standard-AT and can ro-
+bustify a ResNet-50 in under 13 hours [54]. It not only
+adopts Fast Gradient Sign Method (FGSM) [17] to gener-
+ate adversarial samples during the training but also incorpo-
+rates a cyclic learning rate [43] and mixed-precision arith-
+metic [37] to fully accelerate the AT with just 15 epochs. A
+line of research improves the performance and mitigates the
+catastrophic overfitting problem discovered in the Fast-AT,
+e.g., YOPO [63], GradAlign [2], GAT [45], Sub-AT [30],
+etc., but there are limited explorations on whether these
+recipes are compatible with the full ImageNet [12].
+Although Fast-AT provides competitive PGD results, its
+resulting robustness on ResNet-50 is inferior to that of
+Standard-AT’s as per the AA accuracy on the RobustBench
+leaderboard [7]. Therefore, we use Fast-AT as a rapid in-
+dicator while exploring different architecture components
+and building the RobArch family, and use Standard-AT to
+robustify all members in the RobArch family.
+2.2. Network-level Design
+Depth. In the standard ResNet-50 (D-3-4-6-3), each stage
+downsamples the input features by 2. The downsampling
+in the first stage is replaced by a max-pooling layer in the
+stem stage. We sample 36 architectures based on the depth
+relationship between each pair of stages, i.e., di ≤ di+1
+and di > di+1. The widths in all stages are the same as
+ResNet-50, and when n > 4, we reuse the width in stage
+4. For n = 6, even setting di = 1, i ≤ n leads to 1.83M
+more parameters than ResNet-50. Hence, there is only 1
+data point for the 6-stage network, and we do not continue
+increasing the total stages. Fig. 2a shows the results af-
+ter AT. 4-stage networks attain top natural and adversarial
+accuracies at much lower GMACs than 3-stage networks.
+5-stage and 6-stage networks are significantly less robust.
+These results are expected since shallow stages, in general,
+compute on higher resolutions, and the depth of a 3-stage
+network in shallow stages is deeper than a 4-stage network
+by a large margin for similar total parameters. Hence, we
+select 4-stage networks and further explore the depth rela-
+tionship between stages.
+Huang et al. [26] found that reducing depth in the last
+stage of a 3-stage WideResNet34-10 improves robustness.
+Upon further inspection of our 4-stage models, we observe
+that increasing the stage depths di along with i, then sig-
+nificantly decreasing the depth in the last stage, leads to
+higher robustness. Fig. 2b shows that following such a rule
+(d1 < d2 < d3 > c × d4) leads to a higher accuracy than
+not following it. We set c = 3 and leave the finetuning of
+a larger c to further research. RegNet [40] first discovered
+the depth pattern and applied it to improve benign accuracy.
+Our results extend this discovery to adversarial settings and
+3
+
+show that it helps robustify architectures without incurring
+extra parameters. Overall, we found the optimal stage depth
+ratio is D-5-8-13-1 and listed its performance in Table 1 row
+2.
+Guideline 1: 3-stage ≈ 4-stage > 5-stage ≫ 6-stage
+network in terms of robustness.
+Guideline 2: For a 4-stage network, set d1 < d2 <
+d3 ≫ dn, and D-5-8-13-1 provides the optimal robustness.
+Width. Factors that affect the stage width are pointwise
+convolution channels wi, group convolution groups gi, and
+bottleneck multiplier bi. The width configurations of the
+standard ResNet-50 are W-256-512-1024-2048, G-1-1-1-1,
+BM-0.25-0.25-0.25-0.25. Unless otherwise specified, all
+configurations are kept consistent with ResNet-50 when
+studying one of the factors.
+For bi ∈ {0.125, 0.25, 0.5, 1, 2, 4}, we first test a con-
+stant bi = b in all stages. The accuracy reaches the peak
+when b = 0.25 or 0.5 and significantly decreases when in-
+creasing b from 0.5 to 4, which shows the inverted bottle-
+neck is harmful to robustness. b = 0.25 (ResNet-50) has
+higher natural and PGD10-2 accuracy, while b = 0.5 has
+higher PGD10-4 and PGD10-8 accuracy. Both results are
+shown in Table 1 (rows 1 and 3). Then, we vary bi for dif-
+ferent stages, b1,2 < b3,4 and b1,2 > b3,4. The robustness
+of BM-0.25-0.25-2-2 is better than bi = 2 but worse than
+bi = 0.25. Surprisingly, BM-4-4-0.25-0.25 outperforms
+both bi = 0.25 and bi = 4. We further combine the two op-
+timal bottleneck multipliers and set b1,2 = 0.5, b3,4 = 0.25.
+As shown in Table 1 row 4, this setting attains higher accu-
+racy than both bi = 0.5 and 0.25.
+Next, we study the group convolution groups gi
+∈
+{1, 2, 4, 8, 16, wbi}. gi = wbi is equivalent to the depth con-
+volution. The pointwise convolution width wi is adjusted to
+reach the controlled parameter budget, but bi is always 0.25.
+For a constant gi = g, we observe a significant increase
+from g = 1 (ResNet-50) to g = 2, but then the accuracy
+gradually decreases if we continue to increase g. Similar
+to the bottleneck multiplier study, we vary gi for different
+stages. However, there is no further robustness gain. We list
+the results of g = 2 in Table 1 row 5.
+For the width expansion ratio, we evaluate e
+∈
+{1, 1.5, 2, 2.5, 3}.
+The robustness rises and saturates at
+e = 1.5 and falls for a larger e. We show e = 1.5 in Ta-
+ble 1 row 6. Finally, we combine the optimal configurations
+for all three factors, i.e., b1,2 = 0.5, b3,4 = 0.25, gi = g =
+2, e = 1.5. However, the robustness is inferior to that of just
+using the individual optimal settings. After a close look at
+all the results, we find setting a constant bi = b = 0.25
+works favorably with g and e.
+In addition, we observe
+g = 2, e = 2 and g = 1, e = 1.5 achieve the best two ac-
+curacies. The phenomenon also demonstrates that directly
+combining multiple individual optimal architectural settings
+does not transfer to a better model.
+Guideline 3: Inverted bottleneck harms robustness, es-
+pecially when added to deeper stages.
+Guideline 4:
+For a single modification, b1,2
+=
+0.5, b3,4 = 0.25, gi = 2, and e = 1.5 all show promising
+improvements. However, merging all three configurations
+makes the model less robust, and the optimal width config-
+urations are e = 2, g = 2 or e = 1.5, g = 1 with b = 0.25.
+Combining Depth and Width. In this part, we answer the
+following question: Under a fixed model capacity, does in-
+creasing widths while decreasing depths, or vice versa, im-
+prove robustness?
+We use the optimal depth ratio, D-5-8-13-1. To provide a
+more general understanding and avoid overfitting to specific
+optimal settings, we cross-select e = 1.5, g = 2, b = 0.25
+from the two optimal width configurations from guideline
+4. We proportionally adjust depths and widths to accom-
+modate the fixed budget.
+Fig.
+2c displays the relation-
+ship between depths and widths using PGD10 accuracy. A
+larger bubble size means higher accuracy. The results show
+that increasing depth while decreasing width improves ro-
+bustness in all stages. It is important to note that if we
+continue the trend, catastrophic overfitting [2] occurs dur-
+ing training. Since catastrophic overfitting drastically de-
+creases the robustness, we should deepen the network but
+balance the depth and the width to stabilize the AT pro-
+cess.
+Comparing the top 2 models (dotted lines), both
+PGD10-2 and PGD10-4 accuracies of the deeper model are
+0.10pp (percentage points) higher, but the PGD10-8 accu-
+racy is 0.49pp lower, which is a sign of unstable training.
+Overall, D-5-8-13-1 is selected as the starting point of our
+cumulative model in Sec. 3.1. Compared to ResNet-50
+(D-3-4-6-3), D-5-8-13-1 is much deeper and slimmer with
+significantly higher robustness: ↑ 1.15pp for natural accu-
+racy, ↑ 2.03pp for PGD10-2, ↑ 2.62pp for PGD10-4, and
+↑ 2.75pp for PGD10-8. We observe a similar depth-width
+relationship when scaling up the model in Sec. 3.2.
+Guideline 5: Under a fixed model capacity, first increase
+the network depth proportionally to the optimal depth until
+catastrophic overfitting happens, i.e., a sudden drop in loss
+and increase in training accuracy. The width is adjusted to
+fill the total parameter budget.
+2.3. Stage-level Design
+Stem Stage. The stem stage in a standard ResNet-50 con-
+sists of a convolution layer and a max-pooling layer, each
+of which has a downsampling factor of 2. All 4 tandemly-
+connected body stages downsample the input resolution by
+2 except the first stage. The convolution layer uses a 7 × 7
+kernel and outputs 64-layer features.
+In the stem stage, we modify the following architec-
+tural components: channel width, kernel size, “patchify”
+stem, and downsampling factor. First, we test channel width
+∈ {32, 64, 96} and kernel size ∈ {3, 5, 7}. With less than
+4
+
+Table 1. PGD10 robustness of architecture components. All con-
+figurations trained with Fast-AT and evaluated on full ImageNet
+validation set. We provide ResNet-50 as baseline. Appendix D
+shows detailed results, including PGD10-2 and PGD10-8.
+Idx.
+Configurations
+Natural
+PGD10-4
+1
+ResNet-50
+56.09%
+30.43%
+Network-level Design
+2
+D-5-8-13-1
+57.35%
+33.33%
+3
+BM-0.5-0.5-0.5-0.5
+55.31%
+30.52%
+4
+BM-0.5-0.5-0.25-0.25
+56.11%
+31.26%
+5
+G-2-2-2-2
+57.31%
+32.09%
+6
+W-512-768-1152-1728
+57.17%
+32.04%
+7
+G-2-2-2-2
+56.64%
+31.04%
+BM-05-05-025-025
+W-512-768-1152-1728
+Stage-level Design
+8
+Stem width 96
+57.29%
+32.06%
+9
+Move down (↓) downsampling
+57.08%
+33.08%
+10
+Dense ratio 2
+55.93%
+30.73%
+Block-level Design
+11
+Kernel size 5
+56.73%
+32.77%
+12
+Kernel size 7
+59.70%
+34.67%
+13
+Dilation 2
+52.98%
+28.38%
+14
+Dilation 3
+52.10%
+27.97%
+15
+Act. GELU
+57.48%
+33.12%
+16
+Act. SiLU
+58.19%
+34.07%
+17
+Act. PSiLU
+56.38%
+33.76%
+18
+SE (ReLU)
+57.83%
+32.64%
+19
+Norm-BN-BN-0
+54.15%
+29.59%
+20
+Norm-BN-0-BN
+56.04%
+31.34%
+21
+Norm-0-BN-BN
+56.18%
+31.61%
+0.01M increase in total parameters, switching convolution
+layer width from 32 to 64 and 64 to 96 improve the PGD10-
+4 accuracy by 0.7 and 1.65 percentage points, respectively.
+The “stem width 96” is located in Table 1 row 8. For kernel
+size = 3 or = 5, the training overfits to FGSM and leads
+to a completely non-robust model. The original kernel size
+is 7 in ResNet-50, and increasing it to 9 improves the PGD
+accuracy but leads to a drop in the natural accuracy.
+We study the downsampling factor next. RegNet [40]
+is built based on ResNet, but the max-pooling layer in the
+stem stage is replaced by a stride 2 convolution shortcut
+connection in the first stage. We denote this operation as
+“move down (↓) downsampling.” The evaluation result (Ta-
+ble 1 row 9) manifests 0.99 and 2.65 percentage points in-
+crements in natural and PGD10-4 accuracy. We further dis-
+assemble the operation by only discarding the max-pooling
+layer without adding the stride 2 convolution shortcut. Al-
+though the robustness is slightly lower than “move ↓ down-
+sampling,” it still outperforms ResNet-50 by a large margin.
+Vision Transformer (ViT) [16] first introduced the
+“patchify stem,” and ConvNeXt [34] also incorporated the
+design to improve generalization.
+Motivated by those
+works, we replace the original stem with a 4 × 4 patch, i.e.,
+kernel size = stride = 4, and observe a slight increment
+in robustness. Since moving down the downsampling layer
+boosts robustness, we continue to test a smaller 2×2 patch.
+The accuracy increases as expected, but the gain is slightly
+lower than directly moving down the downsampling layer in
+a ResNet-style stem. Since a small kernel size in the early
+convolution layer leads to a smaller receptive field, a mod-
+erate kernel size of 7 × 7 is preferred. Overall, we select
+“stem width 96” and “move ↓ downsampling” as potential
+candidates while building the cumulative model in Sec. 3.1.
+Guideline 6: Replacing the max-pooling in the stem
+stage with a downsampling shortcut in the first stage sig-
+nificantly improves robustness.
+Guideline 7:
+For the convolution layer in the stem
+stage, directly replacing it with a “patchify” stem design
+contributes to the robustness. However, the optimal con-
+figurations are increasing the channel width and setting
+kernel size = 7.
+Dense Connection. Huang et al. [25] introduced the dense
+connection in DenseNet that concatenates the feature maps
+of all preceding blocks within the stage as the input to the
+current block.
+We extend the definition and experiment
+with different dense ratios i (i ∈ {1, 2, 3, 4, 5}), i.e., i pre-
+ceding feature maps are used to construct the input. Only
+i = 2 shows minor improvements in PGD accuracy, and
+no strong benefits are observed (Table 1 row 10). We fur-
+ther remove the last Rectified Linear Unit (ReLU) since the
+original DenseNet uses the Pre-Activation (PreAct) opera-
+tion [20]. However, the robustness is further degraded, and
+we assume the poor performance of reducing the last acti-
+vation itself (discussed in 2.4) is a potential reason.
+Guideline 8: Dense connection is not beneficial to ro-
+bustness.
+2.4. Block-level Design
+Kernel Size. In this part, we study the kernel size in all
+body stages. Inspired by the large local window size in
+Swin-T [33], ConvNeXt [34] boosts the generalization ac-
+curacy via increasing the kernel size from 3 × 3 to 7 × 7.
+A large kernel size can extract more semantic informa-
+tion but implicitly increases the attack area during back-
+propagation. It is unclear whether a larger kernel size can
+bring higher robustness. We evaluate kernel size ∈ {3, 5, 7}
+and find the accuracy grows along with the kernel size (Ta-
+ble 1 row 1, 11 and 12), but the total parameters also in-
+crease significantly: kernel = 3 (25.56M), kernel = 5
+(45.68M), and kernel = 7 (75.86M). Thus, using a large
+kernel size is a potential candidate to optimize the robust-
+5
+
+ness when scaling up the model. We will revisit the design
+in Sec. 3.2.
+Guideline 9: Purely increasing the kernel size raises the
+model capacity but improves robustness significantly. Thus,
+it is a prospective option when scaling up the network.
+Dilation. Dilated convolution supports the exponential ex-
+pansion of the receptive field without loss of resolution [62].
+The operation offers a wider field of view at a similar com-
+putational cost. However, the results in Table 1 (row 1, 13
+and 14) show that a larger dilation factor significantly de-
+creases both natural and PGD accuracy after AT. Connect-
+ing to the previous kernel size section, we hypothesize that a
+larger receptive field facilitates the attacker. We still observe
+the robustness gain in using a large kernel size because the
+huge model capacity mitigates the effect, yet the accuracy
+drops when adjusting dilation since the operation does not
+change the model capacity. In Sec. 3.2, we also notice the
+kernel size is not effective in optimizing robustness if all
+other modifications are considered at the same scale.
+Guideline 10: Increasing dilation factor enlarges the at-
+tacking area, which leads to inferior robustness.
+Activation. We study two factors in the activation layer:
+the activation function and the number of activation layers
+in a block. For the activation function, we replace ReLU,
+which is used in ResNet-50, with two smoother functions,
+GELU and Sigmoid Linear Unit (SiLU). GELU alone sig-
+nificantly improves the robustness (Table 1 row 15), which
+echoes the result in [3]. SiLU further improves the accu-
+racy (Table 1 row 16), which echoes the result in [57]. Re-
+cently, Dai et al. [9] added learnable parameters to origi-
+nal non-parametric functions, and proposed the parametric
+counterparts, e.g., ReLU to Parametric ReLU (PReLU) and
+SiLU to Parametric SiLU (PSiLU) or Parametric Shifted
+SiLU (PSSiLU).
+These parametric functions outperform
+the non-parametric ones on CIFAR-10 [28]. We test these
+functions on ImageNet and observed PSiLU has the high-
+est robustness among all parametric functions, as shown in
+Table 1 row 17. However, compared to the non-parametric
+versions, all parametric functions are less robust. Since the
+original paper only tested on the small-scale dataset, we be-
+lieve such learnable functions are not compatible with the
+large-scale dataset. Next, we reduce the activation layers in
+each block. Neither reducing one nor reducing two activa-
+tion layers show extra benefits to the robustness. The more
+activation layer we reduce, the worse the performance is.
+Guideline 11: Activation function significantly affects
+robustness. The non-parametric SiLU provides a competi-
+tive improvement.
+Guideline 12: Reducing activation layers in a residual
+block severely hurts the robustness.
+Squeeze and Excitation (SE).
+Hu et al. [24] first
+introduced the SE block that explicitly explored inter-
+dependencies between channels, and adaptively recali-
+brated channel-wise feature responses. Inspired by RegNet
+[40], we place the SE block between the last two convolu-
+tions in each block and set the reduction ratio as 1/4. Com-
+pared to ResNet-50, Table 1 row 18 shows that adding SE
+significantly improves the robustness. Directly adding the
+SE module slightly increases the model capacity by 2.17M,
+but in Sec. 3.1, we show that sacrificing the parameters in
+other components by adopting the SE module can still im-
+prove the robustness, which proves the effectiveness of SE.
+Since switching activation functions shows significant
+differences, we also replace ReLU in the SE block with
+SiLU, GELU and their parametric versions. We still ob-
+serve that non-parametric activation functions are better
+than their parametric counterparts. The SiLU is again the
+optimal activation for SE module. However, in Sec. 3.1,
+we find that replacing the activation function in activation
+layers and SE at the same time causes inferior robustness.
+Guideline 13: The SE module significantly contributes
+to robustness.
+Guideline 14: The robustness improves if we just replace
+the activation function in the SE block. But the modification
+does not work favorably with switching the activation func-
+tion in the residual block.
+Normalization. Similar to the activation layer, we exam-
+ine both normalization functions and the number of nor-
+malization layers in a block. For the normalization func-
+tion, we switch the original Batch Normalization (BN) [27]
+in ResNet-50 to IN [51]. The training is extremely hard to
+converge and thus leading to an almost non-robust model
+(PGD10-4: 8.54%). Then, we attempt to reduce the total
+normalization layers in a residual block. In Table 1, row 19
+to 21 show that reducing the first normalization layer in a
+residual block optimizes the robustness. We keep reducing
+2 BNs, and no further benefits are observed.
+Guideline 15: Switching BN to IN harms robustness.
+Guideline 16: Reducing the first BN in a residual block
+benefits robustness.
+3. Experiments
+In this section, we provide a roadmap that outlines the
+path we take to construct the RobArch using the guidelines
+in Sec. 2. Our roadmap combines architecture components
+such that for each combination we only keep components
+that increase robustness. Then, we scale up the resulting
+model and proposed a family of RobArch models. Finally,
+we compare RobArch with other SOTA architectures. See
+Appendix B for the full experimental setup. We also ablate
+Fast-AT and Standard-AT in Appendix C.
+3.1. A Roadmap from ResNet-50 to RobArch-S
+In this section, we cumulatively construct RobArch-S
+from ResNet-50 based on the proposed guidelines. Table
+6
+
+Table 2.
+The roadmap outlines the path we take to cumula-
+tively improve the robustness and construct RobArch-S (∼26M),
+RobArch-M (∼46M), and RobArch-L (∼104M) based on our
+guidelines. PGD10-2 and PGD10-8 show a similar trend of ac-
+curacy improvement as PGD10-4, and detailed results are shown
+in Appendix E.
+Configurations
+Natural PGD10-4
+Small: ResNet-50 → RobArch-S (S7)
+S0
+ResNet-50
+56.09%
+30.43%
+S1
+S0 + D-5-8-13-1
+57.35%
+33.33%
+S2a
+S1 + g = 2, e = 2, b = 0.25
+57.98%
+33.94%
+S2b
+S1 + g = 1, e = 1.5, b = 0.25
+57.52%
+32.83%
+S3
+S2a + Stem width 96
+57.82%
+34.86%
++ Move down (↓) downsampling
+S4
+S3 + SE (ReLU)
+60.57%
+36.61%
+S5
+S4 + Act. SiLU
+62.04%
+39.48%
+S6
+S5 + SE (SiLU)
+60.32%
+38.24%
+S7
+S5 + Norm-0-BN-BN
+62.27%
+39.88%
+Medium: RobArch-S (S7) → RobArch-M (M2)
+M1
+S7 + Kernel size 5
+63.82%
+41.00%
+M2
+S7 + D-7-11-18-1
+64.40%
+42.06%
+M3
+S7 + W-384-760-1504-2944
+63.52%
+41.43%
+Large: RobArch-M (M2) → RobArch-L (L2)
+L1
+M2 + Kernel size 7
+64.08%
+40.70%
+L2
+M2 + W-512-1024-2016-4032
+66.08%
+43.81%
+L3
+M2 + D-8-13-21-2
+64.91%
+43.09%
+L4
+M2 + D-10-16-26-2
+65.28%
+42.85%
+2 (upper) presents the procedures and results at each step
+of network modification. We start with network depth and
+width. Combining guideline 2 and guideline 5, model S1
+selects the optimal depth configuration D-5-8-13-1.
+For
+width, we test the two optimal width configurations in
+guideline 4 and select g = 2, e = 2 (S2a). For the stem
+stage, model S3 increases the width to 96 and replace the
+max-pooling in the stem stage with a downsampling short-
+cut in the first stage according to guidelines 6 and 7. Then,
+we optimize the block settings in each stage. Guideline 13
+suggests inserting a SE block between the last 2 convolu-
+tions. To accommodate the extra parameters in the modi-
+fication, we reduce the width in all stages and build model
+S4. Next, S5 substitutes SiLU for ReLU in all 3 activation
+layers. However, we find that continuing to replace the acti-
+vation function in the SE block lowers the robustness. Thus,
+we discard the modification, reduce the first BN layer, and
+construct S7. The resulting model is named RobArch-S.
+The guidelines are verified by the consistent increase in ro-
+bustness along the network construction process. The total
+model capacity is comparable to ResNet-50, but both natu-
+ral and PGD-4 accuracies have increased by 6.18 and 9.45
+percentage points, respectively.
+40
+45
+50
+55
+60
+65
+Natural Accuracy
+20
+25
+30
+35
+40
+45 PGD10-4
+Accuracy
+All networks trained with Fast-AT
+RobArch Outperforms SOTA Architectures
+ResNet-18 (12M)
+ResNet-50 (26M)
+ResNet-101 (45M)
+ResNet-152 (60M)
+WideResNet50-2 (69M)
+WideResNet101-2 (127M)
+EfficientNet-B0 (5M)
+EfficientNet-B5 (30M)
+EfficientNetV2-S (21M)
+DenseNet-121 (8M)
+DenseNet-161 (29M)
+MobileNet V2 (4M)
+ResNeXt-50 32x4d (25M)
+RegNetY-3.2GF (19M)
+RegNetY-8GF (39M)
+RegNetX-3.2GF (15M)
+RegNetX-8GF (40M)
+Swin-T (28M)
+RobArch-S (26M)
+RobArch-M (46M)
+RobArch-L (104M)
+Figure 3. Our RobArch model family outperforms SOTA archi-
+tectures under the same Fast-AT training method. With a simi-
+lar model capacity, RobArch-S outperforms ResNet-50 [19] and
+ResNeXt-50 32×4d [61] by 9.45 and 6.80 percentage points, re-
+spectively. RobArch-M outperforms ResNet-101 by 8.16 per-
+centage points. Compared to the models with larger parameters,
+RobArch-S is even more robust than WideResNet101-2 despite
+having 4.85× fewer parameters (highlighted in black). Appendix
+E shows detailed results, including other PGD attack budgets.
+3.2. Scaling Up: The RobArch Family
+We extend our investigation to optimize the robustness
+when scaling up the parameter budget. The budgets align
+with the XCiT [1] family since it is the current SOTA on
+the RobustBench ImageNet leaderboard [7]. Guideline 9
+suggests increasing kernel size as a potential improvement
+when scaling up the model.
+Increasing total depth and
+width are another 2 promising directions [26, 60]. For the
+medium-sized budget (∼46M), model M1 enlarges the ker-
+nel size from 3 to 5, model M2 proportionally deepens
+the network by a factor of 1.4, and model M3 widens the
+channels while keeping the depth same as RobArch-S. The
+training results of M1, M2 and M3 are shown in Table
+2 (middle). In general, all three models are more robust
+than RobArch-S. But in terms of accuracy, increasing depth
+(M2) > increasing width (M3) > increasing kernel size
+(M1). Therefore, we set M2 as RobArch-M.
+For the large-sized budget (∼104M), model L1 increases
+the kernel size from 3 to 7, but leads to a drop in robustness,
+as shown in Table 2 (bottom). RobArch-M increases the
+depth of S7, and according to the depth-width trade-off in
+Fig. 2c, consistently increasing the depth can lead to un-
+stable training. Therefore, model L2 increases the width
+in RobArch-M, and the robustness rises by a large margin.
+7
+
+Table 3. Our RobArch model outperforms ConvNets and Trans-
+formers with similar total parameters against ℓ∞ = 4/255 AA.
+Using the same training configurations as Salman et al. [41], our
+model outperforms both ResNet-50 and WideResNet50-2. Every
+RobArch model outperforms its XCiT counterpart at a similar ca-
+pacity. Appendix F.2 shows the detailed results including PGD100
+for ϵ ∈ {2, 4, 8}.
+Architecture
+#Param
+AutoAttack
+Natural
+ResNet-18 [41]
+12M
+25.32%
+52.49%
+PoolFormer-M12 [11]
+22M
+34.72%
+66.16%
+DeiT-S [3]
+22M
+35.50%
+66.50%
+DeiT-S+DiffPure [39]
+22M
+43.18%
+73.63%
+ResNet-50 [41]
+26M
+34.96%
+63.87%
+ResNet-50+DiffPure [39]
+26M
+40.93%
+67.79%
+ResNet-50+GELU [3]
+26M
+35.51%
+67.38%
+XCiT-S12 [11]
+26M
+41.78%
+72.34%
+RobArch-S
+26M
+44.14%
+70.17%
+XCiT-M12 [11]
+46M
+45.24%
+74.04%
+RobArch-M
+46M
+46.26%
+71.88%
+WideResNet50-2 [41]
+69M
+38.14%
+68.41%
+WideResNet50-2
+69M
+44.39%
+71.16%
++DiffPure [39]
+Swin-B [38]
+88M
+38.61%
+74.36%
+XCiT-L12 [11]
+104M
+47.60%
+73.76%
+RobArch-L
+104M
+48.94%
+73.44%
+We further deepen L2 to explore whether guideline 5 holds
+true when scaling up the model budget. L3 and L4 increase
+the depth by 1.6× and 2× and reduce the width to fit the
+total parameters. The results in Table 2 (bottom) show a
+decline in accuracy along with an increase in depth. The
+phenomenon extends guideline 5 that the depth-width rela-
+tionship also applies to scaling up the models. Finally, we
+set L2 as RobArch-L based on the above discussions, and
+provide the following guidelines:
+Guideline 17: When scaling up the model, increasing
+the kernel size, depth, and width all contribute to the ro-
+bustness. But proportionally increasing the optimal depth
+configuration is most effective.
+Guideline 18: There exists a saturation point for purely
+increasing the depth to fill the parameter budget. We should
+enlarge channel widths when such a degradation happens.
+3.3. Results
+In Fig. 3, we compare RobArch with a series of SOTA
+architectures.
+All architectures are trained with Fast-AT
+for a fair comparison, and we discover a similar trend for
+PGD10-2, PGD10-4, and PGD10-8. Below we provide a
+few observations based on PGD10-4 accuracy:
+1) Under a similar model capacity, RobArch-S outper-
+forms ResNet-50 [19] and ResNeXt-50 32×4d [61] by 9.45
+and 6.80 percentage points, respectively. RobArch-M out-
+performs ResNet-101 by 8.16 percentage points.
+2) Compared to the models with larger parameters,
+RobArch-S is even more robust than WideResNet101-2 de-
+spite having 4.85× fewer parameters.
+3) Increasing the total parameters in general leads to higher
+robustness, and the natural accuracy is positively correlated
+with the adversarial accuracy after AT. Lightweight mod-
+els, e.g., MobileNet V2 and SqueezeNet-1.1, are among the
+least robust. The accuracies of RobArchs consistently grow
+when scaling up the model sizes.
+4) Transformers, e.g., Swin-T [33], and Transformer-based
+architectures, e.g., ConvNeXt-T [34], are non-robust us-
+ing Fast-AT.
+The phenomenon can be attributed to the
+differences in optimizers and learning rates, where most
+Transformer-related architectures use AdamW [35] and tiny
+learning rates.
+As introduced in Sec. 2.1, we then train all RobArchs
+using Standard-AT. All three RobArchs outperform their
+XCiT [1] counterparts (Table 3).
+Using the same train-
+ing configurations as Salman et al. [41], RobArch-S sur-
+passes ResNet-50 AA accuracy by 9.18 percentage points,
+and is even more robust than WideResNet50-2 with 2.6×
+fewer parameters.
+The robustness continues to improve
+when scaling up the model, and RobArch-L achieves the
+new SOTA AA [8] accuracy on RobustBench. RobArch’s
+performance advantage also extrapolates to the PGD attack.
+Overall, the proposed RobArchs outperform both ConvNets
+and Transformers with similar total parameters.
+4. Related Work
+A huge number of AT variants have been proposed,
+e.g., TRADES [64], AWP [56], ADT [15], DART [52],
+MART [53], CAS [4], Max-Margin AT [14], etc. For the ro-
+bust DNN research, only a few studies explored how archi-
+tectures affect robustness [13, 36, 46, 48], e.g., depths [60],
+widths [55] and activation functions [9, 57]. However, the
+total model capacity is unconstrained along with the archi-
+tecture modifications. Besides, simply combining multiple
+individual optimal architectures does not transfer to a bet-
+ter model, e.g., Huang et al. [26] studied depths and widths,
+and found the combination of the optimal depth and width
+ratios is less robust than just using the optimal width ratio.
+5. Conclusion
+In this work, we present the first large-scale system-
+atic study on the robustness of architecture components un-
+der fixed parameter budgets.
+Through our investigation,
+we distill 18 actionable robust network design guidelines
+that empower model developers to gain deep insights. Our
+RobArch models instantiate the guidelines to build a fam-
+ily of top-performing models across parameter capacities
+against strong adversarial attacks.
+8
+
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+2019.
+11
+
+A. Network Configurations
+A.1. Overview of ResNet-style ConvNets
+A standard ResNet-style ConvNet includes a stem stage,
+several body stages, and a classification head, as shown in
+Fig. 4. A typical body stage consists of multiple resid-
+ual blocks, where each of them has a shortcut connection
+that skips other layers and feeds the output of the previous
+layer to the current output of the block [19]. The stem stage
+proceeds the input image through a convolution layer and a
+max-pooling that downsample the resolution by 4 in total.
+The final classification head passes the extracted features
+from body stages through an average pooling and a linear
+layer that outputs the predictions. Table 4 lists ResNet-50
+configurations written in notations defined in the paper.
+A.2. RobArch Architecture
+The RobArch follows the ResNet-style ConvNet design.
+We display block designs for ResNet-50 and RobArch-S in
+Fig. 5. Following the RegNet design [40], we add the SE
+block after the 3 × 3 convolution layer in each block. The
+SE reduction ratio is 0.25.
+Stage 1
+Block 1
+Block 3
+Block 2
+Stage 2
+Input (224x224x3)
+7x7, 64, /2
+3x3 Max-Pool, /2
+Avg-Pool
+Linear
+Predictions
+Stage 3
+Stage 4
+ResNet-style ConvNets
+Body Stages
+Stem 
+Stage
+Body
+Stages
+Classification
+Head
+Block 1
+Block 3
+Block 2
+...
+Figure 4. An overview of ResNet-style ConvNet design, which in-
+cludes a stem stage, several body stages, and a classification head.
+B. Experimental Settings
+We use Fast-AT as a rapid indicator while exploring dif-
+ferent architecture components and building the RobArch
+Table 4. ResNet-50 configurations written in notations defined in
+the paper. The left column lists architecture components, and the
+right column shows notations. ResNet-50 does not have SE block,
+so the configuration is “N/A”. For activation, ReLU-ReLU-ReLU
+represents the three activation layers in a residual block. The same
+also applies to normalization.
+Notation
+Depth
+D-3-4-6-3
+Width
+W-256-512-1024-2048
+G-1-1-1-1
+BM-0.25-0.25-0.25-0.25
+Stem stage
+Stem width 64
+Stem kernel 7
+Downsample factor 4
+Dense connection
+Dense ratio 1
+Kernel size
+Kernel size 3
+Dilation
+Dilation 1
+Activation
+Act. ReLU
+ReLU-ReLU-ReLU
+SE
+N/A
+Normalization
+Norm. BN
+BN-BN-BN
+ResNet Block
+RobArch Block
+1x1, 64
+3x3, 64
+1x1, 256
+BN, ReLU
+BN, ReLU
+ReLu
+256
+BN
+1x1, 288
+BN
+1x1, 72
+3x3, 72, g = 2
+1x1, 72
+Scale
+SiLU
+BN, SiLU
+SiLU
+ReLU
+288
+Sigmoid
+Global pooling
+1x1, 72
+Figure 5. Block designs for a ResNet and a RobArch. For simplic-
+ity, “1 × 1, 64” means pointwise convolution with 64-layer output
+channels. “g = 2” means 2 group convolution groups, and the
+default group is 1.
+family. We follow the same 3-phase training as proposed in
+the Fast-AT paper [54]. Fast-AT sets training ϵ = 1.25×
+test ϵ and finds catastrophic overfitting happens when train-
+ing ϵ goes beyond 10.
+Therefore, we set training ϵ ∈
+{2.5, 5.0, 7.5} corresponding to test ϵ ∈ {2, 4, 6} and show
+12
+
+Table 5. Determine training ϵ for Fast-AT using ResNet-50. Train-
+ing ϵ = 2.5 shows the highest natural and PGD10-2 accuracies,
+while training ϵ = 7.5 shows the highest PGD10-8 accuracy.
+Overall, training ϵ = 5.0 is selected for all Fast-AT experiments
+since it exhibits a balanced performance on all natural and attack
+budgets.
+Training ϵ
+Natural
+PGD10-2
+PGD10-4
+PGD10-8
+2.5
+60.04%
+43.06%
+25.34%
+6.49%
+5.0
+56.09%
+42.66%
+30.43%
+12.61%
+7.5
+49.80%
+36.86%
+26.95%
+13.87%
+the results in Table 5. Larger training ϵ exhibits higher ro-
+bustness against strong attacks at the cost of lowering the
+accuracies of natural and weak attacks. We select training
+ϵ = 5.0 for its balanced performance on natural and various
+attack budgets. We use Standard-AT to robustify all mem-
+bers in the RobArch family, and follow the same training
+configurations as Salman et al. [41].
+Our RobArchs are evaluated against the two strongest
+adversarial attacks, PGD [36] and AA [8]. All PGD attacks
+are tested on the full ImageNet validation set. AA is an en-
+semble of four different parameter-free attacks, three white-
+and one black-box. We use the same 5000 ImageNet valida-
+tion subset provided by the RobustBench [7] for AA com-
+parison.
+C. Ablations on Adversarial Training
+We ablate Fast-AT and Standard-AT for two purposes:
+1) verify the robustness order is consistent under two dif-
+ferent AT methods, 2) compute whether the two approaches
+exhibit comparable robustness increases when subjected to
+the same ablation.
+Since Standard-AT incurs longer training time, we ran-
+domly select one small budget model S4 and show the re-
+sults in Table 6. For natural, PGD10-4 and AA runs, S4
+outperforms ResNet-50 but is inferior to RobArch-S, which
+demonstrates the robustness order is consistent under Fast-
+AT and Standard-AT.
+Then, we compute the robustness
+gain using PGD10-4 as an example. From ResNet-50, S4 to
+RobArch-S, accuracy increases by 6.18 and 3.27 percentage
+points under Fast-AT, and increases by 6.01 and 2.52 per-
+centage points under Standard-AT. Both training methods
+show comparable robustness increases on the same archi-
+tecture against the same attack. The observation also extrap-
+olates to natural and AA accuracies. As expected, Standard-
+AT displays higher robustness than Fast-AT. Hence, we
+conclude that Fast-AT serves as a good indicator when ex-
+ploring different architecture components and building the
+RobArch family. Standard-AT can fully robustify all mem-
+bers in the RobArch family after finalizing the architectures.
+Table 6. Ablations on Fast-AT and Standard-AT. We randomly
+select one small budget model S4 from the roadmap and train it
+with both methods. The results show that the robustness order
+is consistent under two different AT methods, and the scales of
+robustness increment are also comparable.
+Model
+Fast-AT
+Standard-AT
+Natural PGD10-4
+Natural PGD10-4
+AA
+S0 (ResNet-50)
+56.09%
+30.43% 63.87%
+39.66% 34.96%
+S4
+60.57%
+36.61% 68.88%
+45.67% 41.44%
+S7 (RobArch-S) 62.27%
+39.88% 70.17%
+48.19% 44.14%
+Table 7. Our RobArch model family outperforms SOTA archi-
+tectures under the same Fast-AT training method. The results are
+consistent across natural and different attack budgets. We high-
+light all three RobArchs for easy comparisons.
+Architecture
+#Param Natural PGD10-2 PGD10-4 PGD10-8
+SqueezeNet 1.1
+1 M
+0.10 %
+0.10 %
+0.10 %
+0.10 %
+MobileNet V2
+4 M 41.60%
+31.23%
+21.89%
+8.94 %
+EfficientNet-B0
+5 M 48.78%
+37.74%
+26.90%
+10.92%
+ShuffleNet V2 2.0× 7 M 49.99%
+0.01 %
+0.01 %
+0.02 %
+DenseNet-121
+8 M 52.29%
+40.06%
+28.72%
+12.23%
+ResNet-18
+12 M 46.59%
+35.05%
+24.64%
+9.95 %
+RegNetX-3.2GF
+15 M 57.26%
+45.74%
+33.85%
+15.37%
+RegNetY-3.2GF
+19 M 59.15%
+47.09%
+34.82%
+15.51%
+EfficientNetV2-S 21 M 57.64%
+45.89%
+33.48%
+14.03%
+ResNeXt-50
+25M 57.33%
+45.46%
+33.08%
+14.45%
+32×4d
+ResNet-50
+26 M 56.09%
+42.66%
+30.43%
+12.61%
+RobArch-S
+26 M 62.27%
+51.67%
+39.88%
+18.99%
+Swin-T
+28 M 38.83%
+28.08%
+18.49%
+6.20 %
+ConvNeXt-T
+29 M 21.35%
+15.39%
+10.51%
+4.07 %
+DenseNet-161
+29 M 59.80%
+47.60%
+35.35%
+15.77%
+EfficientNet-B5
+30 M 55.90%
+44.80%
+33.26%
+14.53%
+RegNetY-8GF
+39 M 63.61%
+52.26%
+40.15%
+19.21%
+RegNetX-8GF
+40 M 60.26%
+48.98%
+36.89%
+17.22%
+ResNet-101
+45 M 58.04%
+45.72%
+33.90%
+15.93%
+RobArch-M
+46 M 64.40%
+53.97%
+42.06%
+20.98%
+ResNet-152
+60 M 61.55%
+48.50%
+35.85%
+15.87%
+WideResNet50-2
+69 M 60.66%
+46.99%
+34.10%
+15.37%
+RobArch-L
+104M 66.08%
+55.52%
+43.81%
+22.50%
+WideResNet101-2127M 61.63%
+49.10%
+36.23%
+16.14%
+D. Robust Architecture Design Results
+This section presents the detailed results for all architec-
+ture components, using five tables. In each table, we use a
+bold font to highlight the results that have been presented
+in the paper, and in the caption, we describe the additional
+information that we are introducing here. Table 8 is depth-
+only, Table 13 is width-only, Table 14 is depth-width com-
+bination, Table 9 includes all stage-level designs, and Table
+10 includes all block-level designs. For each component, its
+table includes architecture configurations, total parameters,
+13
+
+Table 8. PGD10 robustness of depth. Bold font means the re-
+sults have been presented in the paper.
+All configurations are
+trained with Fast-AT and evaluated on full ImageNet validation
+set. ResNet-50 serves as the baseline. We presented D-5-8-13-1
+in the main paper, and provide results for all 3-, 4-, 5- and 6-stage
+networks here.
+Config
+#Param Natural PGD10-2 PGD10-4 PGD10-8
+ResNet-50
+25.56M 56.09%
+42.66%
+30.43%
+12.61%
+3-stage Network
+D-16-16-16
+25.02M 57.15%
+41.35%
+29.57%
+14.37%
+D-10-18-16
+25.15M 56.47%
+43.32%
+31.52%
+14.57%
+D-3-22-16
+25.78M 56.77%
+44.99%
+33.24%
+15.14%
+D-16-25-14
+25.30M 56.69%
+44.53%
+32.63%
+14.39%
+D-2-16-18
+26.26M 57.31%
+44.97%
+32.72%
+14.00%
+D-3-29-14
+25.51M 57.27%
+44.74%
+33.02%
+14.88%
+D-3-4-20
+25.21M 57.69%
+45.27%
+32.95%
+14.46%
+D-8-2-20
+25.00M 57.12%
+44.32%
+31.90%
+13.50%
+4-stage Network
+D-1-5-6-3
+25.70M 55.98%
+43.54%
+31.46%
+13.54%
+D-5-2-6-3
+25.14M 52.93%
+40.41%
+29.14%
+12.60%
+D-1-4-7-3
+26.53M 56.60%
+43.62%
+31.51%
+13.76%
+D-6-4-4-3
+23.53M 54.19%
+42.11%
+30.40%
+13.30%
+D-3-5-2-4
+25.83M 53.98%
+41.44%
+30.08%
+13.13%
+D-4-3-10-2
+25.35M 55.62%
+43.15%
+31.32%
+14.03%
+D-2-7-13-1
+25.22M 57.19%
+44.16%
+31.91%
+13.89%
+D-2-9-13-1
+25.78M 57.89%
+45.08%
+32.84%
+14.63%
+D-2-13-8-2
+25.78M 55.86%
+42.91%
+30.96%
+13.40%
+D-1-1-15-1
+25.71M 55.74%
+43.41%
+31.45%
+13.51%
+D-2-5-14-1
+25.78M 56.49%
+44.13%
+32.58%
+14.73%
+D-5-8-13-1
+25.71M 57.35%
+44.83%
+33.33%
+15.46%
+D-2-12-12-1
+25.51M 55.89%
+43.39%
+31.45%
+13.56%
+D-4-8-1-4
+25.62M 54.84%
+42.44%
+30.23%
+12.86%
+D-1-4-2-4
+25.41M 52.46%
+40.25%
+28.80%
+12.22%
+D-2-1-3-4
+25.76M 53.23%
+41.50%
+29.76%
+12.51%
+D-3-24-5-2
+25.58M 57.41%
+44.66%
+32.65%
+14.42%
+D-2-8-5-3
+25.49M 56.43%
+43.65%
+31.70%
+13.62%
+D-6-4-2-4
+25.76M 53.48%
+42.04%
+31.07%
+13.70%
+D-10-6-5-3
+25.49M 57.17%
+43.65%
+31.45%
+13.25%
+D-10-2-2-4
+25.48M 53.03%
+41.01%
+30.32%
+13.45%
+D-1-2-3-4
+25.97M 53.68%
+41.05%
+29.21%
+11.92%
+5-stage Network
+D-1-1-3-1-2
+25.42M 48.85%
+36.89%
+25.98%
+10.37%
+D-1-1-3-2-1
+25.42M 50.14%
+37.33%
+26.11%
+10.35%
+D-3-6-2-2-1
+25.85M 51.64%
+39.12%
+28.23%
+12.24%
+D-2-3-7-1-1
+26.06M 52.16%
+39.79%
+28.40%
+11.72%
+D-3-4-6-2-1
+29.76M 53.67%
+41.25%
+29.88%
+12.65%
+6-stage Network
+D-1-1-1-1-1-1 27.39M 40.82%
+29.46%
+20.00%
+7.52%
+natural, PGD10-2, PGD10-4, and PGD10-8 accuracies.
+Table 9. PGD10 robustness of all stage-level designs. Bold font
+means the results have been presented in the paper. All configu-
+rations are trained with Fast-AT and evaluated on full ImageNet
+validation set. ResNet-50 serves as the baseline. We presented
+“Stem width 96” and “Move down (↓) downsampling” for the stem
+stage, and “Dense ratio 2” for the dense connection in the main pa-
+per. We complete the results by providing all other configurations,
+and PGD attack budgets here.
+Config
+#Param Natural PGD10-2 PGD10-4 PGD10-8
+ResNet-50
+25.56M 56.09%
+42.66%
+30.43%
+12.61%
+Stem Stage
+Stem width 32
+25.54M 55.89%
+41.64%
+29.73%
+13.25%
+Stem width 96
+25.57M 57.29%
+44.55%
+32.06%
+13.74%
+Stem kernel 3
+25.55M 38.93%
+0.46%
+0.55%
+0.30%
+Stem kernel 5
+25.55M 59.59%
+0.38%
+0.09%
+0.04%
+Stem kernel 9
+25.56M 55.75%
+43.00%
+31.19%
+13.63%
+Move down (↓) 25.56M 57.08%
+45.19%
+33.08%
+14.50%
+downsampling
+Downsample
+25.56M 56.03%
+44.48%
+32.86%
+14.71%
+factor 2
+“Patchify 4”
+25.55M 55.40%
+43.45%
+31.68%
+13.80%
+“Patchify 2”
+25.55M 56.38%
+44.21%
+31.91%
+13.48%
+Dense Connection
+Dense ratio 2
+25.56M 55.93%
+42.85%
+30.73%
+12.67%
+Dense ratio 3
+25.56M 53.45%
+40.70%
+29.39%
+12.84%
+Dense ratio 4
+25.56M 55.02%
+42.44%
+30.52%
+12.98%
+Dense ratio 5
+25.56M 54.45%
+41.96%
+30.07%
+12.49%
+Dense ratio 5
+25.56M 49.68%
+37.32%
+26.15%
+10.28%
+ReLU-ReLU-0
+E. Roadmap Results
+This section presents detailed results for the roadmap
+we take to construct the RobArch family using Table 11.
+We demonstrate each architecture component in the cumu-
+lative RobArch construction process improves natural and
+PGD10-4 in the main paper. In Table 11, we show that the
+accuracy gain is also consistent on PGD10-2 and PGD10-8.
+F. SOTA Architecture Comparisons
+F.1. Fast-AT Comparisons
+This section presents the detailed results of RobArch and
+other SOTA architectures after Fast-AT using Table 7. With
+a similar model capacity, RobArch-S outperforms ResNet-
+50 and ResNeXt-50 43×4d, and RobArch-M outperforms
+ResNet-101. Compared to models with larger parameters,
+RobArch-S is even more robust than WideResNet101-2 de-
+spite having 4.85× fewer parameters. The accuracy con-
+tinues to increase while scaling up the RobArch models,
+with RobArch-L achieving the highest natural and adver-
+sarial accuracies.
+14
+
+Table 10. PGD10 robustness of all block-level designs. Bold font
+means the results have been presented in the paper. All configu-
+rations are trained with Fast-AT and evaluated on full ImageNet
+validation set. ResNet-50 serves as the baseline. Bold means re-
+sults have already appeared in the main paper. We complete the
+results by providing all other configurations and PGD attack bud-
+gets here. For activation, 0-0-ReLU means only the last activation
+layer is preserved in a block and the first two are discarded. The
+same also applies to normalization.
+Config
+#Param Natural PGD10-2 PGD10-4 PGD10-8
+ResNet-50
+25.56M 56.09%
+42.66%
+30.43%
+12.61%
+Kernel Size
+Kernel size 5
+45.68M 56.73%
+44.55%
+32.77%
+14.62%
+Kernel size 7
+75.86M 59.70%
+47.28%
+34.67%
+14.99%
+Dilation
+Dilation 2
+25.56M 52.98%
+40.38%
+28.38%
+11.79%
+Dilation 3
+25.56M 52.10%
+39.69%
+27.97%
+11.05%
+Activation
+Act. GELU
+25.56M 57.48%
+45.05%
+33.12%
+14.80%
+Act. SiLU
+25.56M 58.19%
+46.21%
+34.07%
+14.68%
+Act. PReLU
+25.56M 55.81%
+42.52%
+30.38%
+12.76%
+Act. PSiLU
+25.56M 56.38%
+44.90%
+33.76%
+15.40%
+Act. PSSiLU
+25.56M 57.43%
+44.44%
+32.22%
+13.71%
+ReLU-ReLU-0 25.56M 51.54%
+38.69%
+27.05%
+10.94%
+ReLU-0-ReLU 25.56M 53.91%
+41.22%
+29.62%
+12.30%
+0-ReLU-ReLU 25.56M 54.81%
+42.10%
+30.34%
+12.86%
+0-0-ReLU
+25.56M 51.03%
+39.12%
+28.15%
+12.09%
+0-ReLU-0
+25.56M 47.18%
+34.85%
+24.12%
+9.51%
+ReLU-0-0
+25.56M 44.21%
+32.34%
+22.24%
+8.77%
+Squeeze and Excitation (SE)
+SE (ReLU)
+27.73M 57.83%
+45.09%
+32.64%
+14.01%
+SE (SiLU)
+27.73M 58.49%
+45.79%
+33.63%
+14.51%
+SE (GELU)
+27.73M 58.27%
+45.66%
+33.55%
+14.56%
+SE (PSiLU)
+27.73M 56.98%
+44.19%
+32.19%
+13.68%
+SE (PSSiLU)
+27.73M 57.55%
+45.27%
+33.33%
+14.73%
+Normalization
+Norm. IN
+25.51M 17.15%
+12.49%
+8.54%
+3.55%
+BN-BN-0
+25.53M 54.15%
+41.12%
+29.59%
+12.36%
+BN-0-BN
+25.55M 56.04%
+43.29%
+31.34%
+13.37%
+0-BN-BN
+25.55M 56.18%
+43.64%
+31.61%
+13.47%
+0-0-BN
+25.54M 54.47%
+41.91%
+30.13%
+12.65%
+0-BN-0
+25.52M 54.55%
+41.94%
+30.06%
+12.62%
+BN-0-0
+25.52M 54.44%
+41.47%
+29.72%
+12.50%
+F.2. Standard-AT Comparisons
+This section compares our RobArchs with other SOTA
+models against both PGD and AA in Table 12.
+For
+AA, all three RobArchs outperform their XCiT counter-
+parts. Using the same training configurations as Salman
+et al. [41], RobArch-S surpasses ResNet-50 AA accuracy
+by 9.18 percentage points, and is even more robust than
+WideResNet50-2 with 2.6× fewer parameters. The robust-
+ness continues to scale with model capacity, and RobArch-
+L achieves the new SOTA AA accuracy on the Robust-
+Bench leaderboard.
+It is important to note that ResNet-
+50+DiffPure [39] designed a novel AT method via using
+diffusion models [22] for adversarial purification. Although
+the method improves the AA accuracy by 5.97 percent-
+age points, our architecture modifications show stronger ro-
+bustness even without finetuning the Standard-AT method.
+We believe a carefully designed training recipe can further
+improve RobArchs’ robustness. For PGD, the RobArch-
+S again outperforms ResNet-50 and even WideResNet50-
+2 using the same Standard-AT configurations.
+Overall,
+our RobArchs outperform both ConvNets and Transformers
+with similar total parameters.
+15
+
+Table 11. The roadmap outlines the path we take to cumulatively improve the robustness and construct RobArch-S (∼26M), RobArch-M
+(∼46M), and RobArch-L (∼104M) based on our guidelines. Natural and PGD10-4 accuracies were already shown in the main paper.
+PGD10-2 and PGD10-8 show similar trends of accuracy improvement as PGD10-4.
+Configurations
+#Param
+Natural
+PGD10-2
+PGD10-4
+PGD10-8
+Small: ResNet-50 → RobArch-S (S7)
+S0
+ResNet-50
+25.71M
+56.09%
+42.66%
+30.43%
+12.61%
+S1
+S0 + D-5-8-13-1
+25.56M
+57.35%
+44.83%
+33.33%
+15.46%
+S2a
+S1 + g = 2, e = 2, b = 0.25
+25.84M
+57.98%
+46.00%
+33.94%
+15.27%
+S2b
+S1 + g = 1, e = 1.5, b = 0.25
+25.53M
+57.52%
+44.60%
+32.83%
+14.23%
+S3
+S2a + Stem width 96 + Move down (↓) downsampling
+25.85M
+57.82%
+46.37%
+34.86%
+15.92%
+S4
+S3 + SE (ReLU)
+26.15M
+60.57%
+49.05%
+36.61%
+16.43%
+S5
+S4 + Act. SiLU
+26.15M
+62.04%
+51.41%
+39.48%
+18.95%
+S6
+S5 + SE (SiLU)
+26.15M
+60.32%
+49.74%
+38.24%
+18.18%
+S7
+S5 + Norm-0-BN-BN
+26.14M
+62.27%
+51.67%
+39.88%
+18.99%
+Medium: RobArch-S (S7) → RobArch-M (M2)
+M1
+S7 + Kernel size 5
+45.95M
+63.82%
+52.89%
+41.00%
+19.90%
+M2
+S7 + D-7-11-18-1
+45.90M
+64.40%
+53.97%
+42.06%
+20.98%
+M3
+S7 + W-384-760-1504-2944
+46.16M
+63.52%
+53.11%
+41.43%
+20.27%
+Large: RobArch-M (M2) → RobArch-L (L2)
+L1
+M2 + Kernel size 7
+103.89M
+64.08%
+52.92%
+40.70%
+19.61%
+L2
+M2 + W-512-1024-2016-4032
+104.07M
+66.08%
+55.52%
+43.81%
+22.50%
+L3
+M2 + D-8-13-21-2
+104.13M
+64.91%
+54.64%
+43.09%
+21.81%
+L4
+M2 + D-10-16-26-2
+104.14M
+65.28%
+54.49%
+42.85%
+21.42%
+Table 12. Our RobArch model outperforms ConvNets and Transformers with similar total parameters against ℓ∞ = 4/255 AA and
+ℓ∞ = 2/255, 4/255, 8/255 PGD attacks. Using the same training configurations as Salman et al. [41], our model outperforms both
+ResNet-50 and WideResNet50-2. Every RobArch model outperforms its XCiT counterpart at a similar capacity.
+Architecture
+#Param
+Natural
+AA
+PGD10-4
+PGD50-4
+PGD100-4
+PGD100-2
+PGD100-8
+ResNet-18 [41]
+12M
+52.49%
+25.32%
+30.06%
+29.61%
+29.61%
+40.98%
+11.57%
+RobNet-large [18]
+13M
+61.26%
+-
+37.16%
+37.15%
+37.14%
+-
+-
+PoolFormer-M12 [11]
+22M
+66.16%
+34.72%
+-
+-
+-
+-
+-
+DeiT-S [3]
+22M
+66.50%
+35.50%
+41.03%
+40.34%
+40.32%
+-
+-
+DeiT-S+DiffPure [39]
+22M
+73.63%
+43.18%
+-
+-
+-
+-
+-
+ResNet-50 [41]
+26M
+63.87%
+34.96%
+39.66%
+38.98%
+38.96%
+52.15%
+15.83%
+ResNet-50+DiffPure [39]
+26M
+67.79%
+40.93%
+-
+-
+-
+-
+-
+ResNet50+SiLU [57]
+26M
+69.70%
+-
+43.00%
+41.90%
+-
+-
+-
+ResNet50+GELU [3]
+26M
+67.38%
+35.51%
+40.98%
+40.28%
+40.27%
+-
+-
+ResNet-50-R [26]
+26M
+56.63%
+-
+-
+31.14%
+-
+-
+-
+XCiT-S12 [11]
+26M
+72.34%
+41.78%
+-
+-
+-
+-
+-
+RobArch-S
+26M
+70.17%
+44.14%
+48.19%
+47.78%
+47.77%
+60.06%
+21.77%
+XCiT-M12 [11]
+46M
+74.04%
+45.24%
+-
+-
+-
+-
+-
+RobArch-M
+46M
+71.88%
+46.26%
+49.84%
+49.32%
+49.30%
+61.89%
+23.01%
+WideResNet50-2 [41]
+69M
+68.41%
+38.14%
+42.51%
+41.33%
+41.24%
+55.86%
+16.29%
+WideResNet50-2+DiffPure [39]
+69M
+71.16%
+44.39%
+-
+-
+-
+-
+-
+Swin-B [38]
+88M
+74.36%
+38.61%
+-
+-
+-
+-
+-
+XCiT-L12 [11]
+104M
+73.76%
+47.60%
+-
+-
+-
+-
+-
+RobArch-L
+104M
+73.44%
+48.94%
+51.72%
+51.04%
+51.03%
+63.49%
+25.31%
+16
+
+Table 13. PGD10 robustness of width. Bold font means the results have been presented in the paper. All configurations are trained with Fast-
+AT and evaluated on full ImageNet validation set. ResNet-50 serves as the baseline. In the main paper, we presented BM-0.5-0.5-0.5-0.5
+and BM-0.5-0.5-0.25-0.25 for bottleneck multiplier, G-2-2-2-2 for group convolution groups, W-512-768-1152-1728 for expansion
+ratio, and the combined model. We complete the results by providing all other configurations, and PGD attack budgets here.
+Channel
+Group
+Bottleneck Multiplier
+#Param
+Natural PGD10-2 PGD10-4 PGD10-8
+ResNet-50
+25.56M
+56.09%
+42.66%
+30.43%
+12.61%
+Bottleneck Multiplier
+W-320-672-1456-3136
+G-1-1-1-1
+BM-0.125-0.125-0.125-0.125
+25.47M 53.47%
+41.42%
+30.11%
+13.40%
+W-128-256-568-1304
+G-1-1-1-1
+BM-0.5-0.5-0.5-0.5
+25.57M 55.31%
+42.48%
+30.52%
+13.23%
+W-64-144-320-720
+G-1-1-1-1
+BM-1-1-1-1
+25.61M 53.07%
+40.93%
+29.54%
+12.70%
+W-32-72-168-384
+G-1-1-1-1
+BM-2-2-2-2
+25.72M 51.17%
+38.79%
+27.32%
+11.22%
+W-16-32-88-200
+G-1-1-1-1
+BM-4-4-4-4
+26.19M 47.67%
+35.93%
+25.30%
+10.32%
+W-256-512-168-384
+G-1-1-1-1
+BM-0.25-0.25-2-2
+26.42M 52.33%
+39.79%
+28.52%
+12.30%
+W-24-48-1024-2048
+G-1-1-1-1
+BM-4-4-0.25-0.25
+25.20M 55.78%
+43.09%
+30.79%
+12.89%
+W-128-256-1024-2048
+G-1-1-1-1
+BM-0.5-0.5-0.25-0.25
+24.83M 56.11%
+43.38%
+31.26%
+13.47%
+Group Convolution Groups
+W-256-512-1080-2504
+G-2-2-2-2
+BM-0.25-0.25-0.25-0.25
+26.02M 57.31%
+44.25%
+32.09%
+13.91%
+W-288-576-1248-2592
+G-4-4-4-4
+BM-0.25-0.25-0.25-0.25
+25.58M 56.28%
+44.00%
+31.52%
+13.33%
+W-256-512-1280-2816
+G-8-8-8-8
+BM-0.25-0.25-0.25-0.25
+25.81M 56.54%
+42.49%
+30.07%
+12.86%
+W-256-576-1344-2816
+G-16-16-16-16
+BM-0.25-0.25-0.25-0.25
+25.61M 54.83%
+42.92%
+31.03%
+13.28%
+W-304-640-1384-2848
+G-76-160-337-712
+BM-0.25-0.25-0.25-0.25
+25.52M 55.17%
+42.34%
+30.45%
+12.72%
+W-256-512-1040-2112
+G-8-8-1-1
+BM-0.25-0.25-0.25-0.25
+26.13M 55.49%
+42.42%
+30.78%
+12.82%
+W-256-512-1248-2784
+G-1-1-8-8
+BM-0.25-0.25-0.25-0.25
+25.69M 55.94%
+43.28%
+31.15%
+13.92%
+W-256-512-1248-2592
+G-2-2-4-4
+BM-0.25-0.25-0.25-0.25
+25.41M 57.13%
+43.88%
+31.44%
+13.48%
+Channel / Expansion Ratio
+W-1112-1112-1112-1112
+G-1-1-1-1
+BM-0.25-0.25-0.25-0.25
+25.70M 56.77%
+43.18%
+31.08%
+13.68%
+W-512-768-1152-1728
+G-1-1-1-1
+BM-0.25-0.25-0.25-0.25
+25.95M 57.17%
+44.05%
+32.04%
+14.06%
+W-144-360-904-2264
+G-1-1-1-1
+BM-0.25-0.25-0.25-0.25
+26.01M 53.89%
+41.83%
+30.33%
+13.38%
+W-88-264-792-2376
+G-1-1-1-1
+BM-0.25-0.25-0.25-0.25
+25.81M 52.39%
+40.60%
+29.36%
+12.58%
+Combined
+W-512-768-1152-1728
+G-2-2-2-2
+BM-0.5-0.5-0.25-0.25
+24.43M 56.64%
+43.56%
+31.04%
+13.17%
+Table 14. PGD10 robustness of combining depth and width. We use a bold font to highlight results that have been presented in the paper.
+Specifically, the paper uses a scatter plot to visualize how the PGD10-4 accuracy changes as we vary depth and width. Here, we additionally
+show the results for PGD10-2 and PGD10-8. All configurations are trained with Fast-AT and evaluated on full ImageNet validation set.
+Depth
+Width
+#Param Natural PGD10-2 PGD10-4 PGD10-8
+D-1-2-4-1
+W-768-1152-1712-2560
+25.69M 54.28%
+41.16%
+29.10%
+11.83%
+D-2-4-7-1
+W-648-968-1456-2160
+25.55M 57.25%
+43.60%
+31.52%
+13.59%
+D-4-6-10-1
+W-576-848-1280-1904
+25.51M 57.08%
+44.18%
+32.32%
+14.46%
+D-5-8-13-1
+W-512-768-1152-1728
+25.18M 57.24%
+44.69%
+33.05%
+15.36%
+D-8-12-20-2
+W-424-632-944-1416
+25.37M 57.74%
+44.79%
+33.15%
+14.87%
+D-10-16-26-2
+W-376-568-856-1280
+25.56M 61.36%
+44.92%
+27.23%
+5.67%
+D-20-32-52-4
+W-272-416-616-928
+25.52M 55.76%
+43.28%
+31.31%
+13.03%
+17
+

CdFQT4oBgHgl3EQfOTbk/content/tmp_files/2301.13275v1.pdf.txt

@@ -0,0 +1,451 @@
+On selfconsistency in quantum field theory
+K. Scharnhorst†
+Vrije Universiteit Amsterdam, Faculty of Sciences, Department of Physics and
+Astronomy, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
+Abstract
+A bootstrap approach to the effective action in quantum field theory is dis-
+cussed which entails the invariance under quantum fluctuations of the effective
+action describing physical reality (via the S-matrix).
+†E-mail: [email protected],
+ORCID: http://orcid.org/0000-0003-3355-9663
+arXiv:2301.13275v1  [hep-th]  30 Jan 2023
+
+1
+Introduction
+These are times of contemplation and reorientation in quantum field theory. With
+the experimental detection of the Higgs boson in 2012 finally the finishing stone of
+the Standard Model of elementary particle physics [1] surfaced. On the theoretical
+side, the Standard Model is based on the concept of renormalized local quantum field
+theory. The confidence in this concept originally and primarily relies on the extraor-
+dinary success of the centerpiece of the Standard Model, quantum electrodynamics
+(QED), which exhibits an impressive agreement between theory and experiment
+(Cf., e.g., [2], for more comprehensive reviews see [3].). The successful application
+of renormalized local quantum field theory to the other components of the Standard
+Model, the electroweak theory and to quantum chromodynamics (QCD), have fur-
+ther advanced this confidence. On the other hand, many practicing quantum field
+theorists are aware of the many shortcomings and deficiencies of the concept of renor-
+malized local quantum field theory which, by the way, has changed and developed in
+a multifold way in the decades since its birth at the end of the 1940’s. To name a few
+of these issues we mention here the occurrence of ultraviolet (UV) divergencies, the
+cosmological constant problem, hierarchy and naturalness problems, Haag’s theo-
+rem (For an instructive illustration of the views of a number of well-known quantum
+field theorist see, e.g., the conference volume [4].). It should, however, be pointed
+out that in the quantum field theory community there is no unified view which of
+these issues constitute features and which are problematic aspects of renormalized
+local quantum field theory. Correspondingly, opinions on which direction should be
+chosen for the future conceptual and technical development of quantum field theory
+are diverse (For a recent account of the current situation see [5].). While many
+active researchers might favour new ideas which have not been discussed in the past
+a certain fraction of the quantum field theory community might be willing to not
+completely disregard past ideas which have largely been bypassed so far. In the
+present article, it is our intention to bring together a couple of thoughts and ideas
+(supplemented by the corresponding references) that have emerged in the past. We
+hope that the collection of this information in a single place will be beneficial to
+those readers who consider voices from the past as an inspiration for future research
+rather than purely as a matter for historians of science.
+Let us start with pointing out that with reference to the UV divergency prob-
+lem in QED some of the very fathers of this theory have repeatedly expressed their
+dissatisfaction with their own creation up to the end of their lives. So, Richard
+Feynman stated 1965 in his Nobel Prize speech quite frankly : “. . . , I believe there
+is really no satisfactory quantum electrodynamics, but I’m not sure. . . . , I think
+that the renormalization theory is simply a way to sweep the difficulties of the di-
+vergences of electrodynamics under the rug.” ([6], Science p. 707, Phys. Today pp.
+43/44, Prix Nobel p. 189, Nobel Lectures p. 176, Selected Papers p. 30). Now one
+might think that Feynman has later, after the development of the Wilsonian view
+on renormalization in the early 1970’s and the emergence of the effective field theory
+2
+
+concept changed his view. However, this seems not to be the case and one can read
+in Feynman’s popular science book “QED – The Strange Theory of Light and Mat-
+ter” the passage (𝑛 and 𝑗 are the bare counter parts of the physical electron mass 𝑚
+and electron charge 𝑒, respectively.): “The shell game that we play to find 𝑛 and 𝑗 is
+technically called “renormalization.” But no matter how clever the word, it is what
+I would call a dippy process! Having to resort to such hocus-pocus has prevented
+us from proving that the theory of quantum electrodynamics is mathematically self-
+consistent. It’s surprising that the theory still hasn’t been proved self-consistent
+one way or the other by now; I suspect that renormalization is not mathematically
+legitimate. What is certain is that we do not have a good mathematical way to
+describe the theory of quantum electrodynamics: such a bunch of words to describe
+the connection between 𝑛 and 𝑗 and 𝑚 and 𝑒 is not good mathematics.” ([7], 1st ed.
+1985, pp. 128/129, 2nd ed. 2006, p. 127/128). In a similar way Paul Dirac stated
+in a lecture in 1975 (published 1978): “Hence most physicists are very satisfied with
+the situation. They say: “Quantum electrodynamics is a good theory, and we do
+not have to worry about it any more.” I must say that I am very dissatisfied with
+the situation, because this so called “good theory” does involve neglecting infinities
+which appear in its equations, neglecting them in an arbitrary way. This is just not
+sensible mathematics. Sensible mathematics involves neglecting a quantity when it
+turns out to be small – not neglecting it just because it is infinitely great and you do
+not want it.” ([8], p. 36). Few years later, in 1980 (published in 1983), Dirac repeats
+his critical view: “Some new relativistic equations are needed; new kinds of interac-
+tions must be brought into play. When these new equations and new interactions
+are thought out, the problems that are now bewildering to us will get automatically
+explained, and we should no longer have to make use of such illogical processes as
+infinite renormalization. This is quite nonsense physically, and I have always been
+opposed to it. It is just a rule of thumb that gives results. In spite of its successes,
+one should be prepared to abandon it completely and look on all the successes that
+have been obtained by using the usual forms of quantum electrodynamics with the
+infinities removed by artificial processes as just accidents when they give the right
+answers, in the same way as the successes of the Bohr theory are considered merely
+as accidents when they turn out to be correct.” ([9], p. 55). The weight one might
+be tempted to assign to these views certainly will depend on the scientific taste of
+each theoretician, however, at least one should take note of them.
+It often happens in the course of the development of science that early consid-
+erations and ideas are more fundamental than those emerging later. This is easily
+explainable by the fact that at the early stages of the development of a subject
+one enters largely unchartered territory and simple and structural ideas are then
+needed to choose the right road to scientific progress. Sometimes conflicting ideas
+are competing with each other. Initial dominance of one idea does not necessarily
+mean that less successful concepts should be written off. It happens from time to
+time that these disregarded concepts make a surprising return for one reason or
+the other. Consequently, a look into the past (science history) may be helpful for
+3
+
+shaping the future. For the following considerations, we will depart from such an
+element of science history.
+2
+Extending early thoughts of Wolfgang Pauli
+Let us start our concrete discussion with a statement made by Wolfgang Pauli in
+a private letter (in German) to Victor Weisskopf (by then, assistant to Wolfgang
+Pauli at the ETH Zurich) in 1935. The comment of Wolfgang Pauli concerns the
+self-energy of the electron in QED, a theory which was under development in those
+days. In the context of the struggle with the UV divergencies of QED Wolfgang
+Pauli expresses the following conviction (English translation in brackets: K. S.):
+“. . . (Ich glaube allerdings, daß in einer vernünftigen Theorie die Selbstenergie nicht
+nur endlich, sondern Null sein muß . . . [. . . (However, I believe that in a sensible the-
+ory the self-energy has not only to be finite but zero . . . ]” ([10], p. 779 of: Letter
+[425b] of December 14, 1935 from Pauli to Weisskopf. Part of: Nachtrag zu Band
+I: 1919–1929 und II: 1930–1939, pp. 733-826). How can we understand this expec-
+tation of a future correct quantum electrodynamical theory expressed by Wolfgang
+Pauli? If one starts quantizing the theory (in this case, charged fermions interacting
+with the electromagnetic field) on the basis of a Lagrangian with the physical (i.e.,
+measured) value of the mass of the fermions inserted all physical processes that can
+conceivably have an impact on that mass have effectively been taken into account
+already. Consequently, the total impact of all physical processes taken into account
+in the (theoretical) process of quantization (i.e., taking into account quantum fluc-
+tuations) on the mass of these fermions should vanish (nonrenormalization). This
+statement can be reformulated by saying that the fermion mass should not receive
+any radiative corrections under quantization. One can now extend this early point of
+view of Wolfgang Pauli and consider starting quantization not only with the physical
+value of the fermion mass in the initial Lagrangian but choosing as initial Lagrangian
+(in an arbitrary theory now) the (effective) Lagrangian which describes the physi-
+cal world (with all its – infinitely many – nonlocal and nonpolynomial terms). In
+principle (in theory, not in practice, of course), this can be read off from scattering
+experiments (For the connection between the scattering matrix and the effective
+action see, e.g., [11], sec. 2.4, [12].). If one now starts the process of quantization
+with this “true” Lagrangian all radiative corrections to it should vanish because any
+quantum fluctuations described by this Lagrangian have already been taken into ac-
+count in this Lagrangian. Consequently, the physical (effective) Lagrangian should
+be invariant under the process of quantization, all radiative corrections should van-
+ish. This view of the process of quantization amounts to bootstrapping the effective
+action of a theory. Quantities denoted within standard local renormalizable quan-
+tum field theory as bare and renormalized ones, respectively, then coincide.
+Before continuing our verbal discussion, let us make the above statements more
+4
+
+precise in terms of equations. We consider within a path integral framework La-
+grangian quantum field theory in flat (𝑛-dimensional Minkowski) space-time and
+a (one-component) scalar field theory to pursue the discussion (for the following
+equations cf., e.g., [13], chap. 9). A generalization to more complicated theories
+is straightforward. The generating functional of Green functions of the scalar field
+𝜑(𝑥) is given by the equation
+𝑍[𝐽] = 𝐶
+∫︁
+𝐷𝜑 e 𝑖Γ0[𝜑] + 𝑖
+∫︁
+𝑑𝑛𝑥 𝐽(𝑥)𝜑(𝑥)
+,
+(1)
+where Γ0[𝜑] is the so-called classical action of the theory and 𝐶 some fixed normal-
+ization constant. Then, the generating functional of the connected Green functions
+is
+𝑊[𝐽] = −𝑖 ln 𝑍[𝐽]
+.
+(2)
+The effective action Γ[¯𝜑], which also is the generating functional of the one-particle-
+irreducible (1PI) Green functions, is obtained as the first Legendre transform of
+𝑊[𝐽],
+Γ[¯𝜑] = 𝑊[𝐽] −
+∫︁
+𝑑𝑛𝑥 𝐽(𝑥)¯𝜑(𝑥)
+.
+(3)
+Here
+¯𝜑(𝑥) = 𝛿𝑊[𝐽]
+𝛿𝐽(𝑥)
+(4)
+which in turn leads to
+𝛿Γ[¯𝜑]
+𝛿 ¯𝜑(𝑥) = − 𝐽(𝑥)
+(5)
+in analogy to the classical field equation for Γ0[𝜑]. Equivalently, using the above
+expressions
+e 𝑖Γ[¯𝜑]
+= 𝐶
+∫︁
+𝐷𝜑 e 𝑖Γ0[𝜑 + ¯𝜑] + 𝑖
+∫︁
+𝑑𝑛𝑥 𝐽(𝑥)𝜑(𝑥)
+(6)
+can be considered as the defining relation for the effective action, where the r.h.s.
+of the above equation has to be calculated using a current 𝐽(𝑥), given by Eq. (5),
+which is a functional of ¯𝜑. Eq. (1) defines a map, 𝑔1 : Γ0[𝜑] −→ 𝑍[𝐽], from the class
+of functionals called classical actions to the class of functionals 𝑍. Furthermore,
+we have mappings, 𝑔2 : 𝑍[𝐽] −→ 𝑊[𝐽], (Eq. (2)) and 𝑔3 : 𝑊[𝐽] −→ Γ[¯𝜑] (Eq.
+(3)). These three maps together define a map 𝑔3 ∘ 𝑔2 ∘ 𝑔1 = 𝑓 : Γ0[𝜑] −→ Γ[¯𝜑] (Eq.
+(6)) from the set of so-called classical actions to the set of effective actions. It is
+understood that in order to properly define the map a regularization scheme for the
+scalar field theory is applied. Up to renormalization effects, the classical action Γ0[𝜑]
+determines the effective action Γ[¯𝜑] uniquely via the map 𝑓 which encodes quantum
+5
+
+principles. In this standard scheme, the (quantum) effective action is built directly
+from classical physics and exhibits no independence in its own right.
+The terms of the difference ∆Γ[¯𝜑] = Γ[¯𝜑] − Γ0[¯𝜑] are denoted as ’radiative correc-
+tions’. The above verbal reasoning in generalization of early thoughts of Wolfgang
+Pauli leads to the equation
+∆Γ[¯𝜑] = 0 ,
+(7)
+expressing the vanishing of all radiative corrections, i.e.,
+Γ[¯𝜑] = Γ0[¯𝜑] .
+(8)
+The equation for the complete effective action which is equivalent to the fixed point
+condition for the map 𝑓 reads (𝐶′ is some normalization constant)
+e 𝑖Γ[¯𝜑]
+= 𝐶′
+∫︁
+𝐷𝜑 e 𝑖Γ[𝜑 + ¯𝜑] + 𝑖
+∫︁
+𝑑𝑛𝑥 𝐽(𝑥)𝜑(𝑥)
+,
+(9)
+where
+𝐽(𝑥) = − 𝛿Γ[¯𝜑]
+𝛿 ¯𝜑(𝑥)
+.
+(10)
+The above selfconsistency equation (9) defines the (finite) effective action (including
+its coupling constants and mass ratios) without any reference to classical physics ex-
+clusively via quantum principles encoded in the map 𝑓. The fixed points of the map
+𝑓 then describe physical reality. From out this perspective, the standard formulation
+of quantum field theory represented by eq. (1) can roughly be understood as the first
+step of an iterative solution of the nonlinear functional integro-differential equation
+(9) by applying the map 𝑓 to some initial (in this case ‘classical’) action Γ0[𝜑]. For
+the first time, the above equation (9) to be taken as basis of quantum field theory
+has been proposed in 1972 by L. V. Prokhorov [14]. Not being aware of the earlier
+work by Prokhorov, the same proposal has been made by the present author in 1993
+[15]. In a somewhat different (Hamiltonian) setting (coupled cluster methods), J. S.
+Arponen has expressed similar ideas in 1990 ([16], p. 173, paragraph starting with
+the words: “The possible solution corresponds to a system which suffers no change
+under quantization.”).
+3
+Further discussion
+Given the mature state of standard renormalizable quantum field theory, the above
+point of view (defining the effective action as a fixed point of the map 𝑓) faces
+myriads of objections. Some of them may be misunderstandings, others are com-
+pletely justified concerns, others yet are possibly prejudices. Misunderstandings can
+6
+
+be dealt with most easily – by clarifications. For example, one might ask: Given
+the crucial role of radiative corrections within the standard formulation of quantum
+field theory in correctly describing physical reality (for example, in QED) how could
+one ever possibly think of a theory of physical reality characterized by the vanishing
+of all radiative corrections (in an interacting theory)? The difficulty here, however,
+is just a terminological one. Of course, also a modified formulation of quantum field
+theory needs to deliver those kind of terms in the effective action we denote as ra-
+diative corrections within the established standard approach. While the analytical
+expression yielded from a modified formulation of quantum field theory may differ
+from those within the standard formulation, the numerical results for experimentally
+accessible quantities (e.g., the anomalous magnetic moment of the electron) need to
+be (almost – within experimental limits) the same. The point is that in the modified
+view of quantum field theory represented by eq. (9) those terms denoted in the stan-
+dard formulation as radiative corrections are already incorporated in the action to
+be quantized. But, as the action to be quantized is supposed to be invariant under
+quantization (according to eq. (9)) no new terms may emerge, consequently, in the
+modified formulation of quantum field theory no radiative corrections (relative to
+the initial action to be quantized) occur.
+Certainly, one elementary and justified concern with respect to eq. (9) consists in
+the question of whether eq. (9) allows any non-trivial (i.e., non-Gaussian) solutions
+(Free field theories, of course, always obey eq. (9).). In fact, it has been shown by
+example in a finite-dimensional Grassmann algebra analogue of eq. (9) (i.e., within
+a fermionic zero-dimensional field theory) that eq. (9) has exact non-trivial (i.e.,
+non-Gaussian) solutions [17]. For an (bosonic) example from standard analysis see
+[18]. Of course, as has been pointed out by Prokhorov [14] from the outset eq. (9)
+represents a very complicated equation and presently very little can be said about
+its eventual non-trivial solutions in general. Experience from effective action studies
+in quantum field theory tells us that non-trivial solutions of eq. (9) can be expected
+to be nonlocal and nonpolynomial functionals of fields. Whether these nonlocal and
+nonpolynomial actions Γ solving eq. (9) preserve unitarity and causality can only be
+decided once they are found. However, it has been shown for a wide class of nonlo-
+cal and nonpolynomial (scalar) quantum field theories in the past [19, 20] that they
+respect these two principles – a fact, from which one may derive certain optimism.
+It is of course conceivable that the general version of eq. (9) (written down for an
+arbitrary but fixed collection of fluctuating fields) does not have any non-Gaussian
+solution at all for the certain field content one has chosen. If this was the case the
+existence of a non-Gaussian solution to the generalised version of eq. (9) could be
+applied as a theory selection criterion perhaps in the same way as the (stationary)
+Schrödinger equation selects energy (eigen)values of quantum mechanical systems.
+In a certain sense, at the end of the day only non-Gaussian solutions of eq. (9) are
+physical ones because only they provide the interactions for the structures we ob-
+serve in physical reality. Away from the rigid concept discussed above of the effective
+action as a fixed point of the map 𝑓 non-Gaussian solutions of eq. (9) may be con-
+7
+
+sidered also interesting within the standard lore. While usually perturbation theory
+is built around a Gaussian solution of eq. (9), choosing a non-Gaussian solution of
+eq. (9) as starting point for perturbation theory may also be of some interest. For
+a discussion in this direction see [21].
+From a methodological point of view, the largest difference of the approach rep-
+resented by eq. (9) to the established approach in standard quantum field theory
+consists in the following. Standard local renormalizable quantum field theory starts
+(among other things, e.g., choosing the space-time dimensionality) with the choice
+of the field content of the theory under consideration and the functional form of the
+(classical/bare) action Γ0 (cf. eq. (1)), a quantity which, in principle, is unobservable
+(due to the existence of radiative corrections). This is also true in the different ver-
+sions of the Wilsonian approach to quantum field theory (inspired by the theory of
+phase transitions in statistical mechanics). While in a statistical mechanical system
+(e.g., a spin system modelling a certain microscopic condensed matter structure) the
+structure of the Hamiltonian defined on a lattice with fixed lattice spacing can in
+principle be linked to experimental data, this is not the case within quantum field
+theory where the bare action is not related to observation (For a related discussion
+see, e.g., [22].). Consequently, in the established standard approach to quantum
+field theory the theoretical description of physical reality (i.e., the effective action
+Γ) is inferred from quantities not accessible to experiment in principle. In contrast,
+within the approach represented by eq. (9) only the field content of the quantum
+fluctuations can be chosen, the functional form of the effective action Γ is selfcon-
+sistently restricted by its property to be a solution of this equation. Beyond free
+field theories, i.e., for non-Gaussian solutions of eq. (9), this can be expected to be
+highly restrictive.
+Acknowledgement
+Kind hospitality at the Department of Physics and Astronomy of the Vrije Univer-
+siteit Amsterdam is gratefully acknowledged.
+8
+
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+11
+

CdFQT4oBgHgl3EQfOTbk/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf,len=483
+page_content='On selfconsistency in quantum field theory  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Scharnhorst†  Vrije Universiteit Amsterdam, Faculty of Sciences, Department of Physics and  Astronomy, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands  Abstract  A bootstrap approach to the effective action in quantum field theory is dis-  cussed which entails the invariance under quantum fluctuations of the effective  action describing physical reality (via the S-matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  †E-mail: k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='scharnhorst@vu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='nl,  ORCID: http://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='org/0000-0003-3355-9663  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='13275v1  [hep-th]  30 Jan 2023  1  Introduction  These are times of contemplation and reorientation in quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' With  the experimental detection of the Higgs boson in 2012 finally the finishing stone of  the Standard Model of elementary particle physics [1] surfaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' On the theoretical  side, the Standard Model is based on the concept of renormalized local quantum field  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' The confidence in this concept originally and primarily relies on the extraor-  dinary success of the centerpiece of the Standard Model, quantum electrodynamics  (QED), which exhibits an impressive agreement between theory and experiment  (Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', [2], for more comprehensive reviews see [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' The successful application  of renormalized local quantum field theory to the other components of the Standard  Model, the electroweak theory and to quantum chromodynamics (QCD), have fur-  ther advanced this confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' On the other hand, many practicing quantum field  theorists are aware of the many shortcomings and deficiencies of the concept of renor-  malized local quantum field theory which, by the way, has changed and developed in  a multifold way in the decades since its birth at the end of the 1940’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' To name a few  of these issues we mention here the occurrence of ultraviolet (UV) divergencies, the  cosmological constant problem, hierarchy and naturalness problems, Haag’s theo-  rem (For an instructive illustration of the views of a number of well-known quantum  field theorist see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', the conference volume [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' It should, however, be pointed  out that in the quantum field theory community there is no unified view which of  these issues constitute features and which are problematic aspects of renormalized  local quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Correspondingly, opinions on which direction should be  chosen for the future conceptual and technical development of quantum field theory  are diverse (For a recent account of the current situation see [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' While many  active researchers might favour new ideas which have not been discussed in the past  a certain fraction of the quantum field theory community might be willing to not  completely disregard past ideas which have largely been bypassed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' In the  present article, it is our intention to bring together a couple of thoughts and ideas  (supplemented by the corresponding references) that have emerged in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' We  hope that the collection of this information in a single place will be beneficial to  those readers who consider voices from the past as an inspiration for future research  rather than purely as a matter for historians of science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  Let us start with pointing out that with reference to the UV divergency prob-  lem in QED some of the very fathers of this theory have repeatedly expressed their  dissatisfaction with their own creation up to the end of their lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' So, Richard  Feynman stated 1965 in his Nobel Prize speech quite frankly : “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' , I believe there  is really no satisfactory quantum electrodynamics, but I’m not sure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' , I think  that the renormalization theory is simply a way to sweep the difficulties of the di-  vergences of electrodynamics under the rug.” ([6], Science p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 707, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Today pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  43/44, Prix Nobel p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 189, Nobel Lectures p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 176, Selected Papers p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Now one  might think that Feynman has later, after the development of the Wilsonian view  on renormalization in the early 1970’s and the emergence of the effective field theory  2  concept changed his view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' However, this seems not to be the case and one can read  in Feynman’s popular science book “QED – The Strange Theory of Light and Mat-  ter” the passage (𝑛 and 𝑗 are the bare counter parts of the physical electron mass 𝑚  and electron charge 𝑒, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ): “The shell game that we play to find 𝑛 and 𝑗 is  technically called “renormalization.” But no matter how clever the word, it is what  I would call a dippy process!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Having to resort to such hocus-pocus has prevented  us from proving that the theory of quantum electrodynamics is mathematically self-  consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' It’s surprising that the theory still hasn’t been proved self-consistent  one way or the other by now;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' I suspect that renormalization is not mathematically  legitimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' What is certain is that we do not have a good mathematical way to  describe the theory of quantum electrodynamics: such a bunch of words to describe  the connection between 𝑛 and 𝑗 and 𝑚 and 𝑒 is not good mathematics.” ([7], 1st ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  1985, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 128/129, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 2006, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 127/128).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' In a similar way Paul Dirac stated  in a lecture in 1975 (published 1978): “Hence most physicists are very satisfied with  the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' They say: “Quantum electrodynamics is a good theory, and we do  not have to worry about it any more.” I must say that I am very dissatisfied with  the situation, because this so called “good theory” does involve neglecting infinities  which appear in its equations, neglecting them in an arbitrary way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' This is just not  sensible mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Sensible mathematics involves neglecting a quantity when it  turns out to be small – not neglecting it just because it is infinitely great and you do  not want it.” ([8], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Few years later, in 1980 (published in 1983), Dirac repeats  his critical view: “Some new relativistic equations are needed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' new kinds of interac-  tions must be brought into play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' When these new equations and new interactions  are thought out, the problems that are now bewildering to us will get automatically  explained, and we should no longer have to make use of such illogical processes as  infinite renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' This is quite nonsense physically, and I have always been  opposed to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' It is just a rule of thumb that gives results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' In spite of its successes,  one should be prepared to abandon it completely and look on all the successes that  have been obtained by using the usual forms of quantum electrodynamics with the  infinities removed by artificial processes as just accidents when they give the right  answers, in the same way as the successes of the Bohr theory are considered merely  as accidents when they turn out to be correct.” ([9], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' The weight one might  be tempted to assign to these views certainly will depend on the scientific taste of  each theoretician, however, at least one should take note of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  It often happens in the course of the development of science that early consid-  erations and ideas are more fundamental than those emerging later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' This is easily  explainable by the fact that at the early stages of the development of a subject  one enters largely unchartered territory and simple and structural ideas are then  needed to choose the right road to scientific progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Sometimes conflicting ideas  are competing with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Initial dominance of one idea does not necessarily  mean that less successful concepts should be written off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' It happens from time to  time that these disregarded concepts make a surprising return for one reason or  the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Consequently, a look into the past (science history) may be helpful for  3  shaping the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' For the following considerations, we will depart from such an  element of science history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  2  Extending early thoughts of Wolfgang Pauli  Let us start our concrete discussion with a statement made by Wolfgang Pauli in  a private letter (in German) to Victor Weisskopf (by then, assistant to Wolfgang  Pauli at the ETH Zurich) in 1935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' The comment of Wolfgang Pauli concerns the  self-energy of the electron in QED, a theory which was under development in those  days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' In the context of the struggle with the UV divergencies of QED Wolfgang  Pauli expresses the following conviction (English translation in brackets: K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='):  “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (Ich glaube allerdings, daß in einer vernünftigen Theorie die Selbstenergie nicht  nur endlich, sondern Null sein muß .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (However, I believe that in a sensible the-  ory the self-energy has not only to be finite but zero .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ]” ([10], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 779 of: Letter  [425b] of December 14, 1935 from Pauli to Weisskopf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Part of: Nachtrag zu Band  I: 1919–1929 und II: 1930–1939, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 733-826).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' How can we understand this expec-  tation of a future correct quantum electrodynamical theory expressed by Wolfgang  Pauli?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' If one starts quantizing the theory (in this case, charged fermions interacting  with the electromagnetic field) on the basis of a Lagrangian with the physical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=',  measured) value of the mass of the fermions inserted all physical processes that can  conceivably have an impact on that mass have effectively been taken into account  already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Consequently, the total impact of all physical processes taken into account  in the (theoretical) process of quantization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', taking into account quantum fluc-  tuations) on the mass of these fermions should vanish (nonrenormalization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' This  statement can be reformulated by saying that the fermion mass should not receive  any radiative corrections under quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' One can now extend this early point of  view of Wolfgang Pauli and consider starting quantization not only with the physical  value of the fermion mass in the initial Lagrangian but choosing as initial Lagrangian  (in an arbitrary theory now) the (effective) Lagrangian which describes the physi-  cal world (with all its – infinitely many – nonlocal and nonpolynomial terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' In  principle (in theory, not in practice, of course), this can be read off from scattering  experiments (For the connection between the scattering matrix and the effective  action see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', [11], sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='4, [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' If one now starts the process of quantization  with this “true” Lagrangian all radiative corrections to it should vanish because any  quantum fluctuations described by this Lagrangian have already been taken into ac-  count in this Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Consequently, the physical (effective) Lagrangian should  be invariant under the process of quantization, all radiative corrections should van-  ish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' This view of the process of quantization amounts to bootstrapping the effective  action of a theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Quantities denoted within standard local renormalizable quan-  tum field theory as bare and renormalized ones, respectively, then coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  Before continuing our verbal discussion, let us make the above statements more  4  precise in terms of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' We consider within a path integral framework La-  grangian quantum field theory in flat (𝑛-dimensional Minkowski) space-time and  a (one-component) scalar field theory to pursue the discussion (for the following  equations cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', [13], chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' A generalization to more complicated theories  is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' The generating functional of Green functions of the scalar field  𝜑(𝑥) is given by the equation  𝑍[𝐽] = 𝐶  ∫︁  𝐷𝜑 e 𝑖Γ0[𝜑] + 𝑖  ∫︁  𝑑𝑛𝑥 𝐽(𝑥)𝜑(𝑥)  ,  (1)  where Γ0[𝜑] is the so-called classical action of the theory and 𝐶 some fixed normal-  ization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Then, the generating functional of the connected Green functions  is  𝑊[𝐽] = −𝑖 ln 𝑍[𝐽]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  (2)  The effective action Γ[¯𝜑], which also is the generating functional of the one-particle-  irreducible (1PI) Green functions, is obtained as the first Legendre transform of  𝑊[𝐽],  Γ[¯𝜑] = 𝑊[𝐽] −  ∫︁  𝑑𝑛𝑥 𝐽(𝑥)¯𝜑(𝑥)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  (3)  Here  ¯𝜑(𝑥) = 𝛿𝑊[𝐽]  𝛿𝐽(𝑥)  (4)  which in turn leads to  𝛿Γ[¯𝜑]  𝛿 ¯𝜑(𝑥) = − 𝐽(𝑥)  (5)  in analogy to the classical field equation for Γ0[𝜑].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Equivalently, using the above  expressions  e 𝑖Γ[¯𝜑]  = 𝐶  ∫︁  𝐷𝜑 e 𝑖Γ0[𝜑 + ¯𝜑] + 𝑖  ∫︁  𝑑𝑛𝑥 𝐽(𝑥)𝜑(𝑥)  (6)  can be considered as the defining relation for the effective action, where the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  of the above equation has to be calculated using a current 𝐽(𝑥), given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (5),  which is a functional of ¯𝜑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (1) defines a map, 𝑔1 : Γ0[𝜑] −→ 𝑍[𝐽], from the class  of functionals called classical actions to the class of functionals 𝑍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Furthermore,  we have mappings, 𝑔2 : 𝑍[𝐽] −→ 𝑊[𝐽], (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (2)) and 𝑔3 : 𝑊[𝐽] −→ Γ[¯𝜑] (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' These three maps together define a map 𝑔3 ∘ 𝑔2 ∘ 𝑔1 = 𝑓 : Γ0[𝜑] −→ Γ[¯𝜑] (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  (6)) from the set of so-called classical actions to the set of effective actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' It is  understood that in order to properly define the map a regularization scheme for the  scalar field theory is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Up to renormalization effects, the classical action Γ0[𝜑]  determines the effective action Γ[¯𝜑] uniquely via the map 𝑓 which encodes quantum  5  principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' In this standard scheme, the (quantum) effective action is built directly  from classical physics and exhibits no independence in its own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  The terms of the difference ∆Γ[¯𝜑] = Γ[¯𝜑] − Γ0[¯𝜑] are denoted as ’radiative correc-  tions’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' The above verbal reasoning in generalization of early thoughts of Wolfgang  Pauli leads to the equation  ∆Γ[¯𝜑] = 0 ,  (7)  expressing the vanishing of all radiative corrections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=',  Γ[¯𝜑] = Γ0[¯𝜑] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  (8)  The equation for the complete effective action which is equivalent to the fixed point  condition for the map 𝑓 reads (𝐶′ is some normalization constant)  e 𝑖Γ[¯𝜑]  = 𝐶′  ∫︁  𝐷𝜑 e 𝑖Γ[𝜑 + ¯𝜑] + 𝑖  ∫︁  𝑑𝑛𝑥 𝐽(𝑥)𝜑(𝑥)  ,  (9)  where  𝐽(𝑥) = − 𝛿Γ[¯𝜑]  𝛿 ¯𝜑(𝑥)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  (10)  The above selfconsistency equation (9) defines the (finite) effective action (including  its coupling constants and mass ratios) without any reference to classical physics ex-  clusively via quantum principles encoded in the map 𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' The fixed points of the map  𝑓 then describe physical reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' From out this perspective, the standard formulation  of quantum field theory represented by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (1) can roughly be understood as the first  step of an iterative solution of the nonlinear functional integro-differential equation  (9) by applying the map 𝑓 to some initial (in this case ‘classical’) action Γ0[𝜑].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' For  the first time, the above equation (9) to be taken as basis of quantum field theory  has been proposed in 1972 by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Prokhorov [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Not being aware of the earlier  work by Prokhorov, the same proposal has been made by the present author in 1993  [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' In a somewhat different (Hamiltonian) setting (coupled cluster methods), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  Arponen has expressed similar ideas in 1990 ([16], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 173, paragraph starting with  the words: “The possible solution corresponds to a system which suffers no change  under quantization.”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  3  Further discussion  Given the mature state of standard renormalizable quantum field theory, the above  point of view (defining the effective action as a fixed point of the map 𝑓) faces  myriads of objections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Some of them may be misunderstandings, others are com-  pletely justified concerns, others yet are possibly prejudices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Misunderstandings can  6  be dealt with most easily – by clarifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' For example, one might ask: Given  the crucial role of radiative corrections within the standard formulation of quantum  field theory in correctly describing physical reality (for example, in QED) how could  one ever possibly think of a theory of physical reality characterized by the vanishing  of all radiative corrections (in an interacting theory)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' The difficulty here, however,  is just a terminological one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Of course, also a modified formulation of quantum field  theory needs to deliver those kind of terms in the effective action we denote as ra-  diative corrections within the established standard approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' While the analytical  expression yielded from a modified formulation of quantum field theory may differ  from those within the standard formulation, the numerical results for experimentally  accessible quantities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', the anomalous magnetic moment of the electron) need to  be (almost – within experimental limits) the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' The point is that in the modified  view of quantum field theory represented by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9) those terms denoted in the stan-  dard formulation as radiative corrections are already incorporated in the action to  be quantized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' But, as the action to be quantized is supposed to be invariant under  quantization (according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9)) no new terms may emerge, consequently, in the  modified formulation of quantum field theory no radiative corrections (relative to  the initial action to be quantized) occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  Certainly, one elementary and justified concern with respect to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9) consists in  the question of whether eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9) allows any non-trivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', non-Gaussian) solutions  (Free field theories, of course, always obey eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' In fact, it has been shown by  example in a finite-dimensional Grassmann algebra analogue of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', within  a fermionic zero-dimensional field theory) that eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9) has exact non-trivial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=',  non-Gaussian) solutions [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' For an (bosonic) example from standard analysis see  [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Of course, as has been pointed out by Prokhorov [14] from the outset eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9)  represents a very complicated equation and presently very little can be said about  its eventual non-trivial solutions in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Experience from effective action studies  in quantum field theory tells us that non-trivial solutions of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9) can be expected  to be nonlocal and nonpolynomial functionals of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Whether these nonlocal and  nonpolynomial actions Γ solving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9) preserve unitarity and causality can only be  decided once they are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' However, it has been shown for a wide class of nonlo-  cal and nonpolynomial (scalar) quantum field theories in the past [19, 20] that they  respect these two principles – a fact, from which one may derive certain optimism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  It is of course conceivable that the general version of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9) (written down for an  arbitrary but fixed collection of fluctuating fields) does not have any non-Gaussian  solution at all for the certain field content one has chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' If this was the case the  existence of a non-Gaussian solution to the generalised version of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9) could be  applied as a theory selection criterion perhaps in the same way as the (stationary)  Schrödinger equation selects energy (eigen)values of quantum mechanical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  In a certain sense, at the end of the day only non-Gaussian solutions of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9) are  physical ones because only they provide the interactions for the structures we ob-  serve in physical reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Away from the rigid concept discussed above of the effective  action as a fixed point of the map 𝑓 non-Gaussian solutions of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9) may be con-  7  sidered also interesting within the standard lore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' While usually perturbation theory  is built around a Gaussian solution of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9), choosing a non-Gaussian solution of  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9) as starting point for perturbation theory may also be of some interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' For  a discussion in this direction see [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  From a methodological point of view, the largest difference of the approach rep-  resented by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9) to the established approach in standard quantum field theory  consists in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Standard local renormalizable quantum field theory starts  (among other things, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', choosing the space-time dimensionality) with the choice  of the field content of the theory under consideration and the functional form of the  (classical/bare) action Γ0 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (1)), a quantity which, in principle, is unobservable  (due to the existence of radiative corrections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' This is also true in the different ver-  sions of the Wilsonian approach to quantum field theory (inspired by the theory of  phase transitions in statistical mechanics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' While in a statistical mechanical system  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', a spin system modelling a certain microscopic condensed matter structure) the  structure of the Hamiltonian defined on a lattice with fixed lattice spacing can in  principle be linked to experimental data, this is not the case within quantum field  theory where the bare action is not related to observation (For a related discussion  see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Consequently, in the established standard approach to quantum  field theory the theoretical description of physical reality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', the effective action  Γ) is inferred from quantities not accessible to experiment in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' In contrast,  within the approach represented by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9) only the field content of the quantum  fluctuations can be chosen, the functional form of the effective action Γ is selfcon-  sistently restricted by its property to be a solution of this equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Beyond free  field theories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', for non-Gaussian solutions of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (9), this can be expected to be  highly restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  Acknowledgement  Kind hospitality at the Department of Physics and Astronomy of the Vrije Univer-  siteit Amsterdam is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  8  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Donoghue, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Golowich, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Holstein:  Dynamics of the Standard  Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' : Cambridge Monographs on Particle Physics, Nuclear Physics  and Cosmology, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' Cambridge University Press, Cambridge, 1992 (DOI:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' :  Cambridge Monographs on Particle  Physics, Nuclear Physics and Cosmology, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' Cambridge University Press,  Cambridge, 2014 (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content='  [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Drake (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ):  Springer Handbook of Atomic, Molecular, and Optical Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Springer-Verlag,  New York, NY, 2006, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content='  [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' ): Quantum Electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' World Scientific, Singapore,  1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (DOI:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1142/0495).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  [4] Tian Yu Cao (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ): Conceptual Foundations of Quantum Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Cam-  bridge University Press, Cambridge, 1999, 2004, 2009  (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1017/CBO9780511470813).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' Poland, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Rastelli: Snowmass Topical Summary: Formal QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' To ap-  pear in the Proceedings of the US Community Study on the Future of Par-  ticle Physics (Snowmass 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' [arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='03128 (https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='org/  abs/2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='03128)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Feynman: The development of the space-time view of quantum electrody-  namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Science 153:3737(1966)699-708 (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='3737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='699);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  Physics  Today  19:8(1966)31-44  (DOI:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='3048404).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  The  text  is  freely available online at the Nobel Foundation URL: http://nobelprize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  org/nobel_prizes/physics/laureates/1965/feynman-lecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Also  printed in: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Nobelstiftelsen (Nobel Foundation): Les Prix Nobel en 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  Almqvist & Wiksell International, Stockholm, 1966, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 172-191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' No-  belstiftelsen (Nobel Foundation):  Nobel Lectures, Including Presentation  Speeches and Laureates’ Biographies: Physics, 1963-1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Elsevier, Amster-  dam, 1972, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 155-178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Reprinted: World Scientific Publishing, Singapore,  1998 (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Laurie M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Brown (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ):  Selected Papers of  Richard Feynman, with Commentary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' World Scientific Series in 20th Century  Physics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' World Scientific Publishing, Singapore, 2000, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' Alix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' Princeton University Press, Princeton, NJ, 1985,  1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Penguin Books, London, 1985, 1990, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' Dirac: The origin of quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' In: L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Brown, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  Hoddeson (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ): The Birth of Particle Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Cambridge University Press,  Cambridge, 1983, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 39-55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  [10] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' von Meyenn (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ): Wolfgang Pauli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Wissenschaftlicher Briefwechsel mit  Bohr, Einstein, Heisenberg u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='. Band III: 1940-1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Scientific Correspon-  dence with Bohr, Einstein, Heisenberg a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='. Volume III: 1940-1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Sources in  the History of Mathematics and Physical Sciences, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Springer-Verlag,  Berlin, 1993, 2014 (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1007/978-3-540-78802-7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  [11] Л.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Фаддеев [L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Faddeev], А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Славнов [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Slavnov]: Введение  в квантовую теорию калибровочных полей [Vvedenie v kvantovuyu teoriyu  kalibrovochnykh pole˘ı].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Наука [Nauka], Moscow, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' : 1978, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' & ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' :  1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' УРСС [URSS], Moscow, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' [in Russian] English translation: Gauge  Fields: Introduction to Quantum Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Frontiers in Physics, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' : Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 50,  2nd rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' & ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' : Benjamin/Cummings Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=', Reading, MA,  1980, 2nd rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' & ext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' : Addison-Wesley, Redwood City, CA, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Westview  Press, Boulder, CO, 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' CRC Press, Boca Raton, FL, 2018, 2019 (DOI:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1201/9780429493829).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Jevicki, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Lee: The S-matrix generating functional and effective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  Physical Review D 37:6(1988)1485-1491 (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1485)  [Brown University report BROWN-HET-634].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Zuber: Quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' International Series in Pure  and Applied Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' McGraw-Hill, New York, 1980, 1985, 1986, 1987, 1988,  1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Reprint: Dover Books on Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Dover Publications, Mineola, NY, 2005,  2006, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  [14] Л.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' В.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Прохоров [L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Prokhorov]: Уравнения для эффективных лагранжи-  анов [Uravneniya dlya èffektivnykh lagranzhianov].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Ядерная Физика [Yader-  naya Fizika] 16:4(1972)854-862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' [in Russian] English translation: Equations for  effective Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Soviet Journal of Nuclear Physics 16:4(1973)473-477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' Scharnhorst: Functional integral equation for the complete effective ac-  tion in quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' International Journal of Theoretical Physics  36:2(1997)281-343 (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1007/BF02435737)  [University of Leipzig report NTZ-16-1993, arXiv:hep-th/9312137 (https:  //arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='org/abs/hep-th/9312137)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Arponen: Independent-cluster methods as mappings of quantum theory  into classical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Theoretica Chimica Acta 80:2-3(1991)149-179 (DOI:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1007/BF01119618) [University of Helsinki report HU-TFT-90-75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  10  [17] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Scharnhorst:  A Grassmann integral equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Journal of Mathematical  Physics 44:11(2003)5415-5449 (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1612896, errata sheet at the  URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='vu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='nl/~scharnh/mea30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='pdf)  [arXiv:math-ph/0206006 (https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='org/abs/math-ph/0206006)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  [18] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Pioline: Cubic free field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' In: L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Baulieu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Rabinovici, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Harvey,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Pioline, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Windey (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' ): Progress in String, Field and Particle Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  Proceedings of the NATO Advanced Study Institute, Cargèse, Corsica, France,  May 27-June 8, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' NATO Science Series II: Mathematics, Physics and Chem-  istry, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Kluwer Academic Publishers, Dordrecht, 2003, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 453-456  (DOI: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1007/978-94-010-0211-0_37) [LPTHE Paris preprint LPTHE-P03-01,  arXiv:hep-th/0302043 (https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='org/abs/hep-th/0302043)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  [19] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Alebastrov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Efimov:  A proof of the unitarity of S-matrix in  a nonlocal quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Communications in Mathematical Physics  31:1(1973)1-24 (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1007/BF01645588,  the article is freely available online from the Project Euclid website:  http://projecteuclid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='org/euclid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='cmp/1103858926)  [Institute of Theoretical Physics, Kiev, report ИТФ-72-110Р [ITF-72-110R] [in  Russian]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  [20] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' Alebastrov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Efimov: Causality in quantum field theory with non-  local interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Communications in Mathematical Physics 38:1(1974)11-28  (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1007/BF01651546,  the article is freely available online from the Project Euclid website:  http://projecteuclid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='org/euclid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='cmp/1103859964)  [Joint Institute of Nuclear Resarch, Dubna, report Р2-7572 [R2-7572] [in Rus-  sian]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  [21] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
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+page_content=' Helias: Expansion of the effective action around non-Gaussian the-  ories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Journal of Physics A: Mathematical and Theoretical 51:37(2018)375004,  35 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1088/1751-8121/aad52e), erratum ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' 51:45(2018)459502, 1  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1088/1751-8121/aae1d7) [arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='05599 (https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  org/abs/1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='05599)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Rosaler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Harlander: Naturalness, Wilsonian renormalization, and “fun-  damental parameters” in quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content=' Studies in History and Philos-  ophy of Modern Physics 66(2019)118-134 (DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='shpsb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='003)  [RWTH Aachen report TTK-19-04, ELHC research group report ELHC_2018-  005, arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='09489 (https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='org/abs/1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='09489)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}
+page_content='  11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdFQT4oBgHgl3EQfOTbk/content/2301.13275v1.pdf'}

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+Correct-by-Design Teamwork Plans for Multi-Agent Systems⋆
+Yehia Abd Alrahman and Nir Piterman
+University of Gothenburg, Gothenburg, Sweden
+{yehia.abd.alrahman,nir.piterman}@gu.se
+Abstract. We propose Teamwork Synthesis, a version of the distributed synthesis problem with
+application to teamwork multi-agent systems. We reformulate the distributed synthesis question
+by dropping the fixed interaction architecture among agents as input to the problem. Instead, our
+synthesis engine tries to realise the goal given the initial specifications; otherwise it automatically
+introduces minimal interactions among agents to ensure distribution. Thus, teamwork synthesis
+mitigates a key difficulty in deciding algorithmically how agents should interact so that each obtains
+the required information to fulfil its goal. We show how to apply teamwork synthesis to provide a
+distributed solution.
+1
+Introduction
+Synthesis [24] of correct-by-design multi-agent systems is still one of the most intriguing challenges in
+the field. Traditionally, synthesis techniques targeted Reactive Systems – systems that maintain contin-
+uous interactions with hostile environments. A synthesis algorithm is used to automatically produce a
+monolithic reactive system that is able to satisfy its goals no matter what the environment does. Syn-
+thesis algorithms have been also extended for other domains, e.g., to support rational environments [13],
+cooperation [17,7], knowledge [12], etc.
+A major deficiency of traditional synthesis algorithms is that they produce a monolithic program, and
+thus fail to deal with distribution [8]. In fact, the distributed synthesis problem is undecidable, except
+for specific configurations [25,8]. This is disappointing when the problem we set out to solve is only
+meaningful in a vibrant distributed domain, such as multi-agent systems.
+In this paper, we mount a direct attack on the latter, and especially Teamwork Multi-Agent Systems
+(or Teamwork MAS) [21,26]. Teamwork MAS consist of a set of autonomous agents that share an execution
+context in which they collaborate to achieve joint goals. They are a natural evolution of reactive systems,
+where an agent has to additionally collaborate with team members to jointly maintain correct reactions
+to inputs from the context. Thus, being reactive requires being prepared to respond to inputs coming
+from the context and interactions from the team.
+The context is uncontrolled and can introduce uncertainties for individuals that may disrupt the joint
+behaviour of the team. For instance, a change in sensor readings of agentk that some other agentj cannot
+observe, but is required to react to, etc. Thus, maintaining correct (and joint) reactions to contextual
+changes requires a highly flexible coordination structure [29]. This implies that fixing all interactions
+within the team in advance is not useful, simply because the required level of connectivity changes
+dynamically.
+Despite that flexible coordination mechanisms are undeniably effective to counter uncertainties, the
+literature on distributed synthesis and control is primarily focused on fixed coordination, e.g., Distributed
+synthesis [25,8]), Decentralised supervision [30,27], and Zielonka synthesis [31,9]. This reality, however,
+is due to the fact that there is no canonical model to describe distributed computations, and hence the
+focus is on well-known models with fixed structures. It is widely agreed that the undecidability result is
+mainly due to partial (or lack of) information. The latter can also be rephrased as “lack of coordination”.
+Note that the decidability of a distributed synthesis problem is conditioned on the right match between
+the given concurrency model and its formulation [20].
+We are left in the middle of these extremes: Distributed synthesis [25,8], Zielonka synthesis [31,9], and
+Decentralised supervision [30]. All are undecidable except for specific configurations. Zielonka synthesis
+is decidable if synchronising agents are allowed to share their entire state, and this produces agents that
+are exponential in the size of the joint deterministic specification.
+We propose Teamwork Synthesis, a decidable reformulation of the distributed synthesis problem.
+We reformulate the synthesis question by dropping the fixed interaction architecture among agents.
+⋆ This work is funded by the Swedish research council grant: SynTM (No. 2020-03401) (Led by the first author)
+and the ERC consolidator grant D-SynMA (No. 772459)(Led by the second author).
+arXiv:2301.01257v1  [cs.LO]  3 Jan 2023
+
+2
+Y. Abd Alrahman and N. Piterman
+Instead, our approach dynamically introduces minimal interactions when needed to maintain correctness.
+Teamwork synthesis consider a set of agent interfaces, an environment model that specifies assumptions
+on the context and (possibly) partial interactions among agents, and a formula over the joint goal of the
+team within the context. A solution for teamwork synthesis is a set of reconfigurable programs, one per
+agent such that their dynamic composition satisfies the formula under the environment model.
+The contributions in this paper are threefold:
+(i) we introduce the Shadow transition system (or
+Shadow TS for short) which distills the essential features of reconfigurable multicast from CTS [3],
+augments, and disciplines them to support teamwork synthesis;
+(ii) we propose a novel parametric
+bisimulation that is able to abstract unnecessary interactions, and thus helps producing Shadow TSs with
+least amount of coordinations, and with size that is, in the worst case, equivalent to the joint deterministic
+specification. This is a major improvement on the Zielonka approach and with less coordination;
+(iii)
+lastly, we present teamwork synthesis and show how to reduce it to a single-agent synthesis. The solution
+is used to construct an equivalent loosely-coupled distributed one. Our synthesis engine will try realise
+the goal given the initial specifications, otherwise it will automatically introduce additional required
+interactions among agents to ensure distributed realisability. Note that those additional interactions are
+strategic, i.e., they are introduced dynamically when needed and disappear otherwise. Thus, teamwork
+synthesis will enable us to mitigate a key difficulty in deciding algorithmically how agents should interact
+so that each obtains the required information to carry out its functionality.
+The paper’s structure is as follows: In Sect. 2, we give an overview on teamwork synthesis. In Sect. 3,
+we present a short background materials, and later in Sect. 4, we present a case study to illustrate our
+approach. In Sect. 5, we present the Shadow TS and the corresponding bisimulation. In Sect. 6, we present
+teamwork synthesis and in Sect. 7, we report our concluding remarks.
+2
+Teamwork Synthesis in a nutshell
+We consider a team of K autonomous agents that execute in a shared context, and pursue a joint goal.
+A context can be a physical space or an external entity that may impact the joint goal.
+Interaction among team members is established based on a set of channels (or event names), denoted
+Y and partitioned among all members. An agent, say agentk, can locally control a subset of event names
+(Yk ⊆ Y ) by being responsible of sending all messages with channels from Yk while other agents may be
+eligible to receive.
+We assume that every agentk, partially observes its context by means of reading local sensor observa-
+tion values xk ∈ Xk. Moreover, agentk may react to new inputs from Xk or messages (with channels from
+other agents, i.e., in (Y \Yk)) by generating local actuation signals o ∈ Ok. That is, the signals agentk
+uses to control its state, e.g., a robot sends signals to its motor to change direction.
+Message exchange is established in a reconfigurable multicast fashion. That is, agentk may send
+messages to interested team members, i.e., agents that currently listen to the sending channel. A receiving
+agent, agentj for j ̸= k, can adjust its actuation signals Oj accordingly. Agents can connect/disconnect
+channels dynamically based on need. An agent only receives messages on channels that listens to in its
+current state, and cannot observe others.
+Agentk starts from a fixed initial state, and in every future execution step it either: observes a new
+sensor input from Xk; receives a message on a channel from (Y \Yk) that agentk listens to in the current
+state; or sends a message on a channel from Yk to interested members. In all cases, agentk may trigger
+individual actuation signals Ok accordingly.
+As a team, every team execution starts from a fixed initial state. Moreover, in every execution step the
+team either observes an aggregate sensor input – some members (i.e., a subset of K) observe an input – or
+exposes a message on channel from Y originated exactly from one member. In both cases, the team may
+trigger an aggregate actuation signal O. Formally, the set of aggregate sensor inputs over {Xk}k∈K is
+X = {x : K �→ �
+k Xk | x(k) /∈ �
+j̸=k Xj and ∃ k ∈ K, s.t. x(k) is defined}. That is, a global observation
+corresponds to having new sensor values for some of the agents. Note that x is a partial function. Similarly,
+the set of aggregate actuation signals over {Ok}k∈K is O = {o : K �→ �
+k Ok | o(k) /∈ �
+j̸=k Oj}. Note
+that unlike X, the set of aggregate output signals O can be empty.
+Thus, teamwork synthesis only requires that aggregate observations X and interactions on channels
+from Y interleave [18] after initialisation θi, see the assumption automaton (A) below:
+The rationale is that we start from an environment model E that specifies both aggregate context
+observations X and (possibly) interactions on channels from Y , i.e., the environment model E may
+centrally specify an interaction protocol on channels from Y . Then we are given a set of agent interfaces
+{⟨Xk, Yk, Ok⟩}k∈K such that X is the set of aggregate observations over {Xk}k∈K, O is the set of
+
+Correct-by-Design Teamwork Plans for Multi-Agent Systems
+3
+Fig. 1: Execution Assumption
+aggregate actuation signals over {Ok}k∈K as defined before, and Y = �
+k Yk; and a formula ϕ over the
+joint goal of the team within E (i.e., the language of ϕ is in ((Y ∪ X) × O)ω).
+Our synthesis engine will try realise the goal given the initial protocol description (which can also
+be empty) on Y , and if this is not possible, it will automatically introduce additional required interac-
+tions among agents to ensure distributed realisability. We use the Shadow TS, with essential features of
+reconfigurable multicast, as the underlying distributed model for teamwork synthesis.
+Formally, a solution for teamwork Synthesis T = ⟨E ∩ A, ϕ, O⟩ is a set of |K|-Shadow TSs, one for
+each ⟨Xk, Yk, Ok⟩ such that their team composition satisfies ϕ under E ∩A, where E ∩A is the standard
+automata intersection of E and the execution assumption A depicted above. We show that the teamwork
+synthesis problem can be reduced to a single-agent synthesis. The solution of the latter can be efficiently
+decomposed into a set of equivalent shadow TSs.
+3
+Background
+We present the background material on symbolic automata for environment’s specifications and linear
+temporal logic (ltl).
+Definition 1 (Environment model). An environment model E is a deterministic symbolic automaton
+of the form E = ⟨Q, Σ, Ψ, q0, ρ⟩,
+• Q is a set of states and q0 ∈ Q is the initial state.
+• Σ is a structured alphabet of the form (Y ∪ X).
+• Ψ is a set of predicates over Σ such that every predicate ψ ∈ Ψ is interpreted as follow: �·� : Ψ →
+(Y ∪ X).
+• ρ : Q × Ψ → Q is the transition function, s.t. for all transitions (q, ψ, q′), (q, ψ′, q′′) ∈ ρ, if ψ ∧ ψ′ is
+satisfiable then q′ = q′′.
+The language of E, denoted by LE, is a set of infinite sequences of letters in (Y ∪ X)ω. Two environ-
+ment models E1 and E2 can be composed by means of standard automata intersection (E1 ∩ E2).
+For goal specifications, we use ltl to specify the goals of individual agents and their joint goals. We
+assume an alphabet of the form (Y ∪ X) × O) as defined before. A model σ for a formula ϕ is an infinite
+sequence of letters in (Y ∪X)×O), i.e., it is in ((Y ∪X)×O))ω. Given a model σ = σ0, σ1, . . ., we denote
+by σi the letter at position i.
+LTL formulas are constructed using the following grammar.
+ϕ ::= v ∈
+�
+k
+(Xk ∪ Ok) | y ∈ Y | ¬ϕ | ϕ1 ∨ ϕ2 | Xϕ | ϕ1 U ϕ2
+For a formula ϕ and a position i ≥ 0, ϕ holds at position i of σ, written σ, i |= ϕ, where σi = (v, o), if:
+• For xk ∈ Xk we have σ, i |= xk iff v ∈ X and and v(k) = xk. That is, xk is satisfied if v(k) is defined
+and equal to xk.1
+• For ok ∈ Ok we have σ, i |= ok iff o(k) = ok
+• For y ∈ Y we have σ, i |= y iff v = y
+• σ, i |= ¬ϕ iff σ, i ̸|= ϕ
+• σ, i |= ϕ ∨ ψ iff σ, i |= ϕ or σ, i |= ψ
+• σ, i |= Xϕ iff σ, i + 1 |= ϕ
+• σ, i |= ϕ U ψ iff there exists k ≥ i such that σ, k |= ψ and σ, j |= ϕ for all j, i ≤ j < k
+If σ, 0 |= ϕ, then ϕ holds on σ (written σ |= ϕ). A set of models M satisfies ϕ, denoted M |= ϕ, if every
+model in M satisfies ϕ. A formula is satisfiable if the set of models satisfying it is not empty.
+We use the usual abbreviations of the Boolean connectives ∧, →, and ↔ and the usual definitions
+for true and false. We introduce the following temporal abbreviations Fφ = true U φ, Gφ = ¬F¬φ, and
+φ1Rφ2 = ¬(¬φ1U¬φ2).
+1 It is possible to say v(k) is defined and not equal to xk by �
+xk̸=x∈Xk x.
+
+0
+Y4
+Y. Abd Alrahman and N. Piterman
+4
+Distributed Product Line Scenario
+We use a distributed product line scenario to illustrate Teamwork Synthesis and its underlying principles.
+The product line, in our scenario, is operated by three robot arms: (i) the tray arm that observes
+inputs on the input-tray and forwards them for processing;
+(ii) the proc arm that is responsible for
+processing the inputs; (iii) and the pkg arm that packages and delivers the final product.
+The operator of the product line is an uncontrollable human, adding inputs, denoted by (in), to the
+input-tray. The operator serves as the execution context in which the three robot arms operate. Only the
+tray arm can observe the input (in).
+The specifications of the robot arms are as follows: The interface of the tray is of the form Inttray =
+⟨{in}, {fwd}, {rFwd}⟩. That is, the tray arm can observe the input in on the input-tray, it can also send
+a message on channel fwd, and it has one actuation signal rFwd to instruct its motor to get ready to
+forward the input. The f-automaton below specifies its part of the interaction protocol.
+That is, the tray arm can forward by sending a message on fwd only after it observes an input in.
+The safety goals of the tray are:
+ϕ1 = G(in → rFwd) &
+G((rFwd ∧ (X¬fwd)) → (XrFwd))
+That is, the motor gets ready to forward whenever an input is observed. Moreover, the motor remains
+ready to forward as long as forwarding did not happen.
+The
+interface
+of
+the
+proc
+arm
+is
+of
+the
+form
+Intproc
+=
+⟨∅, {proc}, {rProc}⟩ . That is, the proc arm cannot observe any input, but it can send a message on
+proc, and it has one actuation signal rProc to instruct its motor to get ready to process the input. The
+p-automaton below and the ltl formula pd specify the arm part in the interaction protocol.
+pd = G((proc ∧ (Xin) ∧ (XXfwd)) → X(dlv R¬proc))
+Namely, the proc arm can process by sending a message on proc only after a forward has happened.
+Moreover, the arm cannot process twice in row without a deliver in between. We will use A(pd) to denote
+the automaton representing pd.
+The safety goals of the proc arm are as follows:
+ϕ2 = G(fwd → rProc) &
+G((rProc ∧ (X¬proc)) → (XrProc))
+That is, the motor gets ready to process whenever forward happens. Moreover, the motor remains
+ready to process as long as processing did not happen.
+The interface of the pkg arm is Intpkg = ⟨∅, {dlv}, {rDlv}⟩ . That is, the pkg arm cannot observe
+any input, but it can deliver by sending a message on dlv, and it has one actuation signal rDlv to instruct
+its motor to get ready to package and deliver the input. The d-automaton below specifies its part of the
+interaction protocol.
+The pkg arm can send a message on dlv only after processing has happened. The safety and liveness
+goals of the pkg arm are:
+ϕ3 = G(proc → rDlv) &
+G((rDlv ∧ (X¬dlv)) → (XrDlv))
+That is, the motor gets ready to deliver whenever process happens. Moreover, the motor remains
+ready to deliver as long as delivering did not happen. We also require GF(rDlv), i.e., the motor must also
+be ready to delivering infinitely often.
+
+ifwaaim
+1.
+Rwd0
+1
+pxroxcpnox
+dl ajpoxc
+0)
+1
+dlyCorrect-by-Design Teamwork Plans for Multi-Agent Systems
+5
+We have the following assumption on the operator op:
+Namely, after a first input the operator waits for processing to happen before it puts a new input.
+Finally, we require GF(in), i.e., the operator must supply input infinitely often.
+We assume that all signals are initially off. That is:
+θ = ¬in ∧ ¬fwd ∧ ¬proc ∧ ¬dlv ∧ ¬rFwd ∧ ¬rProc ∧ ¬rDlv
+Notice that these specifications are written from a central point of view. For instance, the formula pd
+of the proc arm predicates on (in, fwd, and dlv) even if it cannot observe them. To be able to enforce
+this formula, we need to be able to automatically introduce strategic and minimal interactions among
+agents at run-time, only when needed (!), and this is the role of teamwork synthesis.
+The instance of teamwork synthesis T = ⟨E ∩ A, ϕ, O⟩ is:
+(i) E = f ∩ p ∩ d ∩ A(pd) ∩ op
+(ii) A = is an instance of the automaton depicted in Fig. 1
+(iii) ϕ = θ ∧ ϕ1 ∧ ϕ2 ∧ ϕ3 ∧ (GF(in) → GF(rDlv))
+(iv) O =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∅, {(tray �→ rFwd)},
+{(proc �→ rProc)}, {(pkg �→ rDlv)},
+{(tray �→ rFwd), (proc �→ rProc)},
+{(tray �→ rFwd), (pkg �→ rDlv)},
+{(proc �→ rProc), (pkg �→ rDlv)},
+{(tray �→ rFwd), (proc �→ rProc), (pkg �→ rDlv)}
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+A solution for T = ⟨E ∩ A, ϕ, O⟩ is a 3-Shadow TSs, one for each ⟨Xk, Yk, Ok⟩k∈{1,2,3} such that
+T1∥T2∥T3 |= ϕ under E ∩ A.
+We will revisit the scenario, at the end of Sect. 6, to show the distributed realisation of this problem
+and its features.
+5
+Shadow Transition Systems
+We formally present the Shadow Transition System and we use it to define the behaviour of individual
+agents. We also define how to compose different agents to form a team.
+Definition 2 (Shadow TS). A shadow TS is of the form Tk = ⟨Sk, Intk, Actk, ∆k
+e, ∆k, Lk, lsk, sk
+0⟩,
+where:
+• Sk is the set of states of Tk and sk
+0 ∈ Sk its initial state.
+• Intk = ⟨Xk, chk, Ok⟩ is the interface of Tk, where
+• Xk is an observation alphabet, chk is a set of interaction channels, and Ok is an output (or
+actuation) alphabet. We use i to range over elements in Xk or chk;
+• lsk : Sk → 2chk is a channel listening function. That is, lsk defines (per state) the channels that
+Tk listens to.
+• Actk ⊆ (chk ×{!, ?}×Υk) is the set of messages. Intuitively, a message consists of a channel ch ∈ chk,
+a type (send ! or receive ?), and a load (or contents) υ ∈ Υk.
+• Lk : Sk → (chk ∪ (Xk ⊎ {⊥})) × (Ok ⊎ {⊥}) is a labelling function where ⊥ denotes undefined label,
+i.e., Lk labels states with input (output) letters that were observed (correspondingly produced).
+• ∆k
+e ⊆ Sk × (chk ∪ Xk) denotes the environment potential moves from Sk, i.e., ∆k
+e can be thought of as
+a ghost transition relation denoting the instantaneous perception of Tk of its environment.
+• ∆k ⊆ Sk × Actk × Sk is the transition relation of Tk. The relation ∆k can be thought of as a shadow
+transition relation of ∆k
+e. That is, for every potential move in ∆k
+e, there must be a corresponding shadow
+transition in ∆k as follows:
+• For every state s ∈ Sk and every letter i ∈ (chk∪Xk), if (s, i) ∈ ∆k
+e then there exists o ∈ Ok, s′ ∈ Sk
+such that Lk(s′) = (i, o) and (s, a, s′) ∈ ∆k for some a ∈ Actk
+
+0
+iin,6
+Y. Abd Alrahman and N. Piterman
+• For any state s ∈ Sk, if for every letter i ∈ (chk ∪ Xk), (s, i) /∈ ∆k
+e and there exists o ∈ Ok, s′ ∈ Sk
+such that (s, a, s′) ∈ ∆k for some a ∈ Actk then a must be a receive.
+Shadow TSs can be composed to form a team as in Def. 3 below. We use projx
+k to denote the projection
+of a team label into Xk of agentk, and similarly for projch
+k and projo
+k, i.e., for projection on chk and OK
+respectively. We use ⊥ to denote that the projection is undefined.
+Definition 3 (Team). Given a set K = {1, . . . , n} of shadow TSs Tk = ⟨Sk, Intk, Actk, ∆k
+e, ∆k, Lk, lsk, sk
+0⟩
+where k ∈ K, their composition ∥kTk is the team T = ⟨S, Int, Act, ∆e, ∆, L, ls, s0⟩,
+• S = (s1, . . . , sn),
+s0 = (s1
+0, . . . , sn
+0),
+Act = �
+k Actk,
+Υ = �
+k Υk
+• Int = ⟨X, ch, O⟩ such that ch = �
+k chk,
+X = {x : K �→ �
+k Xk | x(k) /∈ �
+j̸=k Xj},
+and O = {o : K �→ �
+k Ok | o(k) /∈ �
+j̸=k Oj}
+∆ =
+�
+�
+�
+�
+�
+(s1, . . . , sn),
+(c, !, υ),
+(s′
+1, . . . , s′
+n)
+�
+�
+������
+∃k ∈ {1, n} . (sk, (c, !, υ), s′
+k) ∈ ∆k and ∀j ̸= k.
+(1) (sj, (c, ?, υ), s′
+j) ∈ ∆j and c ∈ lsj(sj) or
+(2) c /∈ lsj(sj) and s′
+j = sj
+�
+�
+�
+∆e =
+�
+�
+�
+�(s1, . . . , sn), i�
+������
+((s1, . . . , sn), (c, !, υ), (s′
+1, . . . , s′
+n)) ∈ ∆
+for some (c, !, υ) ∈ Act such that
+L((s′
+1, . . . , s′
+n)) = (i, o) for some o ∈ O
+�
+�
+�
+• Given a state (s1, . . . , sn), let x = �
+k{k �→ projx
+k(Lk(sk)) | projx
+k(Lk(sk)) ̸= ⊥}, ch = �
+k projch
+k (Lk(sk)),
+and
+o = �
+k{k �→ projo
+k(Lk(sk)) |
+projo
+k(Lk(sk)) ̸= ⊥}, then L((s1, . . . , sn)) = (x, o) if x ̸= ∅ and
+(ch, o) otherwise. In the systems we construct we achieve that ch is always a unique value in ch.
+• ls((s1, . . . , sn)) = �
+k lsk(sk)
+Note that the composition in Def. 3 does not necessarily produce a shadow TS. However, our synthesis
+engine will generate a set of shadow TSs such that their composition is also a shadow TS.
+Intuitively, multicast channels are blocking. That is, if there exists an agentk with a send transition
+(sk, (c, !, v), s′
+k) ∈ ∆k on channel c then every other parallel agentj, s.t.j ̸= k (that listens to c in its current
+state, i.e., c ∈ ls(sj)) must supply a matching receive transition (sj, (c, ?, v), s′
+j) ∈ ∆j or otherwise the
+sender is blocked. Other parallel agents that do not listen to c simply cannot observe the interaction,
+and thus cannot block it. We restrict attention to the set of shadow TSs T that satisfy the following
+property:
+Property 1 (Local broadcast). ∀T ∈ T , s ∈ ST , we have thatc ∈ ls(s) iff (s, (c, ?, υ), s′) ∈ ∆T for all
+c ∈ ch and υ ∈ Υ.
+Thus, a shadow TS cannot block a message send by listening to its channel and not supplying a
+corresponding receive transition. This reduces the semantics to asynchronous local broadcast. That is,
+message sending cannot be blocked, and is sent on local broadcast channels rather than a unique public
+channel (⋆) as in CTS [3].
+A run of T is the infinite sequence r = s0a0s1a1s2 . . . such that for all k ≥ 0 : (sk, ak, sk+1) ∈ ∆ and
+s0 is the initial state. An execution of T is the projection of a run r to state labels. That is, for a run
+r = s0a0s1a1s2 . . . , there is an execution w induced by r such that w = L(s0)L(s1)L(s2) . . . . We use LT
+to denote the language of T, i.e., the set of all executions of T. For a specification spec ⊆ ((X ∪ch)×O)ω,
+we say that T satisfies spec if and only if LT ⊆ spec. Note that the key idea in our work is that we use a
+specification that only refers to aggregate input and output, and is totally insensitive to messages. As we
+will see later, the latter will be used by a synthesis engine to ensure distributed realisability.
+Lemma 1. The composition operator is a commutative monoid.
+Proof. The proof follows directly by Property 1 and the definition of ∥. There, the existential and universal
+quantifications on k are insensitive to the location of sk in (s1, . . . , sn) for k ∈ {1, n}. A sink state, denoted
+by 0, (i.e., a state with zero outgoing transitions and empty listening function, i.e., ls(0) = ∅) is the Id-
+element of ∥ because it cannot influence the composition.
+We define a notion of parameterised bisimulation that we use to efficiently decompose a Shadow TS.
+
+Correct-by-Design Teamwork Plans for Multi-Agent Systems
+7
+E -Bisimulation Consider the TS T with a finite state space S = {s1, . . . , sn} that is composed with the
+TS C (that we call the parameter TS). The latter has a finite state space E = {ϵ1, . . . , ϵm} and will be
+used as the basis to minimise the former. That is, TS C is only agent that T can interact with. This is the
+only bisimulation used in this paper. When we write bisimulation we mean parameterised bisimulation.
+We first introduce some notations:
+• (ϵ
+a
+−→ ϵ′): a parameter state ϵ permits message a = (c, !, υ) iff ϵ can receive a, does not listen to c, or
+is the sender. Formally,
+(i)
+c ∈ ls(ϵ) and (ϵ, (c, ?, υ), ϵ′) ∈ ∆E or
+(ii)
+c ̸∈ ls(ϵ) and (ϵ = ϵ′) or
+(iii)
+(ϵ, (c, !, υ), ϵ′) ∈ ∆E
+Note that this item and Property 1 ensure that message send is autonomous and cannot be restricted
+by the parameter TS.
+• (s
+a!
+−−→ s′) : s sends message a = (c, !, υ) iff (s, (c, !, υ), s′) ∈ ∆
+• (s
+a?
+−−→ s′) : s receives message a = (c, !, υ) and updates iff c ∈ ls(s), (s, (c, ?, υ), s′) ∈ ∆, L(s) ̸= L(s′).
+• (s
+τa
+−−→ s′) : s can discard iff a = (c, !, υ), c ∈ ls(s), (s, (c, ?, υ), s′) ∈ ∆, L(s) = L(s′). Note the state’s
+label did not change by receiving. We drop the name a from τa when a is arbitrary.
+Note that all kinds of receives (
+τa
+−−→ or
+a?
+−−→) cannot happen without a joint message-send.
+• We use (
+τ−→⋆)ϵ′
+ϵ to denote a sequence (possibly empty) of arbitrary discards (
+τa
+−−→ for any a), starting
+when the parameter state is ϵ and ending with ϵ′. We define a family of transitive closures as the
+minimal relations satisfying: (i) s (
+τ−→⋆)ϵ′
+ϵ s; and (ii) if s1 (
+τ−→⋆)ϵ2
+ϵ1 s2, (s2
+τa
+−−→ s3), (ϵ2
+a
+−→ ϵ3), and
+s3 (
+τ−→⋆)ϵ4
+ϵ3 s4 then s1 (
+τ−→⋆)ϵ4
+ϵ1 s4.
+These are the reflexive and transitive closure of
+τ
+−→ while making sure that also the parameter supplies
+the sends that are required.
+• We will use (s
+a
+−→) when s has a transition, and (s ̸
+a
+−→) when s has no a transitions.
+Definition 4 (E -Bisimulation). Let the shadow TS C with finite state space E = {ϵ1, . . . , ϵm} be a
+parameter TS. An E -bisimulation relation R is a symmetric E -indexed family of relations Rϵ ⊆ S × S
+for ϵ ∈ E , such that whenever (s1, s2) ∈ Rϵ then L(s1) = L(s2), and for all a ∈ Act, if ϵ
+a
+−→ ϵ′ then
+1.
+s1
+a!
+−−→ s′
+1
+implies
+∃s′
+2, s2
+a!
+−−→ s′
+2 and (s′
+1, s′
+2) ∈ Rϵ′;
+2.
+s1
+a?
+−−→ s′
+1
+implies
+s2 ̸
+τa
+−−→
+and
+�
+if
+s2
+a?
+−−→
+then ∃s′
+2,
+s2
+a?
+−−→ s′
+2 and (s′
+1, s′
+2) ∈ Rϵ′
+else ∃s′
+2, s′′
+2, ϵ′′,
+s2 (
+τ−→⋆)ϵ
+ϵ′′ s′′
+2
+a?
+−−→ s′
+2, and (s′
+1, s′
+2) ∈ Rϵ′
+�
+3.
+s1
+τa
+−−→ s′
+1
+implies
+s2 ̸
+a?
+−−→
+and
+�
+if s2
+τa
+−−→
+then ∃s′
+2,
+s2
+τa
+−−→ s′
+2 and (s′
+1, s′
+2) ∈ Rϵ′
+else
+∃s′
+2, ϵ′′,
+s2 (
+τ−→⋆)ϵ
+ϵ′′ s′
+2 and (s′
+1, s′
+2) ∈ Rϵ′
+�
+Two states s1 and s2 are equivalent with respect to a parameter state ϵ ∈ E , written s1 ∼ϵ s2, iff there
+exists an E -bisimulatin R such that (s1, s2) ∈ Rϵ. Please note that R is symmetric.
+Def. 4 equates two states with same labelling with respect to the current parameter state ϵ if: (1) they
+supply the same send transitions; (2) they supply same receive transitions or one can discard a number
+of messages and reach a state in which it can supply a matching receive; (3) is similar to (2) except for
+the “else” part where one state can supply an arbitrary number of discard (possibly none). In all cases,
+both states are required to evolve to equivalent states under the next parameter state ϵ′.
+Note that case 2 and 3 (and their symmetrics) in Def. 4 allow an agent to avoid participating in
+interactions that do not affect it.
+We use ∼0 to denote the equivalence under the empty parameter O. That is, O has a singleton sink
+state 0. The parallel composition ∥ in Def. 3 is a commutative monoid, and 0 is the id-element. Thus, we
+have that (s, 0) is equivalent to s for all s ∈ S.
+We need to prove that ∼ϵ is closed under the parallel composition in Def. 3 within a composite
+parameter C . That is, a parameter C of the form C1∥ . . . ∥Cn for some n. For a composite state ϵ =
+(ϵ1, . . . , ϵn) and w = {i1, . . . , ij}, we use (ϵi1, . . . , ϵij) to denote a w-cut of ϵ. That is, a projection of ϵ on
+states (ϵi1, . . . , ϵij). Moreover, we use ϵ\p to denote ϵ without cut p.
+
+8
+Y. Abd Alrahman and N. Piterman
+Theorem 1 (∼ϵ is closed under ∥). For all states s1, s2 of a shadow TS, all composite parameter
+states ϵ ∈ E of the form ϵ = (ϵ1, . . . , ϵn), and all cuts p of length w ≤ n, we have that:
+s1 ∼ϵ s2 implies (s1, p) ∼ϵ\p (s2, p)
+Proof. It is sufficient to prove that for every composite parameter state ϵ, the following relation:
+Rϵ\p = {((s1, p), (s2, p)) | for all states s1, s2, s.t. (s1 ∼ϵ s2)}
+is a (ϵ\p)-bisimulation.
+Recall that ∥ is a commutative monoid, and thus it is closed under commutativity, associativity, and
+Id-element. Thus, the rest of the proof is by induction on the length (w) of the projection with respect to
+the history of the parameter TS. The key idea of the proof is that send actions of the form (c, !, υ) can
+only originate from within the composition, i.e., can be sent by s1 (or s2) or ϵ. Moreover, a receive action
+of the form (c, ?, υ) can only happen jointly with a corresponding send while the latter is autonomous.
+6
+Teamwork Synthesis
+Given an environment model E that specifies both aggregate context observations X and scheduled
+interactions on channels from Y , the execution assumption A automaton depicted in Fig. 1, a formula
+ϕ over the joint goal of the team within E (i.e., L (ϕ) ⊆ ((Y ∪ X) × O)ω), a set of agent interfaces
+{⟨Xk, Yk, Ok⟩}k∈K such that X is the set of aggregate observations of {Xk}k∈K, O is the set of aggregate
+actuation signals of {Ok}k∈K as defined in Sect. 2, and Y = �
+k Yk, a solution for teamwork Synthesis
+T = ⟨E ∩ A, ϕ, O⟩ is a set of |K| of Shadow TSs, one for each ⟨Xk, Yk, Ok⟩ such that T1∥ . . . ∥Tk |= ϕ
+under E ∩ A.
+We show that the teamwork synthesis problem can be reduced to a single-agent synthesis. The so-
+lution of the latter can be efficiently decomposed into a set of loosely coupled shadow TSs, where their
+composition is an equivalent implementation.
+Theorem 2. Teamwork Synthesis whose specification ϕ is a gr(1) formula [5] of the form θ ∧ Gφ ∧
+(�n
+i=1 GF λi → �m
+i=1 GF γi) can be solved with effort O(m.n.(|E ∩ A|.|O|)2), where |E ∩ A| is the number
+of transitions in E ∩ A.
+Proof. We construct ˆE = ⟨ ˆQ,
+ˆΣ, ˆΨ, ˆq0, ˆρ⟩ that extends E ∩ A = ⟨Q, Σ, Ψ, q0, ρ⟩ to include the
+set of aggregate signals in O. To simplify the notations, we freely use o ∈ O to mean the predicate that
+characterises it. The components of ˆE in relation to E ∩ A are:
+•
+ˆQ = Q,
+ˆq0 = q0,
+ˆΣ = Σ × O
+• We extend the interpretation function �·� to include variables in O. That is, �·� : ˆΨ → (Y ∪ X) × O
+• ˆρ =��(q0, θ ∧ θi, q)� �� (q0, θi, q) ∈ ρ �
+∪
+��(q, ψ ∧ o, q′)� �� q ̸= q0, (q, ψ, q′) ∈ ρ and o ∈ O �
+We use the construction above to construct a symbolic fairness-free ds [5]. We transform ˆE into an
+equivalent fairness-free symbolic discrete system ds D = ⟨Vd, ρd, θd⟩ in the obvious way. We use the
+variables X′ where |X′| = �
+k⌈log |Xk|⌉ to encode observations, the variables Y ′ where |Y ′| = log |Y | to
+encode channels, the variables O′ where |O′| = �
+k⌈log |Ok|⌉ to encode outputs, and the variable st2 to
+encode the states ˆQ. That is, D = ⟨Vd, ρd, θd⟩, where:
+• Vd = (X′ ∪ Y ′ ∪ O′ ∪ {st})
+• We define ρd(Vd, Vd
+′) which is a predicate on the current assignment to Vd in relation to the next
+assignment. We use the primed copy Vd
+′ to refer to the next assignment of Vd.
+ρd =
+�
+(q1,ψ1,q2), (q2,ψ2,q3)∈ρ′ ψ1 ∧ (ψ2)′ ∧ (st = q1) ∧ (st′ = q2)
+• θd = θ ∧ (st = q0)
+For a state (s, q) ∈ 2(X′∪Y ′∪O′) × Q where Q is the domain of st, we say that (s, q) |= v iff v ∈ s.
+We naturally generalise satisfaction to boolean combination of (Vd) and (Vd
+′). Now, given a Teamwork
+Synthesis problem T = ⟨D( ˆE), ϕ, O⟩, where D = ⟨Vd, ρd, θd⟩, ϕ is a gr(1) formula divided into a
+liveness assumption �n
+i=1 GF λi, a liveness goal �m
+i=1 GF γi, and a safety goal Gφ, we construct a gr(1)
+game G = ⟨V, X, O, θe, θs, ρe, ρs, ϕg⟩ as follows:
+2 To simplify the notation we will consider st to be a non-boolean variable.
+
+Correct-by-Design Teamwork Plans for Multi-Agent Systems
+9
+We use φ in the safety goal to prune any transition in ρd with condition on O that is in conflict with φ
+( it is unsafe). That is, we construct ρ′
+d from ρd by removing all transitions t ∈ ρd such that ¬(t → φ). The
+initial transitions from state q0 are not subject to check against φ. Clearly, (Xρ′
+d) → Gφ. Moreover, ρ′
+d
+also encodes the environment safety by definition. Now, our game is as follows: G = ⟨Vd, (X′ ∪Y ′), (O′ ∪
+{st}), true, θd, true, ρ′
+d, ϕg⟩, where ϕg = �n
+i=1 GF λi → �m
+i=1 GF γi
+To support response formulas of the form G(x → Fy) and general ltl safety formulas instead, the
+complexity is adjusted as follows: O((m+g).(n+a).(|E ∩ A|.|O|.|ϕ(s)|.2(a+g))2), where m, n are adjusted
+by adding the number of response assumptions a and response guarantees g while |ϕ(s)| is the size of the
+safety goal. This is because the disciplined environment model E ∩ A will be intersected with the safety
+goal and each response formula. A response formula can be encoded in a two-state automaton [23].
+The solution of the gr(1) game can be used to construct a Mealy machine with interface ⟨X, Y, O⟩
+as defined below:
+Definition 5 (Mealy Machine). A Mealy machine M is of the form M = ⟨Q, I, O, q0, δ⟩, where:
+• Q is the set of states of M and q0 ∈ Q is the initial state.
+• I = (X ∪ Y ) is an alphabet, partitioned into a set of aggregate sensor inputs X and a set of channels
+Y , and O is the aggregate output alphabet.
+• δ : Q × (Y ∪ X) → Q × O is the transition function of M.
+The language of M, denoted by LM, is the set of infinite sequences ((Y ∪X)×O)ω that M generates.
+We will use M to build a corresponding shadow TS. Namely, we construct a language equivalent
+shadow TS T with set of states S = δ as shown below. Note that the constructed mealy machine in the
+previous step has exactly one outgoing transition from the initial state q0, i.e., |(q0, (i, o), q) ∈ δ| = 1.
+This is ensured by the execution assumption A automaton depicted in Fig. 1.
+Lemma 2 (From Mealy to Shadow TS). Given the constructed Mealy machine M = ⟨Q, I, O, q0, δ⟩
+then we use a function f : δ → Act, a load Υ, and channels ch to construct a shadow TS T =
+⟨S, Int, Act, ∆e, ∆, L, ls, s0⟩ with |δ| many states, and LT = LM.
+Proof. We construct T as follows:
+• S = {(q, (i, o), q′) | (q, (i, o), q′) ∈ δ}
+• s0 = (q0, (i, o), q) for the unique q ∈ Q, s.t. (q0, (i, o), q) ∈ δ
+• L((q, (i, o), q′)) = (i, o)
+• Int = ⟨X, ch, O⟩ where X = {x : K �→ �
+k Xk |
+x(k) /∈ �
+j̸=k Xj}; ch ⊆ (2K)\{∅} ∪ Y , i.e., the
+maximal set of channels that agent may use to interact, where K is the set of agent identities; and
+O = {o : K �→ �
+k Ok | o(k) /∈ �
+j̸=k Oj}
+• ls(s) = ∅ for all s ∈ S and Act ⊆ (ch × {!} × Υ) where Υ ⊆ X ∪ {∅}. Note that Act is restricted to
+send messages.
+• ∆ =
+�
+�
+�
+�
+�
+(q, (i, o), q′),
+a,
+(q′, (i′, o′), q′′)
+�
+�
+����
+(q, (i, o), q′) ∈ δ, (q′, (i′, o′), q′′) ∈ δ
+and a = f((q′, (i′, o′), q′′))
+�
+�
+�
+• We use projx
+k(i) to project i on Xk, s.t. function f is:
+f((s, (i, o), s′)) =
+�
+(i, !, ∅)
+if i ∈ Y
+(ids, !, i)
+if i ∈ X, ids = {k | projx
+k(i) ̸= ⊥}
+• ∆e =
+��(q, (i, o), q′), i′ � ��(q, (i, o), q′), a, (q′, (i′, o′), q′′)) ∈ ∆ �
+It is not hard to see that LT = LM.
+Lemma 3 (Decomposition).
+A
+shadow
+TS
+T
+=
+⟨S,
+Int,
+Act,
+∆e, ∆, L, ls, s0⟩, as constructed in Lemma 2, can be decomposed into a set of TSs {Tk}k for k ∈ {1, n},
+s.t. T ∼0 (T1∥ . . . ∥Tn).
+Proof. We construct the components of each Tk as follows:
+• Sk = S,
+sk
+0 = s0
+• For each s ∈ S, (s, (c, !, υ), s′) ∈ ∆, and each Tk, we have that
+
+10
+Y. Abd Alrahman and N. Piterman
+1. if k ∈ c and |c| = 1 then (s, (c, !, υ), s′) ∈ ∆k
+2. if
+k
+∈
+c
+and
+|c|
+>
+1
+then
+(s, (c, !, υ), s′)
+∈
+∆k
+and
+(s, (c, ?, υ), s′)
+∈ ∆k
+3. if y = c for some y ∈ Yk then (s, (c, !, υ), s′) ∈ ∆k
+4. otherwise (s, (c, ?, υ), s′) ∈ ∆k
+• ∆k
+e = {(s, i) | (s, (c, !, υ), s′) ∈ ∆k, Lk(s′) = (i, o)}
+• Actk = {a | (s, a, s′) ∈ ∆k}
+• Lk(s) = projk(L(s)), i.e., the projection of L(s) on Agentk
+• lsk(s) = {c | (s, (c, ?, υ), s′) ∈ ∆k}
+It is sufficient to prove that s0 of T is equivalent to s′
+0 of the composition (T ′ = T1∥ . . . ∥Tn) under the
+empty parameter state 0, and that each Tk is indeed a shadow TS. The key idea of the proof is that
+the construction creates isomorphic shadow TSs that are fully synchronous. That is, they have the same
+states and transition structure, and only differ in state labelling, listening function, and transition role
+(send ! or receive ?). Thus, every send in ∆ of T is divided into a set of one (or more sends) ! and exactly
+(n − 1)-receives ?. In case of more than one send (as in item (2)), then for each Tk that implements such
+send there must be a matching receive with same source and target states. By Def. 3, we can reconstruct
+T ′ that is isomorphic to T with ls(s) = ch for all s ∈ S. However, we only need to prove T ∼0 T ′ under
+the O-parameter which cannot interact, and thus ls(s) is not important after constructing T ′.
+Lemma 3 provides an upper bound on number of communications each agent Tk must participate in
+within the team. We will use the E -Bisimulation in Sect. 5 to reduce such number with respect to the
+rest of the composition (or team). That is, given a set K = {1, . . . , n} of agents and for each agentk,
+we minimise the corresponding Tk with respect to the parameter T = ∥j∈(K\{k})Tj. Note that Lemma 3
+produces agents that are of the same size of the deterministic specification. Size wise, we recall that in
+Zielonka synthesis [31,9]) agents are exponential in the size of the deterministic specification. We will
+reduce this size even more by constructing the quotient shadow TS of Tk:
+Definition 6 (Quotient Shadow TS). For a shadow TS Tk = ⟨Sk, Intk, Actk, ∆k
+e, ∆k, Lk, lsk, sk
+0⟩,
+E the state-space of ∥
+j̸=k
+Tj, and E -bisimulation, the quotient shadow TS [Tk]∼ = ⟨S′
+k, Int′
+k,Act′
+k, ∆k���
+e , ∆′
+k, Lk, lsk′, sk′
+0 ⟩
+• S′
+k = {s∼ | s ∈ Sk} with s∼ = {s′ ∈ Sk | s ∼e s′ for e ∈ E }
+• sk′
+0 = sk
+0∼
+• ∆′
+k =��s∼, (c, !, υ), s′
+∼
+� �� (s, (c, !, υ), s′) ∈ ∆k
+�
+∪
+��s∼, (c, ?, υ), s′
+∼
+� �� s∼ ̸= s′
+∼, (s, (c, ?, υ), s′) ∈ ∆k
+�
+• ∆k′
+e = {(s, i) | (s, (c, !, υ), s′) ∈ ∆′
+k, Lk(s′) = (i, o)}
+• Act′
+k = {a | (s, a, s′) ∈ ∆′
+k}
+• lsk′(s) = {c | (s, (c, ?, υ), s′) ∈ ∆′
+k}
+The definition of E -Bisimulation in Sect. 5 can be easily converted to an algorithm (see [22]) that
+efficiently (almost linear) computes E -Bisimulation as the largest fixed point.
+Scenario Revisited The distributed realisation of the teamwork synthesis instance, in Sect. 4, is
+depicted in Fig. 2, where each arm is supplied with a shadow TS that represents its correct behaviour.
+For a shortcut, we only use the “first letter” of a channel name y ∈ Yk and the “first two letters” of an
+output letter name o ∈ Ok in the figures, e.g., we use the shortcuts f for fwd, rF for rFwd, etc.
+Note that every state of Tk for k ∈ {1, 2, 3} is labelled with input/output letter and a set of channels
+that Tk listens to in this state, e.g., T1 in Fig. 2 (i) initially has empty input/output letter (the initial
+state is labelled with ⊥/⊥) and is not listening to any channel (the listening function is initially ∅).
+Moreover, T2 and T3 in Fig. 2 (i) and (ii) are initially listening to channel f and respectively p.
+Transitions are labelled with either message send (!) or receive (?). For instance, T1 can initially send
+the message ({1}, !, in) on channel {1} independently and move alone to the next state in which T1 reads
+the letter (in) from the input-tray, and consequently signals its motor to get ready to forward, i.e., by
+signalling (rF). Recall that this is a shadow transition of the potential trigger of the environment from
+the initial state. This is akin to say that once T1 senses a trigger from the environment, it immediately
+
+Correct-by-Design Teamwork Plans for Multi-Agent Systems
+11
+(i) T1: The Tray Arm Agent
+(ii) T2: The processing Arm Agent
+(iii) T3: The Packaging Arm Agent
+Fig. 2: The distributed Realisation of the Product line
+permits it by providing a shadowing transition. Note that T2 and T3 are initially busy waiting to receive
+a message on f and respectively on p to kick start their executions.
+Clearly, message ({1}, !, in) on channel {1} is a strategic interaction that our synthesis engine added
+to ensure distributed realisability. Notice that T2 and T3 do not initially listen to {1} and they cannot
+observe the interaction on it, but later they will listen to it when they need (e.g., after (p, !, ∅)).
+By composition, as defined in Def. 3, initially T1 move independently, and from the next state T1 sends
+the message (f, !, ∅) in which T2 participates while T3 stays disconnected. Indeed, T3 only gets involved in
+the third step. Notice how the listening functions of these TSs change dynamically during execution, and
+allowing for loosely coupled distributed implementation. The latter has a feature that in every execution
+step one can send a message, and the others are either involved (i.e., they receive) or they cannot observe
+it (i.e., they do not listen).
+Recall that state labels are the elements of executions and the transition labels are complimented by
+the synthesis engine to ensure distributed realisability. As one can see, all TSs initially start from states
+that satisfy the initial condition θ. Indeed, there is no signal initially enabled. Moreover, the composite
+labelling of states in future execution steps satisfies the formula ϕ under the environment model E and
+the execution assumption A.
+Note that the machines in Fig. 2 is everything we need. That is, unlike supervisory control [27] where
+the centralised controller is finally composed with the environment model, and the composition is checked
+against the goal, we do not have such requirement. Indeed, the machines in Fig. 2 fully distribute the
+control.
+The results in this paper are unique, and aspire to unlock distributed synthesis for multi-agent systems
+for the first time.
+7
+Concluding Remarks
+We introduced teamwork synthesis which reformulates the original distributed synthesis problem [25,8]
+and casts it on teamwork multi-agent systems. Our synthesis technique relies on a flexible coordination
+model, named Shadow TS, that allow agents to co-exist and interact based on need, and thus limits the
+interaction to interested agents (or agents that require information to proceed).
+
+(0)
+L/dF
+(0)
+f/. 
+p?
+(p?
+(d.?0)
+in/R
+..0
+作/ 1
+(d.?D
+(p)
+(d)
+1d.7
+(p.?.O)
+(p)/cP
+L/P
+f.?.O
+Q.
+.U0
+10/
+L
+dt
+J儿/P
+/rD
+((1),?n)
+(d,)
+/p
+(f?0)
+/rD
+LrD
+(p)
+(f,?)
+O..
+/p)12
+Y. Abd Alrahman and N. Piterman
+Unlike the existing distributed synthesis problems, our formulation is decidable, and can be reduced to
+a single-agent synthesis. We efficiently decompose the solution of the latter and minimise it for individual
+agents using a novel notion of parametric bisimulation. We minimise both the state space and the set
+of interactions each agent requires to fulfil its goals. The rationale behind teamwork synthesis is that we
+reformulate the original synthesis question by dropping the fixed interaction architecture among agents as
+input to the problem. Instead, our synthesis engine tries to realise the goal given the initial specifications;
+otherwise it automatically introduces minimal interactions among agents to ensure distributed realisabil-
+ity. Teamwork synthesis shows algorithmically how agents should interact so that each is well-informed
+and fulfils its goal.
+Related works We report on related works with regards to concurrency models used for distributed
+synthesis, bisimulation relations, and also other formulations of distributed synthesis.
+Shadow TS adopts the reconfigurable semantics approach from CTS [3,4,2], but it is actually weaker in
+terms of synchronisation. Indeed, the requirement in Property 1 lifts out the blocking nature of multicast,
+and thus the semantics of the Shadow TS is reduced to a local broadcast. That is, message sending can no
+longer be blocked, and is broadcasted on local channels rather than a unique public channel ⋆ (broadcast
+to all) as in CTS [3]. The advantage is that the semantics of Shadow TS is asynchronous and no agent can
+force other agents to wait for it. It is definitely weaker than shared memory models as in [25,8] and it is
+also weaker than the synchronous automata of supervision [27] and Zielonka automata [31,9]. Note that
+the last two adopt the multi-way synchronisation (or blocking rendezvous) of Hoare’s CSP calculus [11].
+Thus, the synchronisation dependencies are lifted out in our model. Intuitively, an agent can, at most,
+block itself to wait for a message from another agent, but in no way can block the executions of others
+unwillingly.
+Our notion of bisimulation in Def. 4 is novel with respect to existing literatures on bisimulation [6,19,28].
+To the best of our knowledge, it is the only bisimulation that is able to abstract actual messages, and
+thus reduce synchronisations. It treats receive transitions in a sophisticated way that allows it to judge
+when a receive or a discard transition can be abstracted safely. It has a branching nature like in [10], but
+is stronger because the former cannot distinguish different τ transitions. It is parametric like in [15], but
+is weaker in that it can abstract actual receive transitions.
+When it comes to distributed synthesis, there is a plethora of formulations. Here, we only relate to the
+ones that consider hostile environments. These are: Distributed synthesis [25,8], Zielonka synthesis [31,9],
+and Decentralised supervision
+[30]. Unlike teamwork synthesis, all are, in general, undecidable except
+for specific configurations (mostly with a tower of exponentials [14,16]). Zielonka synthesis is decidable if
+synchronising agents are allowed to share their entire state, and this produces agents that are exponential
+in the size of the joint deterministic specification. Teamwork synthesis produces agents that are, in the
+worst case, the size of the joint deterministic specification.
+Future works We want to generalise the execution assumption A of teamwork synthesis depicted in
+Fig. 1 to a more balanced scheduling between interaction events Y and context events X, inspired by RTC
+control [1]. That is, we want to provide a more relaxed built-in transfer of control between the interaction
+and the context events. The latter would majorly simplify writing specifications. We want also to extend
+the Shadow TS to allow multithreaded agents, and thus eliminates interaction among co-located threads.
+Clearly, the positive results in this paper makes it feasible to provide tool support for Teamwork
+synthesis, and with a more user-friendly interface.
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+

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@@ -0,0 +1,557 @@
+Citation
+R. Benkert, M. Prabhushankar, and G. AlRegib, “Forgetful Active Learning With Switch Events: Efficient Sampling for
+Out-of-Distribution Data,” in IEEE International Conference on Image Processing (ICIP), Bordeaux, France, Oct. 16-19
+2022
+Review
+Date of acceptance: June 2022
+Bib
+@ARTICLE{benkert2022 ICIP,
+author={R. Benkert, M. Prabhushankar, and G. AlRegib},
+journal={IEEE International Conference on Image Processing},
+title={Forgetful Active Learning With Switch Events: Efficient Sampling for Out-of-Distribution Data},
+year={2022}
+Copyright
+©2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses,
+in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,
+creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of
+this work in other works.
+Contact
+[email protected] OR [email protected]
+http://ghassanalregib.info/
+arXiv:2301.05106v1  [cs.LG]  12 Jan 2023
+
+FORGETFUL ACTIVE LEARNING WITH SWITCH EVENTS: EFFICIENT SAMPLING FOR
+OUT-OF-DISTRIBUTION DATA
+Ryan Benkert, Mohit Prabhushankar, and Ghassan AlRegib
+OLIVES at the Center for Signal and Information Processing,
+School of Electrical and Computer Engineering,
+Georgia Institute of Technology,
+Atlanta, GA, 30332-0250, USA
+{rbenkert3, mohit.p, alregib}@gatech.edu
+ABSTRACT
+This paper considers deep out-of-distribution active learning.
+In practice, fully trained neural networks interact randomly
+with out-of-distribution (OOD) inputs and map aberrant sam-
+ples randomly within the model representation space. Since
+data representations are direct manifestations of the training
+distribution, the data selection process plays a crucial role in
+outlier robustness.
+For paradigms such as active learning,
+this is especially challenging since protocols must not only
+improve performance on the training distribution most effec-
+tively but further render a robust representation space. How-
+ever, existing strategies directly base the data selection on the
+data representation of the unlabeled data which is random
+for OOD samples by definition. For this purpose, we intro-
+duce forgetful active learning with switch events (FALSE) -
+a novel active learning protocol for out-of-distribution active
+learning. Instead of defining sample importance on the data
+representation directly, we formulate ”informativeness” with
+learning difficulty during training. Specifically, we approxi-
+mate how often the network “forgets” unlabeled samples and
+query the most “forgotten” samples for annotation. We report
+up to 4.5% accuracy improvements in over 270 experiments,
+including four commonly used protocols, two OOD bench-
+marks, one in-distribution benchmark, and three different ar-
+chitectures.
+Index Terms— Active Learning, Forgetting Events, Out-
+of-Distribution.
+1. INTRODUCTION
+A major issue in neural network deployment is their sensitiv-
+ity to unknown inputs [1, 2, 3, 4]. Within the context of deep
+learning, unknown inputs typically refer to samples originat-
+ing from acquisition setups that significantly differ from the
+training environment. Due to the probabilistic nature of neu-
+ral network predictions, samples originating from the training
+environment are called in-distribution while unknown inputs
+are called out-of-distribution, or OOD in short. In practice,
+OOD samples result in unpredictable or even random predic-
+tions and represent a major challenge for real-world deploy-
+ment [5, 6]. The root cause of the perceived unpredictabil-
+ity lies within the structure of the model representation [7].
+During training, the structure is established by imposing con-
+straints (through e.g. a loss function) on known training sam-
+ples and the training distribution is mapped to distinguish-
+able targets. During test phase, the model maps known in-
+distribution samples to their respective targets if they origi-
+nate from the same training distribution. In contrast, OOD
+samples are not constrained during training and the target rep-
+resentation is undefined. This results in unpredictable behav-
+ior at the test phase.
+Fig. 1. Toy example of out-of-distribution (OOD) samples
+within an active learning setting. In both active learning pro-
+tocols the model exhibits similar test performance. However,
+the accuracy on the out-of-distribution samples significantly
+improves when the training set is selected with a different pro-
+tocol.
+
+A field in which OOD properties are especially impor-
+tant is active learning. Active learning is a machine learning
+paradigm in which the model iteratively selects the training
+set from an unlabeled data pool to improve annotation effi-
+ciency. Due to its intuitive practicality, active learning is es-
+pecially popular in applications where annotations are costly
+and has been successfully deployed in various industrial sec-
+tors [8]. In active learning, out-of-distribution performance
+is especially relevant because it is not directly apparent from
+the target performance. For instance, a model may perform
+significantly better on OOD data if the training set contains
+samples that impose constraints on outliers (Figure 1). For
+this purpose, active learning protocols must not only maxi-
+mize performance on in-distribution data but further insure
+robustness on outlier distributions.
+Despite the high relevance for the field, existing active
+learning approaches rarely consider OOD settings. In fact,
+most strategies select the next set of training samples based
+on importance metrics directly derived from the model rep-
+resentation. Since model representations for OOD data are
+undefined by definition, this trait poses a clear inefficiency
+for OOD test performance. For this purpose, we introduce
+FALSE - a novel query strategy that not only shows favor-
+able performance on in-distribution data but is further robust
+to OOD samples.
+Instead of directly defining sample im-
+portance on model representations directly, we define impor-
+tance with statistics gathered from the model during training.
+Specifically, we approximate how often unlabeled samples
+are “forgotten” by the model and consider the “most forgot-
+ten” samples as the most informative. We benchmark our
+method against four commonly used active learning proto-
+cols on two popular OOD benchmarks, as well as one in-
+distribution benchmark with three architectures. Overall, we
+note a significant improvement in terms of test accuracy on
+in-distribution, as well as out-of-distribution data.
+2. BACKGROUND AND RELATED WORK
+2.1. Active learning
+In active learning [9], the goal is to maximize generalization
+performance with minimal data annotations. We consider a
+dataset D, an initial training set Dtrain, as well as an unla-
+beled data pool Dpool from which we select the next sam-
+ple batch for the training set. For batch active learning, the
+algorithm selects a batch of b samples X∗ = {x∗
+1, ..., x∗
+b}
+that are the most informative based on a predefined defini-
+tion. The selection of X∗ is called the query strategy and typ-
+ically queries the batch that maximizes the acquisition func-
+tion a(x1, ..., xb|fw), where fw represents the deep neural
+network with parameters w. In active learning, the definition
+of importance is encoded within the acquisition function a.
+In this regard, typical approaches involve querying samples
+that are the most difficult for generalization [10, 11], maxi-
+mize data diversity [12, 13], or perform a mixture of general-
+ization difficulty and data diversity [14, 15]. Even though a
+large variety of strategies exist, they define sample importance
+on the representation manifold directly. Since representations
+are notoriously unpredictable for OOD data this characteris-
+tic represents a clear inefficiency. Apart from rare exceptions
+[16], out-of-distribution active learning settings are not con-
+sidered in existing literature.
+2.2. Forgetting Events
+Our approach most closely relates to the study of continual
+learning or more specifically catastrophic forgetting [17, 18,
+19, 20]. While several papers study forgetting in the con-
+text of different target tasks [18, 19], other papers focus on
+forgetting within a single task [17]. Within the context of
+active learning, we define informativeness with learning dif-
+ficulty. Specifically, we say a sample is informative if it was
+“forgotten” frequently during the training process and there-
+fore difficult to learn. Within the context of deep learning, a
+sample is considered “forgotten” if it was correctly classified
+(“learned”) at time t and misclassified (“forgotten”) at a later
+time t′ > t. More formally, we consider recognition tasks
+where the model calculates a prediction ˜yi for each sample xi
+where the model prediction is correct when ˜yi is equal to the
+ground truth label yi. The accuracy of a sample at epoch t can
+be expressed as
+ct
+i = 1˜yt
+i=yi.
+(1)
+Here, 1˜yt
+i=yi represents a binary variable that reduces to
+one if the model prediction is correct and zero otherwise.
+We further define a sample as “forgotten” if the accuracy de-
+creases in two subsequent epochs:
+et
+i = int(ct
+i < ct−1
+i
+) ∈ 1, 0
+(2)
+Similar to [17], we call the binary event et
+i a forgetting
+event. Within the context of active learning, we quantify in-
+formativeness by the amount of forgetting events that occur
+for a given sample each active learning round. In contrast to
+existing literature, our definition of sample importance is not
+directly dependent on the data representation of the model but
+relies on artifacts from representation shifts. We reason that
+this characteristic is potentially favorable for out-of distribu-
+tion active learning.
+3. FORGETFUL ACTIVE LEARNING WITH
+SWITCH EVENTS
+Even though forgetting events are simple to formulate and do
+not rely on the representation directly, the computation is not
+tractable in practice. Speci��cally, the accuracy evaluation in
+Equation 1 requires labels which are not present in Dpool by
+definition. To alleviate this issue, we approximate forgetting
+
+Fig. 2. Workflow of FALSE. Within the active learning round N, we gather switch events for every samples in Dpool and
+aggregate them over all epochs K. Subsequently. we query b samples with the most switch events.
+events with prediction switches. A prediction switch occurs
+when the prediction of the model changes between two sub-
+sequent epochs. Formally, we write
+st
+i = int(˜yt
+i ̸= ˜yt−1
+i
+) ∈ 1, 0.
+(3)
+Similar to our previous definition, we call the binary event
+st
+i a switch event at time t. Since samples with a larger amount
+of switch events are considered more informative, we select
+the batch X∗ with the most switch events in each round:
+X∗ =
+argmax
+x1,...,xb∈Dpool
+Σb
+i=1ΣK
+t=1st
+i
+(4)
+Here, K refers to the total amount of epochs during an
+active learning round. We call our method Forgetful Active
+Learning with Switch Events or FALSE in short. We show
+the workflow during each active learning round in Figure 2.
+First, we count the switch events for every sample in Dpool
+occurring within round N. Subsequently, we select b samples
+with the most switch events and append them to the training
+set Dtrain.
+4. EXPERIMENTS
+4.1. Experimental Setup
+We compare FALSE to four popular strategies in active learn-
+ing literature: Entropy sampling [10], coreset [12], least con-
+fidence sampling [10], and active learning by learning [15].
+We choose this constellation as it contains two strategies
+based on generalization difficulty (entropy and least confi-
+dence sampling), one based on data diversity (coreset), and
+a hybrid protocol that combines generalization difficulty and
+Algorithms
+CIFAR10-C
+CINIC10
+CIFAR10
+ResNet-18
+FALSE
+21.85
+18.06
+21.80
+ALBL [15]
+10.19
+5.66
+7.25
+Coreset [12]
+-41.87
+-35.07
+-41.04
+Entropy [10]
+7.83
+8.58
+0.45
+L. Conf. [10]
+16.23
+13.47
+9.93
+DenseNet-121
+FALSE
+13.92
+16.50
+15.47
+ALBL [15]
+7.69
+6.67
+3.28
+Coreset [12]
+-23.40
+-14.36
+-21.74
+Entropy [10]
+1.77
+2.52
+-7.76
+L. Conf. [10]
+-0.20
+3.81
+-3.57
+ResNet-34
+FALSE
+22.20
+23.30
+27.45
+ALBL [15]
+7.56
+9.67
+13.96
+Coreset [12]
+-8.25
+-7.18
+-5.21
+Entropy [10]
+14.80
+15.31
+12.51
+L. Conf. [10]
+17.52
+16.60
+15.99
+Table 1. Area under difference curve with reference to ran-
+dom sampling over several datasets, query strategies, and ar-
+chitectures. Positive, and higher numbers are better. Negative
+numbers imply that the strategy has a lower accuracy curve
+than the random baseline.
+data diversity by switching between the coreset and least con-
+fidence strategy (active learning by learning). For our OOD
+experiments, we consider CIFAR10-C [1] (a corrupted ver-
+sion of CIFAR10), as well as CINIC10 [21] (a larger dataset
+with the same classes as CIFAR10). In addition, we validate
+on the CIFAR10 in-distribution test set. In all experiments,
+we train on the CIFAR10 training set. For CIFAR10-C, we
+test on all corruptions except ”labels”, ”shot noise”, and
+”speckle noise”, and consider the difficulty levels 2 and 5.
+
+st = int(yt ±yt-1)Fig. 3. FALSE in comparison to different query strategies on the level 2 CIFAR10-C [1] corruptions “JPEG Compression”,
+“Elastic Transform”, and “Implulse Noise”. We compare FALSE to active learning by learning (ALBL), coreset, least confi-
+dence sampling (Least Conf.), and entropy sampling (Entropy).
+We choose this setup as it considers 17 realistic corruptions at
+an intermediate, as well as high difficulty. For CINIC10, we
+test on the CINIC10 test set. We start with an initial training
+pool size of randomly chosen 128 samples and query 1024
+samples each round. Since query strategy performance can
+be sensitive to architecture choices, we perform our experi-
+ments on resnet-18, resnet-34, as well as densenet-121. We
+optimize with the adam variant of SGD with a learning rate of
+1e − 4 and train our models until a training accuracy of 98%
+is reached. Furthermore, we use pretrained model weights
+each round as this represents a realistic practical scenario
+where data may be redundant in one domain but scarce in an-
+other. Finally, we retrain our model form scratch each round
+to prevent warm starting [22]. Due to space limitations, we
+show the learning curves of three level 2 corruptions of the
+CIFAR10-C dataset in Figure 3. Additionally, we summarize
+the performance of each protocol over the first 20 rounds by
+calculating the curve distance to randomly selecting samples
+each round (Table 1). For each experiment constellation, we
+subtract the accuracy curve of the respective strategy from
+the random selection baseline curve and calculate the area
+under the difference curve. In other words, the table shows
+the distance of each strategy to randomly selecting samples
+each round.
+A negative result implies that the strategy is
+overall worse than random selection, while a higher score
+means that the respective strategy outperforms random selec-
+tion by a larger margin over the first 20 rounds. All results
+are averaged over five random seeds. In total, this amounts to
+270 separate active learning experiments.
+4.2. Discussion
+From both Figure 3 and Table 1 we observe that FALSE is
+a good choice for OOD active learning.
+In particular, we
+observe that FALSE is especially favorable in early rounds
+where few labeled data samples are available for training.
+We reason that the lack of training samples results in insuffi-
+ciently calibrated representations and the importance rankings
+are more inaccurate when defined on the representation di-
+rectly. For later rounds, enough training samples are available
+to learn a more conclusive representation and FALSE outper-
+forms by a smaller margin (Figure 3). On average, we observe
+up to 4.5% improvements in accuracy across existing strate-
+gies. We further note, that FALSE is more robust to different
+datasets and architectures for both in-distribution and OOD
+experiments.
+In Table 1, we see that FALSE consistently
+outperforms random selection by a large margin over differ-
+ent datasets and architectures. This is evident in the consis-
+tent large area under the difference curve. In contrast several
+strategies, outperform random selection on some datasets or
+architectures but match or underperform the baseline in other
+settings. For instance, least confidence sampling exhibits a
+high area score on CIFAR10-C when resnet-34 is used, but
+slightly underperforms the random baseline with densenet-
+121. Furthermore, entropy sampling performs well on the in-
+distribution test set when resnet-34 is used but underperforms
+random by a significant margin when densenet-121 is used.
+We further observe, that FALSE also shows the strongest per-
+formance overall. This is evident in the largest area across all
+settings in Table 1.
+5. CONCLUSION
+In this paper, we introduced FALSE as a novel query strat-
+egy for OOD active learning. Specifically, we derived ”infor-
+mativeness” from the learning dynamics during training time.
+We approximated how often the network ”forgot” unlabeled
+samples by counting switch events during each active learn-
+ing round. We empirically analyzed our method with exhaus-
+tive experiments, and noted a clear improvement over existing
+methods in in-distribution, and out-of-distribution settings.
+
+60
+55
+50
+FALSE
+FALSE
+FALSE
+ALBL
+ALBL
+ALBL
+55 
+50 
+Coreset
+45 
+Coreset
+Coreset
+Entropy
+Entropy
+Entropy
+50 
+Least Conf.
+45
+Least Conf.
+Least Conf.
+40
+45 
+Accuracy
+35
+35
+30
+30
+30 
+25
+25
+25
+20
+20
+20
+1000
+2000
+3000
+4000
+5000
+6000
+1000
+2000
+3000
+4000
+5000
+6000
+1000
+2000
+3000
+4000
+5000
+6000
+Samples
+Samples
+Samples6. REFERENCES
+[1] Dan Hendrycks and Thomas Dietterich, “Benchmarking
+neural network robustness to common corruptions and
+perturbations,” Proceedings of the International Con-
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+[2] Gukyeong Kwon, Mohit Prabhushankar, Dogancan
+Temel, and Ghassan AlRegib,
+“Backpropagated gra-
+dient representations for anomaly detection,” in Euro-
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+pp. 206–226.
+[3] Jinsol Lee and Ghassan AlRegib, “Gradients as a mea-
+sure of uncertainty in neural networks,” in 2020 IEEE
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+[4] Gukyeong Kwon, Mohit Prabhushankar, Dogancan
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+ing unreal and real environments for traffic sign recog-
+nition,” arXiv preprint arXiv:1712.02463, 2017.
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+abs/1910.08475, 2019.
+

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+page_content='Citation  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Benkert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Prabhushankar, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' AlRegib, “Forgetful Active Learning With Switch Events: Efficient Sampling for  Out-of-Distribution Data,” in IEEE International Conference on Image Processing (ICIP), Bordeaux, France, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' 16-19  2022  Review  Date of acceptance: June 2022  Bib  @ARTICLE{benkert2022 ICIP,  author={R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Benkert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Prabhushankar, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' AlRegib},  journal={IEEE International Conference on Image Processing},  title={Forgetful Active Learning With Switch Events: Efficient Sampling for Out-of-Distribution Data},  year={2022}  Copyright  ©2022 IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Personal use of this material is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Permission from IEEE must be obtained for all other uses,  in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,  creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of  this work in other works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  Contact  rbenkert3@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='edu OR alregib@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='edu  http://ghassanalregib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='info/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='05106v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='LG]  12 Jan 2023  FORGETFUL ACTIVE LEARNING WITH SWITCH EVENTS: EFFICIENT SAMPLING FOR  OUT-OF-DISTRIBUTION DATA  Ryan Benkert, Mohit Prabhushankar, and Ghassan AlRegib  OLIVES at the Center for Signal and Information Processing,  School of Electrical and Computer Engineering,  Georgia Institute of Technology,  Atlanta, GA, 30332-0250, USA  {rbenkert3, mohit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='p, alregib}@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='edu  ABSTRACT  This paper considers deep out-of-distribution active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  In practice, fully trained neural networks interact randomly  with out-of-distribution (OOD) inputs and map aberrant sam-  ples randomly within the model representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Since  data representations are direct manifestations of the training  distribution, the data selection process plays a crucial role in  outlier robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  For paradigms such as active learning,  this is especially challenging since protocols must not only  improve performance on the training distribution most effec-  tively but further render a robust representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' How-  ever, existing strategies directly base the data selection on the  data representation of the unlabeled data which is random  for OOD samples by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' For this purpose, we intro-  duce forgetful active learning with switch events (FALSE) -  a novel active learning protocol for out-of-distribution active  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Instead of defining sample importance on the data  representation directly, we formulate ”informativeness” with  learning difficulty during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Specifically, we approxi-  mate how often the network “forgets” unlabeled samples and  query the most “forgotten” samples for annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' We report  up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='5% accuracy improvements in over 270 experiments,  including four commonly used protocols, two OOD bench-  marks, one in-distribution benchmark, and three different ar-  chitectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  Index Terms— Active Learning, Forgetting Events, Out-  of-Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' INTRODUCTION  A major issue in neural network deployment is their sensitiv-  ity to unknown inputs [1, 2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Within the context of deep  learning, unknown inputs typically refer to samples originat-  ing from acquisition setups that significantly differ from the  training environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Due to the probabilistic nature of neu-  ral network predictions, samples originating from the training  environment are called in-distribution while unknown inputs  are called out-of-distribution, or OOD in short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' In practice,  OOD samples result in unpredictable or even random predic-  tions and represent a major challenge for real-world deploy-  ment [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' The root cause of the perceived unpredictabil-  ity lies within the structure of the model representation [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  During training, the structure is established by imposing con-  straints (through e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' a loss function) on known training sam-  ples and the training distribution is mapped to distinguish-  able targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' During test phase, the model maps known in-  distribution samples to their respective targets if they origi-  nate from the same training distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' In contrast, OOD  samples are not constrained during training and the target rep-  resentation is undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' This results in unpredictable behav-  ior at the test phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Toy example of out-of-distribution (OOD) samples  within an active learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' In both active learning pro-  tocols the model exhibits similar test performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' However,  the accuracy on the out-of-distribution samples significantly  improves when the training set is selected with a different pro-  tocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  A field in which OOD properties are especially impor-  tant is active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Active learning is a machine learning  paradigm in which the model iteratively selects the training  set from an unlabeled data pool to improve annotation effi-  ciency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Due to its intuitive practicality, active learning is es-  pecially popular in applications where annotations are costly  and has been successfully deployed in various industrial sec-  tors [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' In active learning, out-of-distribution performance  is especially relevant because it is not directly apparent from  the target performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' For instance, a model may perform  significantly better on OOD data if the training set contains  samples that impose constraints on outliers (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' For  this purpose, active learning protocols must not only maxi-  mize performance on in-distribution data but further insure  robustness on outlier distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  Despite the high relevance for the field, existing active  learning approaches rarely consider OOD settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' In fact,  most strategies select the next set of training samples based  on importance metrics directly derived from the model rep-  resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Since model representations for OOD data are  undefined by definition, this trait poses a clear inefficiency  for OOD test performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' For this purpose, we introduce  FALSE - a novel query strategy that not only shows favor-  able performance on in-distribution data but is further robust  to OOD samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  Instead of directly defining sample im-  portance on model representations directly, we define impor-  tance with statistics gathered from the model during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  Specifically, we approximate how often unlabeled samples  are “forgotten” by the model and consider the “most forgot-  ten” samples as the most informative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' We benchmark our  method against four commonly used active learning proto-  cols on two popular OOD benchmarks, as well as one in-  distribution benchmark with three architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Overall, we  note a significant improvement in terms of test accuracy on  in-distribution, as well as out-of-distribution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' BACKGROUND AND RELATED WORK  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Active learning  In active learning [9], the goal is to maximize generalization  performance with minimal data annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' We consider a  dataset D, an initial training set Dtrain, as well as an unla-  beled data pool Dpool from which we select the next sam-  ple batch for the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' For batch active learning, the  algorithm selects a batch of b samples X∗ = {x∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=', x∗  b}  that are the most informative based on a predefined defini-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' The selection of X∗ is called the query strategy and typ-  ically queries the batch that maximizes the acquisition func-  tion a(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=', xb|fw), where fw represents the deep neural  network with parameters w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' In active learning, the definition  of importance is encoded within the acquisition function a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  In this regard, typical approaches involve querying samples  that are the most difficult for generalization [10, 11], maxi-  mize data diversity [12, 13], or perform a mixture of general-  ization difficulty and data diversity [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Even though a  large variety of strategies exist, they define sample importance  on the representation manifold directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Since representations  are notoriously unpredictable for OOD data this characteris-  tic represents a clear inefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Apart from rare exceptions  [16], out-of-distribution active learning settings are not con-  sidered in existing literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Forgetting Events  Our approach most closely relates to the study of continual  learning or more specifically catastrophic forgetting [17, 18,  19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' While several papers study forgetting in the con-  text of different target tasks [18, 19], other papers focus on  forgetting within a single task [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Within the context of  active learning, we define informativeness with learning dif-  ficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Specifically, we say a sample is informative if it was  “forgotten” frequently during the training process and there-  fore difficult to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Within the context of deep learning, a  sample is considered “forgotten” if it was correctly classified  (“learned”) at time t and misclassified (“forgotten”) at a later  time t′ > t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' More formally, we consider recognition tasks  where the model calculates a prediction ˜yi for each sample xi  where the model prediction is correct when ˜yi is equal to the  ground truth label yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' The accuracy of a sample at epoch t can  be expressed as  ct  i = 1˜yt  i=yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  (1)  Here, 1˜yt  i=yi represents a binary variable that reduces to  one if the model prediction is correct and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  We further define a sample as “forgotten” if the accuracy de-  creases in two subsequent epochs:  et  i = int(ct  i < ct−1  i  ) ∈ 1, 0  (2)  Similar to [17], we call the binary event et  i a forgetting  event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Within the context of active learning, we quantify in-  formativeness by the amount of forgetting events that occur  for a given sample each active learning round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' In contrast to  existing literature, our definition of sample importance is not  directly dependent on the data representation of the model but  relies on artifacts from representation shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' We reason that  this characteristic is potentially favorable for out-of distribu-  tion active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' FORGETFUL ACTIVE LEARNING WITH  SWITCH EVENTS  Even though forgetting events are simple to formulate and do  not rely on the representation directly, the computation is not  tractable in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Specifically, the accuracy evaluation in  Equation 1 requires labels which are not present in Dpool by  definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' To alleviate this issue, we approximate forgetting  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Workflow of FALSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Within the active learning round N, we gather switch events for every samples in Dpool and  aggregate them over all epochs K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' we query b samples with the most switch events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  events with prediction switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' A prediction switch occurs  when the prediction of the model changes between two sub-  sequent epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Formally, we write  st  i = int(˜yt  i ̸= ˜yt−1  i  ) ∈ 1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  (3)  Similar to our previous definition, we call the binary event  st  i a switch event at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Since samples with a larger amount  of switch events are considered more informative, we select  the batch X∗ with the most switch events in each round:  X∗ =  argmax  x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=',xb∈Dpool  Σb  i=1ΣK  t=1st  i  (4)  Here, K refers to the total amount of epochs during an  active learning round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' We call our method Forgetful Active  Learning with Switch Events or FALSE in short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' We show  the workflow during each active learning round in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  First, we count the switch events for every sample in Dpool  occurring within round N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Subsequently, we select b samples  with the most switch events and append them to the training  set Dtrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' EXPERIMENTS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Experimental Setup  We compare FALSE to four popular strategies in active learn-  ing literature: Entropy sampling [10], coreset [12], least con-  fidence sampling [10], and active learning by learning [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  We choose this constellation as it contains two strategies  based on generalization difficulty (entropy and least confi-  dence sampling), one based on data diversity (coreset), and  a hybrid protocol that combines generalization difficulty and  Algorithms  CIFAR10-C  CINIC10  CIFAR10  ResNet-18  FALSE  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='85  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='06  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='80  ALBL [15]  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='19  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='66  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='25  Coreset [12]  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='87  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='07  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='04  Entropy [10]  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='83  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='45  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' [10]  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='23  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='47  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='93  DenseNet-121  FALSE  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='92  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='50  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='47  ALBL [15]  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='69  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='67  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='28  Coreset [12]  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='40  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='36  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='74  Entropy [10]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='77  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='52  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='76  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' [10]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='81  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='57  ResNet-34  FALSE  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='20  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='30  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='45  ALBL [15]  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='56  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='67  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='96  Coreset [12]  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='25  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='18  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='21  Entropy [10]  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='80  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='31  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='51  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' [10]  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='52  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='60  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='99  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Area under difference curve with reference to ran-  dom sampling over several datasets, query strategies, and ar-  chitectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Positive, and higher numbers are better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Negative  numbers imply that the strategy has a lower accuracy curve  than the random baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  data diversity by switching between the coreset and least con-  fidence strategy (active learning by learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' For our OOD  experiments, we consider CIFAR10-C [1] (a corrupted ver-  sion of CIFAR10), as well as CINIC10 [21] (a larger dataset  with the same classes as CIFAR10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' In addition, we validate  on the CIFAR10 in-distribution test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' In all experiments,  we train on the CIFAR10 training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' For CIFAR10-C, we  test on all corruptions except ”labels”, ”shot noise”, and  ”speckle noise”, and consider the difficulty levels 2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  st = int(yt ±yt-1)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' FALSE in comparison to different query strategies on the level 2 CIFAR10-C [1] corruptions “JPEG Compression”,  “Elastic Transform”, and “Implulse Noise”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' We compare FALSE to active learning by learning (ALBL), coreset, least confi-  dence sampling (Least Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' ), and entropy sampling (Entropy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  We choose this setup as it considers 17 realistic corruptions at  an intermediate, as well as high difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' For CINIC10, we  test on the CINIC10 test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' We start with an initial training  pool size of randomly chosen 128 samples and query 1024  samples each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Since query strategy performance can  be sensitive to architecture choices, we perform our experi-  ments on resnet-18, resnet-34, as well as densenet-121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' We  optimize with the adam variant of SGD with a learning rate of  1e − 4 and train our models until a training accuracy of 98%  is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Furthermore, we use pretrained model weights  each round as this represents a realistic practical scenario  where data may be redundant in one domain but scarce in an-  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Finally, we retrain our model form scratch each round  to prevent warm starting [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Due to space limitations, we  show the learning curves of three level 2 corruptions of the  CIFAR10-C dataset in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Additionally, we summarize  the performance of each protocol over the first 20 rounds by  calculating the curve distance to randomly selecting samples  each round (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' For each experiment constellation, we  subtract the accuracy curve of the respective strategy from  the random selection baseline curve and calculate the area  under the difference curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' In other words, the table shows  the distance of each strategy to randomly selecting samples  each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  A negative result implies that the strategy is  overall worse than random selection, while a higher score  means that the respective strategy outperforms random selec-  tion by a larger margin over the first 20 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' All results  are averaged over five random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' In total, this amounts to  270 separate active learning experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Discussion  From both Figure 3 and Table 1 we observe that FALSE is  a good choice for OOD active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  In particular, we  observe that FALSE is especially favorable in early rounds  where few labeled data samples are available for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  We reason that the lack of training samples results in insuffi-  ciently calibrated representations and the importance rankings  are more inaccurate when defined on the representation di-  rectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' For later rounds, enough training samples are available  to learn a more conclusive representation and FALSE outper-  forms by a smaller margin (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' On average, we observe  up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='5% improvements in accuracy across existing strate-  gies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' We further note, that FALSE is more robust to different  datasets and architectures for both in-distribution and OOD  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  In Table 1, we see that FALSE consistently  outperforms random selection by a large margin over differ-  ent datasets and architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' This is evident in the consis-  tent large area under the difference curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' In contrast several  strategies, outperform random selection on some datasets or  architectures but match or underperform the baseline in other  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' For instance, least confidence sampling exhibits a  high area score on CIFAR10-C when resnet-34 is used, but  slightly underperforms the random baseline with densenet-  121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Furthermore, entropy sampling performs well on the in-  distribution test set when resnet-34 is used but underperforms  random by a significant margin when densenet-121 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  We further observe, that FALSE also shows the strongest per-  formance overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' This is evident in the largest area across all  settings in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' CONCLUSION  In this paper, we introduced FALSE as a novel query strat-  egy for OOD active learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Specifically, we derived ”infor-  mativeness” from the learning dynamics during training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  We approximated how often the network ”forgot” unlabeled  samples by counting switch events during each active learn-  ing round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' We empirically analyzed our method with exhaus-  tive experiments, and noted a clear improvement over existing  methods in in-distribution, and out-of-distribution settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  60  55  50  FALSE  FALSE  FALSE  ALBL  ALBL  ALBL  55  50  Coreset  45  Coreset  Coreset  Entropy  Entropy  Entropy  50  Least Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  45  Least Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  Least Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  40  45  Accuracy  35  35  30  30  30  25  25  25  20  20  20  1000  2000  3000  4000  5000  6000  1000  2000  3000  4000  5000  6000  1000  2000  3000  4000  5000  6000  Samples  Samples  Samples6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' REFERENCES  [1] Dan Hendrycks and Thomas Dietterich, “Benchmarking  neural network robustness to common corruptions and  perturbations,” Proceedings of the International Con-  ference on Learning Representations, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content='  [2] Gukyeong Kwon, Mohit Prabhushankar, Dogancan  Temel, and Ghassan AlRegib,  “Backpropagated gra-  dient representations for anomaly detection,” in Euro-  pean Conference on Computer Vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
+page_content=' Springer, 2020,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE4T4oBgHgl3EQffQ16/content/2301.05106v1.pdf'}
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+arXiv:2301.04912v1  [astro-ph.SR]  12 Jan 2023
+MNRAS 000, 1–5 (2022)
+Preprint 13 January 2023
+Compiled using MNRAS LATEX style file v3.0
+δ Scuti pulsations in the bright Pleiades eclipsing binary HD 23642
+John Southworth 1, S. J. Murphy2, K. Pavlovski3
+1 Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK
+2 Centre for Astrophysics, University of Southern Queensland, Toowoomba, QLD 4350, Australia
+3 Department of Physics, Faculty of Science, University of Zagreb, Bijenicka cesta 32, 10000 Zagreb, Croatia
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We announce the discovery of pulsations in HD 23642, the only bright eclipsing system in the Pleiades, based on light curves
+from the Transiting Exoplanet Survey Satellite (TESS). We measure 46 pulsation frequencies and attribute them to δ Scuti
+pulsations in the secondary component. We find four ℓ = 1 doublets, three of which have frequency splittings consistent with
+the rotation rate of the star. The dipole mode amplitude ratios are consistent with a high stellar inclination angle and the stellar
+rotation period agrees with the orbital period. Together, these suggest that the spin axis of the secondary is aligned with the
+orbital axis. We also determine precise effective temperatures and a spectroscopic light ratio, and use the latter to determine the
+physical properties of the system alongside the TESS data and published radial velocities. We measure a distance to the system
+in agreement with the Gaia parallax, and an age of 170 ± 20 Myr based on a comparison to theoretical stellar evolutionary
+models.
+Key words: stars: fundamental parameters — stars: binaries: eclipsing — stars: oscillations
+1 INTRODUCTION
+The Pleiades is one of the closest and most extensively studied star
+clusters, containing ∼1300 stars at a distance of 136 pc (Melis et al.
+2014; Heyl et al. 2022). It is a benchmark for studying theoreti-
+cal modelling (e.g. Vandenberg & Bridges 1984) the kinematics of
+star clusters (Lodieu et al. 2019), binarity (Torres et al. 2021), pul-
+sations (Murphy et al. 2022), rotation (Rebull et al. 2016), and the
+connection between rotation, lithium depletion and stellar inflation
+(Somers & Pinsonneault 2015; Bouvier et al. 2018).
+Eclipsing binary stars (EBs) are another crucial research area
+for improving our understanding of stellar evolution (Torres et al.
+2010). The masses, radii and effective temperatures (Teffs)
+of stars can be determined to precisions approaching 0.2%
+(Maxted et al. 2020; Miller et al. 2020), allowing their use as checks
+and calibrators of theoretical models (e.g. Claret & Torres 2018;
+Tkachenko et al. 2020). In particular, EBs in open clusters are entic-
+ing targets for multiple scientific goals (e.g. Southworth et al. 2004a;
+Brogaard et al. 2011; Torres et al. 2018).
+A third phenomenon well suited to extending our understanding
+of stellar physics is that of pulsations. δ Scuti stars pulsate in pres-
+sure modes of low radial order (n ∼ 1 . . . 10) that are mostly sen-
+sitive to the stellar envelope (Aerts et al. 2010; Kurtz 2022). These
+modes probe the stellar density, making them good indicators of age
+(Aerts 2015), especially when mass and/or metallicity are already
+constrained. Recently, the discovery that δ Scuti stars pulsate in reg-
+ular patterns (Bedding et al. 2020) has allowed pulsation modes to
+be identified in many stars (e.g. Kerr et al. 2022; Currie et al. 2022),
+including five members of the Pleiades (Murphy et al. 2022), facili-
+tating detailed asteroseismic modelling. In some cases, this confers
+an age precision better than 10% (Murphy et al. 2021).
+HD 23642 (V1229 Tau, HII 1431) was found to be a double-
+lined spectroscopic binary by Pearce (1957) and Abt (1958). Its
+eclipsing nature was announced independently by Miles (1999)
+and Torres (2003) using data from the Hipparcos satellite. De-
+tailed analyses of the system using ground-based data have been
+presented by Griffin (1995); Torres (2003); Munari et al. (2004);
+Southworth et al. (2005) and Groenewegen et al. (2007). HD 23642
+was observed in long cadence mode by the K2 satellite (Howell et al.
+2014) in Campaign 41. An analysis of these data was presented by
+David et al. (2016).
+In this work we present the discovery of δ Scuti pulsations in this
+important EB and determine the physical properties of the system to
+high precision for the first time. We note that independent detections
+are presented by Chen et al. (2022) without detailed analysis, and
+by Bedding et al. (2023). Section 2 in the current work outlines the
+observations used, Sections 3 and 4 present our spectroscopic and
+photometric analyses, Section 5 is dedicated to the pulsation analy-
+sis, and our work is concluded in Section 6.
+2 OBSERVATIONS
+HD 23642 was observed using the TESS mission (Ricker et al.
+2015) in three consecutive sectors (42–44) covering 76 days
+(2021/08/20 to 2021/11/06), at a cadence of 120 s. We downloaded
+the data from the Mikulski Archive for Space Telescopes (MAST)
+archive and extracted the simple aperture photometry (SAP) from
+the FITS files (Jenkins et al. 2016); the PDCSAP data are practically
+identical. We included only the data with a QUALITY flag of zero,
+totalling 44 438 points, and converted them into differential magni-
+tude. The data errors were not used as they were much smaller than
+1 HD 23642 was requested as a target by nine Guest Observer proposals
+including GO4028 (PI Southworth) and GO4035 (PI Murphy).
+© 2022 The Authors
+
+2
+Southworth et al.
+Figure 1. TESS simple aperture photometry (SAP) light curve of HD 23642. The sectors are labelled.
+the scatter of the measurements. The full TESS data are shown in
+Fig. 1.
+We observed HD 23642 during two observing runs in Novem-
+ber 2006 using the Nordic Optical Telescope (NOT) and its FIbre
+Echelle Spectrograph (FIES). A total of 27 échelle spectra were ob-
+tained using the medium-resolution fibre in bundle A, which covered
+364–736 nm with a resolving power of R ≈ 47 000. Exposure times
+of 600 s yielded a signal to noise ratio (S/N) of approximately 200,
+although some spectra had a lower S/N due to cloudy conditions.
+The data were reduced using IRAF échelle package routines, with
+particular care taken in the normalisation and merging of the échelle
+orders (Kolbas et al. 2015).
+3 SPECTROSCOPIC ANALYSIS
+We first sought to measure the spectroscopic parameters of the two
+stars, in particular their Teffs and light ratio. To do this we applied
+the method of spectral disentangling (Simon & Sturm 1994) using
+the Fourier approach (Hadrava 1995) and concentrating on the 650-
+670 nm spectral range which includes the Hα line. This wavelength
+interval is contaminated by telluric lines, which we removed before
+the disentangling process. We used the FDBINARY code (Iliji´c et al.
+2004) and our standard methods (Pavlovski & Hensberge 2010;
+Pavlovski et al. 2018). We also fixed the velocity amplitudes of the
+stars to the values measured by Torres et al. (2021), so effectively
+ran in spectral separation mode.
+The Hα profiles of the two stars were then modelled to deter-
+mine the Teffs and light ratio. We fixed the surface gravities and ro-
+tational velocities of the stars to the values determined in Section 4,
+thus avoiding the degeneracy between Teff and log g present in the
+Balmer lines of hot stars. This approach required us to iterate our
+analysis with that described in Section 4 to ensure internal consis-
+tency; the iteration converged within one step. Optimal fitting was
+performed with the STARFIT code (Kolbas et al. 2015), which uses a
+genetic algorithm to search for the best fit within a grid of synthetic
+spectra pre-calculated using the UCLSYN code (Smalley et al. 2001).
+The fractional light contributions of the two components were forced
+to sum to unity (Tamajo et al. 2011) and only the wings of the Hα
+line were fitted (with metallic lines masked). The uncertainties were
+calculated using the MCMC approach described in Pavlovski et al.
+(2018).
+We found Teffs of 10 200 ± 90 K and 7670 ± 85 K for the two
+stars, and a light ratio of ℓB/ℓA(Hα) = 0.313±0.010. HD 23642 A
+is known to be chemically peculiar: Abt & Levato (1978) classified
+its spectrum as A0Vp(Si) + Am. Our light ratio should not be af-
+fected by this because it was obtained from only the Hα line. We
+propagated it to the TESS passband using BT-Settl theoretical spec-
+tra (Allard et al. 2012) to obtain ℓB/ℓA(TESS) = 0.328 ± 0.011
+where the errorbar includes the contributions from ℓB/ℓA(Hα) and
+both Teffs.
+4 LIGHT CURVE AND PHYSICAL PROPERTIES
+The light curve of HD 23642 shows shallow partial eclipses and
+clear reflection and ellipsoidal effects. The stars are well-separated
+and the system is suitable for analysis with the JKTEBOP code
+(Southworth et al. 2004b; Southworth 2013), for which we used ver-
+sion 43. We fitted for the fractional radii of the stars in the form of
+their sum (rA + rB) and ratio (k = rB/rA), the orbital inclina-
+tion (i), the central surface brightness ratio of the two stars (J), the
+amount of third light (L3), the reference time of mid-eclipse (T0),
+and the orbital period (P). Limb darkening was included using the
+power-2 law (Hestroffer 1997) with the scaling coefficient for each
+star (c1 and c2) fitted and the power-law coefficients (α1 and α2)
+fixed at values from Claret & Southworth (2022). We also included
+one quadratic function per TESS half-sector to normalise the light
+curve to zero differential magnitude, to allow for the possibility of
+slow drifts in brightness for either instrumental or astrophysical rea-
+sons. The pulsations are of much lower amplitude than the eclipses
+so were treated as red noise. A circular orbit was assumed.
+Our initial solutions gave measurements of the fractional radii
+to a disappointing precision. This is caused by the ratio of the
+radii being poorly determined for shallow partial eclipses, a well-
+known phenomenon that has been noted before for this system
+(Southworth et al. 2005; David et al. 2016). We therefore imposed
+the spectroscopic light ratio from Section 3 as a Gaussian prior in
+our solution. This significantly improved the precision of the fitted
+parameters, and is the only viable approach in a system like this
+with shallow partial eclipses and a noise limit set by the presence of
+pulsations. Uncertainties in the fitted parameters were determined
+by Monte Carlo and residual-permutation simulations (Southworth
+2008), taking the larger of the two options for each quantity. The
+values and uncertainties of the parameters are given in Table 1. We
+have added an extra uncertainty of ±0.0010 in quadrature to the un-
+certainties in rA and rB to account for the variations in these param-
+MNRAS 000, 1–5 (2022)
+
+Pulsations in Pleiades eclipsing binary HD 23642
+3
+Figure 2. A short section of the TESS light curve, chosen at random, is shown (blue points) along with the JKTEBOP best fit (red line). The residuals are
+displayed offset to the base of the figure and magnified by a factor of 5 to make the pulsations visible.
+Table 1. Properties of the HD 23642 system. The velocity amplitudes K are
+from Torres et al. (2021).
+Quantity and unit
+Star A
+Star B
+Spectroscopic parameters:
+K ( km s−1)
+100.07 ± 0.23
+142.60 ± 0.28
+Teff (K)
+10200 ± 90
+7670 ± 85
+Light ratio at Hα
+0.313 ± 0.010
+Light ratio in TESS passband
+0.328 ± 0.011
+JKTEBOP analysis:
+rA + rB
+0.2664 ± 0.0011
+k
+0.784 ± 0.012
+i (◦)
+78.63 ± 0.09
+J
+0.5561 ± 0.0011
+ℓ3
+0.029 ± 0.014
+c
+0.52 fixed
+0.63 fixed
+α
+0.45 fixed
+0.42 fixed
+P (d)
+2.46113223 ± 0.00000060
+T0 (BJDTDB)
+2459509.282357 ± 0.000007
+Fractional radii
+0.1494 ± 0.0016
+0.1171 ± 0.0017
+Physical properties:
+Mass (MN
+⊙)
+2.273 ± 0.011
+1.595 ± 0.008
+Radius (MN
+⊙)
+1.799 ± 0.019
+1.410 ± 0.021
+log g (c.g.s.)
+4.285 ± 0.009
+4.342 ± 0.13
+Vsynch ( km s−1)
+36.98 ± 0.40
+28.99 ± 0.42
+Luminosity log(L/LN
+⊙)
+1.499 ± 0.018
+0.792 ± 0.023
+Reddening EB−V (mag)
+0.040 ± 0.010
+K-band distance (pc)
+134.7 ± 2.0
+eters between different model choices, specifically about whether or
+not to fit for L3 or limb darkening.
+We determined the physical properties of the system from the re-
+sults of the JKTEBOP analysis and the velocity amplitudes of the
+stars measured by Torres et al. (2021). This was done using the JK-
+TABSDIM code (Southworth et al. 2005) modified to report results
+on the IAU scale (Prša et al. 2016). The full set of measured sys-
+tem properties are given in Table 1. Our masses are in good agree-
+ment with those found by other authors, with minor differences due
+to the newer velocity amplitudes adopted. The rotational velocities
+of the stars determined by Southworth et al. (2005), 37 ± 2 and
+33 ± 3 km s−1, are in agreement with the synchronous values, sug-
+gesting that both stars are rotating synchronously.
+The measured radii, however, are significantly different from
+previous
+values
+(Munari et al.
+2004;
+Southworth et al.
+2005;
+Groenewegen et al. 2007; David et al. 2016) in the sense that RA
+is larger and RB is smaller. This solves an existing problem with
+HD 23642 B, which was previously found to be significantly larger
+than predicted by theoretical evolutionary models for the Pleiades’
+age and metallicity. We attribute this change to differences in the
+spectroscopic light ratios adopted in those studies, which are slightly
+larger than the one measured in the current work. Both components
+of HD 23642 are known to be chemically peculiar so light ratios de-
+termined from metal lines are unreliable. Our new light ratio should
+be preferred because it is based on the Hα line so is not affected by
+the chemical peculiarity, and is also close to the wavelengths trans-
+mitted by the TESS passband. A visual comparison of the masses,
+radii and Teffs of the primary star to predictions from the PAR-
+SEC evolutionary models (Bressan et al. 2012) for solar metallicity
+shows good agreement for an age of 170 ± 20 Myr. We note that
+this is at the upper limit of accepted ages of the Pleiades cluster
+(Gossage et al. 2018; Murphy et al. 2022). The secondary compo-
+nent is 1.5σ smaller than expected: further investigation is needed to
+understand this.
+To determine the distance (d) and interstellar reddening (EB−V )
+to the system we used the BV apparent magnitudes from the Ty-
+cho satellite (Høg et al. 2000), the JHKs apparent magnitudes
+from 2MASS (Skrutskie et al. 2006) converted into the Johnson
+system using transformations from Carpenter (2001), and bolomet-
+ric corrections from Girardi et al. (2002). We determined the value
+of EB−V that results in consistent distances across the BV JHK
+bands, finding EB−V
+=
+0.040 ± 0.010 and d
+=
+134.7 ±
+2.0 pc. This distance is slightly shorter than the 138.3 ± 0.1 pc
+determined by simple inversion of the parallax from Gaia EDR3
+(Gaia Collaboration 2021). This EB−V is specific to HD 23642, is
+consistent with previous studies (Taylor 2008), and is not affected
+by reddening variations between Pleiades members.
+5 ANALYSIS OF THE PULSATIONS
+The residual light curve after subtraction of the JKTEBOP model
+shows several pulsation modes at the 0.1 mmag level (Fig. 3). We
+used the PERIOD04 code (Lenz & Breger 2004) to extract 46 pulsa-
+tions at frequencies f > 20 d−1 down to 0.015 mmag amplitude,
+and used non-linear least squares to optimise the frequencies. The
+table of frequencies is given in the Appendix. From the frequencies
+MNRAS 000, 1–5 (2022)
+
+4
+Southworth et al.
+Figure 3. Fourier transform of the light curve residuals, after subtracting the eclipse model, revealing several pulsations at the 0.1 mmag level. Mode identifica-
+tions (Section 5) are overlaid, showing radial modes as well as a series of rotationally split dipole modes.
+and Teffs of the stars we deduce that they represent δ Scuti pulsa-
+tions in the secondary component, as the primary is hotter than the
+δ Scuti instability strip.
+We used the ECHELLE package (Hey & Ball 2020) to manu-
+ally find values of the asteroseismic large spacing, ∆ν, that give
+vertical patterns in an échelle diagram. At ∆ν
+=
+6.97 d−1,
+there are two parallel ridges on the left-hand side of the échelle
+(Fig. 4), at an x-location suggestive of ℓ = 1 modes (Bedding et al.
+2020; Murphy et al. 2021). With the same ∆ν, part of a radial
+mode series can also be identified, with period ratios consistent
+with low-order radial modes (Netzel et al. 2022). It is notewor-
+thy that the independently-determined ∆ν is consistent with the
+value of five other δ Scuti stars in the Pleiades (6.82–6.99 d−1;
+Murphy et al. 2022), and also suggests that the system is relatively
+young (≲200 Myr).
+Since ℓ
+=
+1 doublets are seen, which are interpretable as
+m = ±1 pairs, the inclination of the pulsating star is not small
+(Gizon & Solanki 2003). However, at some orders, most notably at
+n = 3, there is a peak slightly offset from halfway between the
+two identified ℓ = 1 modes that could be a central component, after
+second-order effects of rotation are accounted for. To evaluate this
+possibility would require the calculation of rotating evolutionary and
+pulsation models. If confirmed, it implies that the stellar inclination
+is also not completely edge-on and, given the measured orbital incli-
+nation, implies that the spin and orbital axes of the pulsating star are
+aligned.
+The identification of four ℓ = 1 doublets offers the chance of a
+consistency check on the mode identification. If the doublets all have
+approximately the same splitting, it strengthens the proposed mode
+identifications. For the doublets at n = 6, 3, 2, and 1 we measure
+splittings of 0.813, 0.813, 0.812, and 0.586 d−1, respectively. Thus,
+the n = 6, 3, and 2 doublets seem secure, but the n = 1 doublet does
+not. Since the Ledoux constant, Cn,ℓ, is close to zero for p modes,
+we can estimate the stellar rotation rate, Ω, to first order based on
+these frequency splittings (Aerts et al. 2010):
+νn,ℓ,m = ν0 + mΩ(1 − Cn,ℓ),
+(1)
+where ν0 is the rest-frame frequency of the pulsation mode (we
+adopt the convention that prograde modes have positive m). The
+stellar rotation frequency is therefore 0.41 d−1, corresponding to a
+rotation period of 2.46 d. This is equal to the orbital period of the sys-
+tem, and the stars are known to rotate approximately synchronously
+(see Section 4), so this supports our inference of the stellar rotational
+inclination.
+Figure 4. Échelle diagram for HD 23642, marked with mode identifications.
+Radial modes are shown as circles, dipole modes are triangles. Modes with-
+out an identifications are shown as crosses, some of which may belong to the
+other star; most will be modes of higher degree (ℓ ≥ 2).
+6 SUMMARY AND CONCLUSIONS
+HD 23642 has it all: youth, eclipses, pulsations, chemical peculiar-
+ity, and membership of the Pleiades. We establish its physical prop-
+erties to high precision for the first time, helped particularly by the
+measurement of a spectroscopic light ratio immune to the chemical
+peculiarity. We determine a distance and interstellar reddening to the
+system in good agreement with previous measurements, and infer an
+age of 170±20 Myr for the Pleiades by comparison to PARSEC evo-
+lutionary model predictions.
+A total of 46 pulsation frequencies are detected to high signifi-
+cance from the three consecutive sectors of TESS photometry. We
+attribute them to δ Scuti pulsations in the secondary star. We use an
+MNRAS 000, 1–5 (2022)
+
+obs1=0
+80
+obsl=1
+X
+unidentified
+70
+60
+Frequency (d-1)
+50
+40
+X4
+XX
+X双XQ
+XXXX
+XLX
+X
+XX
+XX
+30
+XXX X
+XXXXX
+X
+X
+X
+X
+20
+X
+10
+AV = 6.97 i
+0
+1
+2
+3
+4
+5
+6
+7
+8
+Frequency mod Av (d-1)0.200
+radial
+0.175
+prograde
+retrograde
+0.125
+e
+0.100
+litud
+0.075
+idw
+A
+0.050
+0.025
+0.000
+10
+20
+30
+40
+50
+60
+70
+0
+Frequency, d-1Pulsations in Pleiades eclipsing binary HD 23642
+5
+échelle diagram to assign modes to 11 of the pulsations, based on the
+identification of ℓ = 1 doublets. An asteroseismic large spacing of
+∆ν = 6.97 d−1 allows identification of a series of radial modes with
+period ratios consistent with other δ Scuti stars in the Pleiades. The
+stellar rotation rate we find from the mode splittings is in agreement
+with the spectroscopic v sin i and the orbital period of the system.
+This implies that the spin axis of the pulsating secondary compo-
+nent is aligned with the orbital axis of the system. HD 23642 is well
+suited to detailed analysis with evolutionary and pulsation models
+including rotation.
+DATA AVAILABILITY
+The TESS data used in this work are available in the MAST archive
+(https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html).
+The FIES spectra are available in reduced form from the NOT
+archive (http://www.not.iac.es/archive/).
+ACKNOWLEDGEMENTS
+We thank Dominic Bowman and Hiromoto Shibahashi for useful
+discussions. The TESS data presented in this paper were obtained
+from the Mikulski Archive for Space Telescopes (MAST) at the
+Space Telescope Science Institute (STScI). STScI is operated by the
+Association of Universities for Research in Astronomy, Inc. Support
+to MAST for these data is provided by the NASA Office of Space
+Science. Funding for the TESS mission is provided by the NASA
+Explorer Program. SJM was supported by the Australian Research
+Council through Future Fellowship FT210100485.
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+APPENDIX A: MEASURED PULSATION FREQUENCIES
+IN HD 23642
+This paper has been typeset from a TEX/LATEX file prepared by the author.
+MNRAS 000, 1–5 (2022)
+
+6
+Southworth et al.
+Table A1. Extracted frequencies with corresponding amplitudes and provi-
+sional mode identifications for the pulsating component of HD 23642.
+f (d−1)
+amplitude (mmag)
+n
+ℓ
+m
+57.2112
+0.0175
+6
+1
+1
+56.3981
+0.0161
+6
+1
+−1
+36.2304
+0.1029
+3
+1
+1
+35.4177
+0.0768
+3
+1
+−1
+29.3618
+0.0483
+2
+1
+1
+28.5496
+0.0431
+2
+1
+−1
+22.4772
+0.1365
+1
+1
+1
+21.8913
+0.1713
+1
+1
+−1
+39.3467
+0.0938
+4
+0
+0
+33.1444
+0.0869
+3
+0
+0
+26.9261
+0.0260
+2
+0
+0
+53.5132
+0.0188
+–
+–
+–
+39.4291
+0.0302
+–
+–
+–
+39.1434
+0.0157
+–
+–
+–
+39.0450
+0.0199
+–
+–
+–
+38.7352
+0.0514
+–
+–
+–
+38.6162
+0.0407
+–
+–
+–
+38.5353
+0.0362
+–
+–
+–
+38.3290
+0.0296
+–
+–
+–
+37.9217
+0.0272
+–
+–
+–
+37.8036
+0.1171
+–
+–
+–
+36.1084
+0.0165
+–
+–
+–
+35.8956
+0.0370
+–
+–
+–
+35.2961
+0.0289
+–
+–
+–
+34.1939
+0.0286
+–
+–
+–
+33.8544
+0.0206
+–
+–
+–
+32.3307
+0.0221
+–
+–
+–
+29.4768
+0.0239
+–
+–
+–
+29.4484
+0.0198
+–
+–
+–
+29.1269
+0.0368
+–
+–
+–
+28.6681
+0.0302
+–
+–
+–
+28.3104
+0.0203
+–
+–
+–
+28.1144
+0.0378
+–
+–
+–
+28.0038
+0.0230
+–
+–
+–
+27.7391
+0.0347
+–
+–
+–
+27.3022
+0.0181
+–
+–
+–
+26.1835
+0.0203
+–
+–
+–
+25.3700
+0.0252
+–
+–
+–
+24.9365
+0.0264
+–
+–
+–
+24.5100
+0.0159
+–
+–
+–
+24.2900
+0.0160
+–
+–
+–
+23.7500
+0.0281
+–
+–
+–
+22.9355
+0.0181
+–
+–
+–
+22.0604
+0.0403
+–
+–
+–
+20.3501
+0.0157
+–
+–
+–
+MNRAS 000, 1–5 (2022)
+

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+ACCEPTED VERSION
+1
+Explainable, Physics Aware, Trustworthy AI
+Paradigm Shift for Synthetic Aperture Radar
+Mihai Datcu, Fellow, IEEE , Zhongling Huang†, Andrei Anghel, Juanping Zhao, Remus Cacoveanu
+Abstract—The recognition or understanding of the scenes
+observed with a SAR system requires a broader range of cues,
+beyond the spatial context. These encompass but are not limited
+to: imaging geometry, imaging mode, properties of the Fourier
+spectrum of the images or the behavior of the polarimetric
+signatures. In this paper, we propose a change of paradigm for
+explainability in data science for the case of Synthetic Aperture
+Radar (SAR) data to ground the explainable AI for SAR. It aims
+to use explainable data transformations based on well-established
+models to generate inputs for AI methods, to provide knowledge-
+able feedback for training process, and to learn or improve high-
+complexity unknown or un-formalized models from the data.
+At first, we introduce a representation of the SAR system with
+physical layers: i) instrument and platform, ii) imaging formation,
+iii) scattering signatures and objects, that can be integrated with
+an AI model for hybrid modeling. Successively, some illustrative
+examples are presented to demonstrate how to achieve hybrid
+modeling for SAR image understanding. The perspective of
+trustworthy model and supplementary explanations are discussed
+later. Finally, we draw the conclusion and we deem the proposed
+concept has applicability to the entire class of coherent imaging
+sensors and other computational imaging systems.
+Index Terms—SAR image understanding, explainable artificial
+intelligence, deep neural networks, knowledge inspired data
+science
+I. MOTIVATION AND SIGNIFICANCE
+The Earth is facing unprecedented climatic, geomorpho-
+logic, environmental or anthropogenic changes, which require
+global scale, long term observation with Earth Observation
+(EO) sensors. SAR sensors, due to their observation capability
+during day and night and independence on atmospheric effects,
+are the only EO technology to insure global and continuous
+observations. Meanwhile, the SAR observations of Sentinel-1
+satellites in the frame of the European Copernicus program,
+are worldwide freely and openly accessible. This is immensely
+enlarging the SAR Data Science and applications, covering
+a multitude of areas as: urbanization, agriculture, forestry,
+geology, tectonics, oceanography, polar surveys, or biomass
+estimation, only to enumerate a few. Copernicus Open Access
+Hub provides more than 457.59 PB data of satellites covering
+the Earth for more than 570,000 users all around the world. 1
+SAR is a pioneer technology in the field of computational
+sensing and imaging, of which the imaging mechanism is to-
+tally different from optical sensors. A radar instrument carried
+by an airborne or spaceborne platform illuminates the scene by
+side-looking or forward-looking, which allows to discriminate
+objects in the range direction. As the platform moving along
+† Corresponding author ([email protected])
+1https://scihub.copernicus.eu/reportsandstats/
+ground range
+synthetic aperture
+azimuth
+chirp 
+signal
+slant range
+dielectric property
+surface / geometry 
+terrain & object
+wavelength / bandwidth
+ polarization / inc. angle
+sensor & platform
+echo processing
+azimuth focusing
+imaging system
+Doppler variation
+echo
+Knowledge
+Data
+Fig. 1: A simple illustration of how SAR images the world
+(Stripmap Mode). SAR society is facing the big data challenge
+but with limited ground truth. In the meanwhile, the knowledge
+of SAR is equally important. This is also the motivation of the
+physical layers in this paper.
+its track, the SAR sensor is constantly transmitting a sequence
+of chirp signals and receiving echos reflected from objects
+on the ground, as depicted in Fig. 1. When recording all
+individual acquisitions with a short physical antenna and
+mathematically combining them into a synthetic image, a
+much larger synthesized aperture is formed. This allows high
+capacity to distinguish objects in azimuth despite a physically
+small antenna [1]. A high resolution “image” can be processed
+by applying SAR focusing principle, e.g., matching filtering
+[2].
+A deluge of SAR sensors have increased the data availabil-
+ity for various SAR applications. The allure of data-driven
+learning stems from the ability of automatically extracting ab-
+stract features from large data volumes [3]–[6], and therefore,
+many deep learning studies for SAR applications have been
+developed in recent years [7]–[10]. Current popular paradigm
+predominantly follows the blue part in Fig. 2 (a), where SAR
+image data is all that is required to operate an intelligent
+arXiv:2301.03589v1  [eess.IV]  9 Jan 2023
+
+甲
+::ACCEPTED VERSION
+2
+Fig. 2: a The conventional data-driven paradigm for intelligent SAR image understanding based on deep neural networks and
+the proposed paradigm shift integrated and interacted with physical knowledge of SAR. b A bridge can be imaged as multiple
+bright lines, similar as a couple of bridges imaged in the other SAR image, depending on the observation parameters and
+orientations. This positions the load and outmost difficulty of SAR image understanding. c The multipath scattering formation
+in the SAR image.
+network. In addition to data, however, the physical model and
+principles of SAR sensor should not be neglected. In the upper
+example of Fig. 2 (b), A bridge over a placid river that is
+illuminated perpendicular to its primary orientation appears as
+many brilliant lines, resembling the lower SAR image in which
+several bridges are imaged from a different viewing angle. The
+phenomena can be explicable by multi-path scattering [11],
+[12], as illustrated in Fig. 2 (c). Apart from the direct scattering
+from the bridge, the double bounce reflection between the
+bridge and water or vice versa occurs at the corner reflector
+spanned from the smooth vertical bridge facets facing the
+sensor and the water surface. In addition, the triple-bounce
+reflection and maybe some five-path scattering would happen
+between the horizontal plane of bridge and water surface.
+Thus, SAR image implies the causality of multi-path scattering
+phenomena and object characteristics. This positions the load
+of SAR image understanding, and the outmost challenge of
+data science, as new and particular paradigm of Artificial
+Intelligence (AI).
+So far, some researches have discussed the paradigm that
+attempts to bring scientific knowledge and data science models
+together, applied to a broad range of research themes such as
+partial differential equation solving [13] and Earth sciences
+[14]–[16]. In particular for SAR community, however, this
+topic has rarely been systematically analyzed and illustrated.
+Thus, we aim to prospect the hybrid modeling paradigm for
+intelligent SAR image understanding, where deep learning
+is integrated and interacted with SAR physical models and
+principles, to achieve explainability, physics awareness, and
+trustworthiness.
+Explainable AI is a broad concept. A scientific understand-
+ing of explainability is the capacity to clarify the results
+in the context of domain knowledge. The algorithms still
+remain a black box. A different approach is the algorithmic
+explainability. This is constructed such that the results of
+the used model can be described algorithmically. To obtain
+a higher degree of explainability, we aim at the synergy
+of the paradigms: algorithmic and scientific explainability.
+
+Physical Models and Principles of SAR
+a
+Rg/Az FFT & IFFT
+SAR
+Polarization
+Incident angle
+Image Formation
+platform
+Wavelength
+Focused
+image
+ausality
+scattering analysis 
+Shape
+Dielectric
+Roughness
+terrain / object
+integrated and interactive
+(LULC, object, etc)
+(SLC, GRD, etc)
+Deep Neural
+Network
+prediction
+image data
+Data-Driven Learning Paradigm
+b
+Ilumination direction
+c
+→ Direct scattering
+> Double bouncing
+→ Triple/five path scattering
+height
+-
+bridge
+1-2-1
+3
+1-2-3-2-1
+SAR image
+Optical image
+RangeACCEPTED VERSION
+3
+Fig. 3: Physical layer (i): Sensor and Platform. a: The moving platform creates Doppler variations and synthesizes large virtual
+aperture; PolSAR transmits and receives diverse polarized wave, and SAR polarimetric characteristics are depicted. b: Based
+on the physics behind the platform and sensor, the physical layer produces SAR specific representations such as sub-aperture
+synthesis image and polarimetric feature, with specified physical parameters.
+Algorithmic explainability lies in the guarantee of transparency
+to understand how the machine learning algorithm works by
+participation of SAR physical models and principles. Scientific
+explainability ensures the physical consistency of AI output, as
+well as learning of trustworthy results with physical meaning.
+To ground this, we first lay out a representation of SAR
+physical layer in the context of SAR domain knowledge, as
+presented in Section II. Further, we describe how to integrate
+and interact them with popular neural networks to build a
+hybrid and translucent model for SAR applications using illus-
+trative examples, demonstrated in Section III. The perspective
+of trustworthy models and supplementary explanation for SAR
+community are discussed in Section IV and V. The conclusion
+and perspectives are finally given in Section VI.
+II. SAR PHYSICAL LAYERS
+Other than the neural network layers equipped with a
+number of learnable parameters, SAR physical layers are ones
+embedded with physical knowledge of SAR, well-established,
+interpretable, and supported by domain theories. The concept
+of ”physical layer” apart from ”neural network layer” arose in
+literature [16] to make the model more physically realistic. As
+motivated in Fig. 1, three SAR physical layers are highlighted
+specific for SAR applications in this paper, i.e., (i) sensor
+and platform: referring to antenna characteristics and moving
+satellite/aircraft, (ii) imaging system: figuring image formation
+with focusing process and (iii) scattering signature: reflecting
+the physical properties of terrain and objects.
+A. Sensor and Platform
+Fig. 3 demonstrates the physical layer of sensor and plat-
+form that indicates the physics behind the SAR acquisition
+principle, such as aperture synthesizing with moving platform
+and various characteristics of antenna.
+Existing spaceborne EO SAR missions work in a monostatic
+or quasi-monostatic configuration. The simplest illumination
+mode of a SAR system is the stripmap mode in which the
+antenna pointing direction is constant throughout the acqui-
+sition, as shown in Fig. 3 a. The moving platform leads to
+a sliding Doppler spectrum that impacts the complex SAR
+image. Knowing the behaviour of the Doppler centroid to
+create sub-looks is essential for exploiting look angle diversity
+of the input data, especially for very high-resolution SAR
+images.
+It is well-known that in high-resolution SAR image where
+the signals are performed over a broad bandwidth and wide
+angular aperture, the targets are no longer isotropic and non-
+dispersive. Instead, it is more plausible to infer that the
+target’s backscattering is dependent on illumination angle
+and frequencies [17]. The sub-aperture processing can be
+applied to analyze the target scattering variations. Fig. 3 b
+gives an example of a synthesized pseudo color SAR image
+via sub-aperture processing. The complex-valued SAR image
+is first transformed to the azimuth spectral domain by a
+one-dimensional Fourier transform. Then, the full Doppler
+spectrum is equally split into three intervals, named sub-
+apertures or sub-looks, each containing 1/3 range of azimuth
+
+a
+Full aperture
+b
++Sub-aperture
+1
+1
+1
+sub-aperture features
+Azimuthdirection
+1
+1
+R: sub-look 1
+1
+G: sub-look 2
+B: sub-look 3
+1
+Target
+1
+or
+moving platform
+polarimetric features
+1
+Transmitter
+R: [HH-W/2
+Horizontal
+0
+0
+0
+G: 2|HV|2
+1
+B: [HH+VWV/2
+Vertical
+0
+1
+Receiver
+or
+HH
+HV HHHVHH
+Horizontal
+1
+physical params:
+1
+1
+Doppler centroid, beamwidth, Entropy,
+Vertical
+1
+height, incidence angle, etc.
+1
+1
+phase
+backscattering
+1
+1
+complex
+coefficient
+1
+image
+polarimetric sensor
+1ACCEPTED VERSION
+4
+Fig. 4: Physical layer (ii): Image Formation. a. Targets are characterized by sliding bandpass filtering in the Fourier domain. b.
+On the basis of image formation principle and target scattering model, the physical layer generates the rich target description
+with physical meaning.
+angles. Finally, the three sub-apertures are transformed back
+to time-domain using an inverse Fourier transform, coded as
+the R, G, and B channels, respectively. Red, Green, and Blue
+targets respond mainly on the first, second, and third sub-looks,
+respectively, whilst gray targets indicate that they respond
+equivalently in different sub-looks. The pseudo-colored image
+well demonstrates the particular behavior of some targets.
+Given the precise knowledge of the parameters related to
+Doppler variations (e.g., orbit, azimuth steering rate, radiation
+pattern, incidence angle), the physical layer can generate sub-
+look data deterministically and there is no need to design
+a neural network that should learn to create sub-looks from
+various types of SAR training data.
+Sensor characteristics, such as polarization, interferometry
+and tomography, construct physical layer as well. Fig. 3
+b presents a Pauli pseudo RGB image, where R, G, and
+B channels are formed with |HH − V V |2, 2|HV |2, and
+|HH+V V |2, respectively, indicating the polarimetric relation.
+Several physical layers can be stacked to represent rich physics
+of SAR sensor and platform. Early in literature [18], the diver-
+sity in the polarimetric features with the azimuthal look angle
+was exploited. Thus, the moving platform and polarimetric
+sensor are both characterized. Similarly, the stacked physical
+layers can represent polarimetric and interferometric properties
+of PolInSAR data, or any other combinations.
+B. Imaging System
+The second physical layer we suppose delineates the physics
+behind SAR image formation with an imaging system. The
+selected exemplars are illustrated in Fig. 4.
+A pulse-based radar or a frequency modulated continuous
+wave (FMCW) radar is usually used in a SAR system,
+where a range profile is obtained for each transmitted/received
+waveform, either by range compression in the case of a pulse-
+based radar or by applying a Fourier transform to the beat
+signal in the case of an FMCW radar [19]. By a coherent
+processing of the range profiles, the azimuth focusing process
+outputs a SAR image representing a two-dimensional complex
+reflectivity map of the illuminated area. SAR processing,
+taking a simple point target as example, aims to collect the
+dispersed signal energy in range and azimuth into a single
+pixel. Many traditional imaging algorithms are in terms of a
+Fourier synthesis framework [20], as such, Fourier transform
+provides a specific physical meaning for SAR image. This kind
+of physical layer assists AI model to better depict the target
+scattering beyond the ”image” domain.
+Fig. 4 (a) first shows a simple time-frequency analysis of
+target with short-time Fourier transform [21], [22], charac-
+terizing the backscattering intensity variations in 2-D range
+and azimuth frequency domain. Four kinds of backscattering
+behaviors observed in SAR were defined in literature [23], re-
+lated to different objects shown in Fig. 4. In the high-resolution
+case (wide bandwidth chirp signal and broad angular aperture),
+the complex amplitude of a target is frequency and aspect
+dependent [17]. Thus, the image formation can be extended to
+four dimension (called hyperimage) with wavelet transform,
+providing a concise physically relevant description of target
+scattering. This frequency and angular energy response pattern
+is proved useful for discriminating different scatterers, offering
+valuable prior information to AI model, depicted in Fig. 4 b.
+C. Scattering Signatures of Objects
+Thirdly, we introduce the physical layer regarding the scat-
+tering signatures of objects, in which the causality of target
+characteristics and scattering behaviors is involved.
+For optical images, what you see is what you receive, that
+is, the objects depicted on the optical image are in accord
+with human cognition. Targets in SAR images are reflected by
+scattering characteristics, yet they include a wealth of physical
+information that the human eye cannot immediately identify.
+Fig. 5 a shows example of two typical SAR targets of bridge
+and building. The scattering phenomenon that shows several
+parallel lines over the river can be interpreted as single, double,
+and multiple scattering of the bridge based on the domain
+knowledge. The building, with scattering signatures of layover,
+shadow, single and secondary scattering in the high-resolution
+SAR image, can also be reflected as only layover and shadow
+[24], depending on the building orientation and shape. Similar
+
+Optical
+SAR
+Radar
+a
+image
+image
+spectrogram
+Stable Targets
+4D hyperimage
+Data transform:
+Unstable Targets
+SLC
+Wavelet, Fourier,
+data
+or
+Wigner-Ville
+freguency-anguler
+energy response
+physicalparams:
+Azimuth Variant Targets
+frequency,angle,
+wavenumber, ...
+or.
+Range Variant Targets
+Target description with physical meaningACCEPTED VERSION
+5
+Fig. 5: Physical layer (iii): Scattering Signatures of Objects. a. The Golden Gate Bridge revealing multipath scattering
+characteristics in a Gaofen-3 quad-pol SAR image, and a typical single building representing different scattering regions
+in a high-resolution (1m) SAR image [24]. b. The scattering mechanisms indicated by the H/α plane for full-polarized SAR
+data point out the land-use and land-cover classes [25]. c. The physical layer describes the relationship and reasoning between
+the scattering characteristics seen in the SAR image and the object’s features, such as its shape, structure, or even semantics.
+research by Ferro et al. [26] investigated the relationship
+between double bounce and the orientation of buildings in
+VHR SAR images. Fig. 5 b demonstrates the relations between
+the scattering mechanism of H/α plane and the semantics
+of land-cover and land-use classes [25]. Likewise, one can
+deduce the scattering center position and the specific shape
+of distributed target from a SAR image by applying some
+scattering models [27].
+The conventional data-driven convolutional neural network
+can capture the image contents as we ”see” in the SAR image,
+whereas it is not equipped with the ability to ”interpret” the
+scattering phenomenon as we discussed before. This indicates
+the knowledge gap between SAR scattering signatures and hu-
+man vision cognition. The physical layer delivering semantic
+understanding behind the SAR scattering signature permits a
+more thorough interpretation of the SAR image. As shown in
+Fig. 5 c, the physical layer defines the association between
+the scattering characteristics of a SAR image and the object’s
+qualities, such as shape, structure, or semantics. It can be
+written as an objective function or a regularization term that
+constrains the training of neural networks. This will improve
+the intelligence of AI model to master some causality between
+scattering signatures and the object nature.
+III. HYBRID MODELING WITH SAR PHYSICAL LAYERS
+The integration and interaction of neural network layers
+and physical layers construct the hybrid modeling for SAR
+image interpretation. In view of algorithmic explainability, the
+explainable physical models and domain knowledge improves
+the transparency. For scientific explainability, the hybrid mod-
+eling ensures the physical meaning of output in physical
+layers and the prediction can maintain the physical consis-
+tency. In this section, we demonstrate several hybrid modeling
+approaches with SAR physical layer to achieve explainability
+and physics awareness.
+A. Insert for Substitution
+The introduced physical layer can be inserted in a deep
+neural network for substitution, extracting explainable and
+meaningful features, either as the input of a DNN or fused
+with DNN features in intermediate layers. A common way is
+to insert a physical layer into the input layer to obtain the
+polarimetric features for PolSAR image classification, includ-
+ing the elements of coherency matrix, Pauli decomposition
+features, etc [31], [32]. Similarly, the sub-aperture images are
+generated as the input for target detection [33]. The other
+usage of physical layer is for feature fusion, where the features
+obtained by well-established physical model and deep neural
+networks are combined [34], [35].
+Our recent work, a deep learning framework named Deep
+SAR-Net (DSN) [28], addressed both aspects that inserts the
+physical layer into the input and the intermediate position of
+deep model. As shown in Fig. 6, DSN was proposed for clas-
+sifying SAR images with complex values. Instead of the entire
+data-driven method, i.e. the complex-valued convolutional
+neural networks (CV-CNN), the designed DSN encompasses
+three shallow neural network modules and two physical layers.
+The first physical layer generates the high-dimensional radar
+spectrogram based on time-frequency analysis. The second one
+handles the features of the 2-D projection along the frequency
+axises [22] to maintain the location constraint, making it possi-
+ble to be fused with spatial features from intensity image. DSN
+outperformed CV-CNN especially with limited labeled training
+data, and had a remarkable performance in discriminating the
+man-made target scenes compared with the traditional CNN. It
+demonstrates the Fourier process on single-look complex SAR
+image embedded the knowledge like synthesizing the antenna
+
+α()
+1
+a
+90 [
+Scattering Mechanisms
+Range direction
+Range direction
+80
+ Low entropy scattering
+Low entropy dipole
+ Single scattering
+70
+Low entropy surface
+60
+Medium entropy multiple scattering
+double
+Shadoi
+Medium entropy dipole scattering
+scatterino
+Medium entropy surface scattering
+High entropy double bounce
+40
+High entropy multiple scattering
+-
+Azimuth direction 
+30
+一
+20
+Land Cover Types
+10
+Multiple
+1
+△ water
+★ landslide
+■ farmland
+scattering
+ forest
+口 snow
+0
+0.5
+0.91
+I b
+Entropy(H)
+Bridge
+Building
+I c
+Relation and Reasoning
+Scattering
+object
+-
+phenomenon
+property
+shape, orientation,
+scattering center,
+structure, position, ..
+scattering mechanism,
+一
+一ACCEPTED VERSION
+6
+Physical Layer
+complex-valued 
+SAR image
+Sub-look 
+frequency signals
+intensity
+Deep 
+Network 2
+Deep 
+Network 1
+Deep 
+Network 3
+feature 
+fusion
+semantic labels
+An example of Physical Layer (i) with AI model
+Deep SAR-Net 
+(Huang et al. 2020)
+complex-
+valued SAR 
+image
+2D projection in 
+frequency domain
+intensity image
+Deep 
+Network 2
+Deep 
+Network 1
+Deep 
+Network 3
+spatial 
+features
+frequency 
+features
+target position 
+constraint
+feature 
+fusion
+{
+semantic labels
+Physical Layer
+Physical Layer
+TFA
+Fig. 6: Our recent work Deep SAR-Net (DSN) [28] for SAR image classification can be regarded as a typical example of
+inserting the physical layers into a deep model.
+Fig. 7: a. Unsupervised HDEC-TFA method [29]. It automatically discovered the radar spectrogram patterns more than the
+four defined in [23] with deep neural networks. b. Learning the polarimetric features from single-polarized SAR image,
+supervised by Entropy-Alpha-Anisotropy generated from full-pol data [30]. c. The physical layers in a and b play the role of
+input transform (blue) and supervision generation (green) in hybrid modeling. In addition, the physical layer (red) can act as
+feedback to restrict learning and produce physically consistent outcomes.
+well characterizes the physical property of SAR target, and
+the usages of physical layer cut down unnecessary parameters
+in neural network layers to improve the model performance
+with limited ground truth.
+B. Compensation for Imperfect Knowledge with Feedback
+In condition of unknown/inconclusive physical models or
+incomplete knowledge, it is difficult to extract perfect physical
+parameters or physical scattering characteristics of SAR via
+model-based methods. For instance, obtaining the polarimetric
+features from dual-pol, or even single-polarized SAR image.
+Thus, the physical layer interacted with deep neural network
+take effect.
+1) Target Character Identification:
+Some researches have analyzed the energy response pattern
+in frequency dimensions of target varied in SAR image,
+and discussed the nonstationary targets [18], [36]. Spigai
+et al. [23] pointed out four canonical targets with a rough
+definition shown in Fig. 4 a. However, it remains unknown
+for many complicated scene and objects. Fig. 7 show our
+related work of using physical layer and deep neural network
+for compensation of imperfect knowledge. The first is the
+unsupervised hierarchical deep embedding clustering based
+on time-frequency analysis (HDEC-TFA) [29], which was
+proposed to automatically characterize the radar spectrogram
+(or the sub-band scattering pattern defined in [29]) basically
+in urban area, discovering the various scattering pattern more
+than the four specific classes defined in [23]. It offered a
+new perspective to describe the physical properties of single-
+polarized SAR. Furthermore, we used two stacked physical
+layers to obtain the polarimetric and time-frequency patterns
+and analyzed with deep neural network in reference [37].
+
+a
+C
+Physical Layer
+Deep Embedding Clustering
+TFA
+Physical Layer
+Loss = KL(pllq)
+supervision
+SAR image
+Radar
+NN
+Cluster
+weights
+Result map
+spectrogram
+centers
+loss
+Feedback loss
+Update parameters
+b
+Physical Layer
+Physical Layer
+output
+Ground Truth
+Deep Neural Network
+generation
+Flattening
+ImprovedCost
+Measurement
+CXN
+Physical Layer
+610x10
+×12×4xN
+ One-hot Labels
+SAR inputoriginal slc
+fr=BWr/2,fa=BWa/2
+X=N/2,y=N/2
+X=N/2,fr=BWr/2
+y=N/2,fa=BWa/2
+y=N/2,fr=BWr/2
+X=N/2,fa=BWa/2
+0
+0
+20
+10
+10
+10
+10
+40
+20
+20
+20-
+20
+20
+20
+30
+60
+30
+30
+30
+30
+30
+0
+25
+50
+0
+20
+o
+20
+0
+20
+0
+20
+0
+20
+20original slc
+fr=BWr/2.fa=BWa/2
+X=N/2,y=N/2
+X=N/2,fr=BWr/2
+y=N/2,fa=BWa/2
+y=N/2,fr=BWr/2
+X=N/2,fa=BWa/2
+20
+40
+20
+60
+30
+0
+25
+50
+0
+20
+20
+20
+20
+20
+20original slc
+fr=BWr/2,fa=BWa/2
+X=N/2,y=N/2
+X=N/2,fr=BWr/2
+y=N/2,fa=BWa/2
+y=N/2,fr=BWr/2
+X=N/2,fa=BWa/2
+0
+0
+0
+0
+20
+10
+10
+10
+10
+10
+10
+40
+20-
+20
+20-
+20
+20
+60
+30
+30
+0
+25
+50
+0
+20
+o
+20
+0
+20
+0
+20
+0
+20
+0
+20original slc
+fr=BWr/2,fa=BWa/2
+X=N/2,y=N/2
+X=N/2,fr=BWr/2
+y=N/2,fa=BWa/2
+y=N/2,fr=BWr/2
+X=N/2,fa=BWa/2
+20
+10
+20
+20-
+60
+30
+30
+30
+0
+25
+50
+0
+20
+0
+20
+0
+20
+0
+20
+0
+20
+20original slc
+fr=BWr/2,fa=BWa/2
+X=N/2,y=N/2
+X=N/2,fr=BWr/2
+y=N/2,fa=BWa/2
+y=N/2,fr=BWr/2
+X=N/2,fa=BWa/2
+0
+0
+20
+10
+10
+LO
+10
+10
+40
+20
+20
+20
+20
+20+
+20
+60
+30
+30
+30
+C
+30
+0
+25
+50
+0
+20
+0
+20
+0
+20
+20
+20
+20original slc
+fr=BWr/2,fa=BWa/2
+X=N/2,y=N/2
+X=N/2,fr=BWr/2
+y=N/2,fa=BWa/2
+y=N/2,fr=BWr/2
+X=N/2,fa=BWa/2
+20
+10
+10
+0
+10
+40
+20
+20
+20
+60
+30
+0
+25
+50
+0
+20
+0
+20
+0
+20
+0
+20
+20ACCEPTED VERSION
+7
+SOLEIL 
+synchrotron
+“L”-shape 
+building
+“H”-shape 
+building
+“I”-shape 
+building
+Google Earth
+HDEC-TFA
+GD-Wishart
+Quad-Pol
+HH Channel
+Fig. 8: The SOLEIL synchrotron in France and the surrounding buildings with different shapes are depicted in the Gaofen-3
+SAR image. Both the GD-Wishart [38] result on Quad-Polarization SAR data and the HDEC-TFA [29] result on HH channel
+single-polarized SAR can capture the special scattering characteristics of the objects.
+Fig. 8 demonstrates the result compared with the polarimet-
+ric physical model. The SOLEIL synchrotron in France, shown
+as the round building in the Google Earth remote sensing
+image, is surrounded by three different shapes of buildings.
+The HDEC-TFA method can capture the special characteristics
+of the architectures even in single HH channel SAR image, as
+much as the physical model based method GD-Wishart [38]
+on quad-pol SAR. Some other man-made targets examples
+characterized by time-frequency model with neural networks
+are given in [39]. Our experiments in [29] demonstrated
+the trained model varies with different imaging conditions
+since the sub-band scattering pattern is influenced by several
+imaging parameters, which should be taken into consideration
+when transferring the AI model to other situations.
+2) Polarimetric Parameter Extraction:
+By transmitting and receiving waves that are both horizon-
+tally and vertically polarized, the full-pol SAR image captures
+abundant physical characteristics of the imaged objects that
+can lead to various physical parameters. In contrast, single-
+pol and dual-pol SAR data are less informative for physical
+feature extraction. If only one polarization channel is obtained,
+one cannot derive the other polarization channels in principle.
+Once the objects are known, i.e., once the characteristics of
+targets such as geometry, surface roughness, etc, are identified,
+deep learning can be employed to transfer the knowledge
+learned from physical models to reconstruct the physical
+parameters of objects. As shown in Fig. 7 b, Zhao et al. [30]
+proposed a complex-CNN model to learn physical parameters
+(entropy H and α angle) with transfer learning from single-pol
+and dual-pol SAR data, supervised by features obtained with
+a physical layer. Some similar studies include but not limit to
+[40], [41]. Song et al. [40] addressed ”radar image coloriza-
+tion” issue to reconstruct the polarimetric covariance matrix
+with a designed deep neural network, where the supervision
+was also generated with a physical layer.
+When training a data-driven deep neural network, some
+physical consistencies may not be guaranteed. The authors
+pointed that the reconstructed covariance matrix may not be
+positive semi-definite [40], and they proposed an algorithm to
+correct it. In this case, the additional physical layer embedded
+prior constraint acts as post-processing to revise the physically
+inconsistent result of DNNs. Furthermore, this type of physical
+layer is suggested to provide feedbacks during training, as
+demonstrated in Fig. 7 c, the red part. The feedback of physical
+layer aims to prevent the model from learning the physical
+inconsistency.
+The classification results vary 
+with different imaging conditions
+The same area under 
+different imaging conditions
+HDEC-TFA  (Huang et al. 2021, TGRS)
+Deep Neural Network 
+Physical Layer
+SAR 
+image
+Different 
+representations
+x1
+x2
+xn
+SSL loss
+SAR 
+image
+Deep Neural Network 
+Physical Layer
+(i) or (ii)
+Representation-2
+Representation-1
+Relations
+SSL loss
+a
+b
+Physical Layer (iii)
+Fig. 9: The SAR physical layer can be integrated in a self-
+supervised learning framework to guide the neural network
+training without ground truth. a. The physical layer generates
+various modalities of SAR image using well-established physi-
+cal models, such as sub-aperture images, different polarimetric
+features, etc. The self-supervised learning can be conducted
+with contrastive learning paradigm. b. The physical layer
+produces a physical representation of image, serving as a
+guided signal that drives the neural network to learn a similar
+representation.
+
+DeepEmbeddingClustering
+TFA
+Loss=KL(pllq)
+SARimage
+Radar
+NN
+Cluster
+Result map
+spectrogram
+weights
+centers
+Update parameters7.0
+6.5
+6.0
+5.5
+5.0
+4.5
+4.0
+3.5
+1.0
+0.8
+0.0
+0.
+0.4
+range
+0.6
+0.8
+0.0 
+206.5
+6.0
+5.5
+5.0
+4.5
+4.0
+0.0
+0.2
+.
+8'0
+1.05.5
+5.0
+4.5
+4.0
+3.5
+3.0
+2.5
+L.0
+0.8
+0.0
+2.
+0.2
+0.4
+range
+0.6
+0.8
+0.2
+0.0 6.5
+6.0
+5.5
+5.0
+4.5 
+4.0
+3.5
+0.0
+0.2
+0.4
+azimuth
+0.6
+0.8
+1.0ACCEPTED VERSION
+8
+3) SAR Image Generation/Simulation:
+This paradigm can be popularized to other SAR appli-
+cations. SAR target image generation (or simulation) based
+on deep generative model (such as variational auto-encoder
+[44] and generative adversarial network [45]) has attracted
+much attention in recent years. The generated SAR images
+are expected to be used as data supplements to support target
+identification. The authenticity and interpretability of current
+deep SAR image generation is a substantial obstacle that has
+a significant impact on subsequent tasks [46]. Many latest
+studies input important physical parameters into the deep
+generative model or use them as supervision at the output
+layer, such as depression angle and target orientation, that
+facilitated more reliable outcomes [47], [48]. We consider
+them physical layer as shown in Fig. 7 c, the green and blue
+part.
+Coupling the physical layer as a feedback in neural network
+for SAR image generation has yet to be explored. When
+generative model produces a pseudo SAR image, a physical
+layer will be applied to verify whether it is consistent with the
+knowledge base of SAR, e.g. physical parameters derived from
+a well-established model. If not, the current generative model
+will revise the pseudo result to minimize the inconsistency.
+There are some examples to learn from in the field of fluid
+simulation [49], [50]. As such, the physical layer is used for
+constructing physical inconsistency as a feedback that explic-
+itly constrain the generative model to fulfill some quantitative
+conditions, so as to guarantee authenticity. Referred to some
+latest studies in other fields, physical model as a feedback
+or constraint in the loop of deep learning is also applied to
+under water image enhancement [51] and seismic impedance
+inversion [52].
+C. Self Supervised Learning Guidance
+Self supervised learning has been attracted much attention
+in recent years, since it can help reduce the required amount of
+labeling. One can pre-train a model on unlabeled data and fine-
+tune it on a small labeled dataset. It offers great opportunity for
+SAR community where big data volume is available while the
+ground truth is usually difficult to obtain. There is a remarkable
+potential for SAR physical layer to apply for self-supervised
+learning.
+As shown in Fig. 9, two self-supervised learning paradigms
+are given. The physical layer helps to establish a pretext task
+for SAR image. In Fig. 9 a, different SAR image represen-
+tations are generated by physical layer, for instance, the sub-
+aperture images, various polarimetric feature images, etc. As
+similar to SimCLR [53] that conducted the contrastive learning
+based on data-augmentation, or NPID [54] that learned the
+optimal feature via instance-level discrimination, the surrogate
+task can be built to form a self-supervised learning. An
+illustrative example is in reference [55].
+Fig. 9 b illustrates a second line of thought, which we refer
+to as physics guided learning. Firstly, the physical layer is
+used for generating meaningful physical representations, like
+scattering mechanisms (physical layer (i) and (ii) can both
+achieve this). Meanwhile, the neural network extracts hierar-
+chical spatial features from SAR image. The crucial point is
+how to establish a connection between physical properties and
+image features. We propose to exploit physical layer (iii) to
+reveal relationships and thereby design an objective function
+for self-supervised learning.
+Our recent work [37], [42], [43] details the paradigm in
+Fig. 9 b. A physics guided network (PGN) for SAR image
+feature learning was proposed as shown in Fig. 10. First,
+a physical layer is deployed at the beginning, where the
+physical scattering properties are derived. Based on the crucial
+assumption that SAR image features and the abstract physical
+scattering mechanisms should share common attributes in
+semantic level, a surrogate task was established via the other
+physical layer that defines a loss function. The inspiration is
+from reference [56], which indicated the abstract topic mixture
+on scattering properties and the high-level image features are
+with similar semantics. Thus, we built the relation between the
+image semantics and SAR scattering characteristics. A novel
+objective function was designed to instruct self-supervised
+learning guided by physical scattering mechanisms.
+The advantages of this kind of learning paradigm lie in
+two aspects. First, the training process takes all labeled and
+unlabeled data so that the learned features generalize well in
+test set. Second, the guidance of physical information leads
+to physics awareness of features learned by neural networks,
+i.e., the DNN feature maintains physical consistency. In a
+word, the prior physical knowledge is embedded in the neural
+network. The experiments in [43] verified this quantitatively
+and qualitatively.
+Additionally, the outputs of the physically interpretable
+deep model can be further explained, which in turn inspires
+algorithm improvement. We illustrate with an example of
+sea-ice classification in polar area [42]. The physics guided
+learning is driven by physical signals that reflect the scattering
+properties of SAR image. The guided physical signals are
+visualized with t-sne in Fig. 11, where different colors in
+(a) represent semantic labels of sea-ice and each color in (b)
+indicates samples with similar physical scattering properties.
+One characteristic that can be seen is that young ice and
+water bodies have extremely similar physical representations,
+which would impede semantic discrimination. It can explain
+the physics guided learning result in [42] that about 23% test
+samples in water bodies class were predicted as young ice.
+The explanation will motivate us to improve the algorithm by,
+for instance, relaxing the physical constraints between the two
+classes.
+Similarly, a very recent work [57] was proposed for SAR
+target recognition inspired by our work [43]. The authors
+proposed a CNN under the guidance of SAR target physi-
+cal model, attributed scattering center (ASC), to extract the
+significant target features, that were successively injected into
+the classification network for more robust and interpretable
+results.
+IV. TRUSTWORTHY MODELING
+A. Why Trustworthy Modeling Needed
+The results obtained by applying AI techniques in SAR
+processing can be validated using in situ measurements of
+
+ACCEPTED VERSION
+9
+An example of Physical Layer (iii) with AI model
+image contents
+(Computer vision 
+understanding)
+CNN
+Latent 
+Dirichlet 
+Allocation
+Scattering 
+characteristics
+(Expert knowledge)
+Physics guided network (Huang et al. 2021, IGARSS)
+Topic mixture
+loss 
+function
+High-level 
+image features
+ResBlk-1
+ConvBlk
+Topic Model
+Phy. Info
+Physics-Aware 
+Features
+Physical Model
+ResBlk-2
+ResBlk-3
+Phy. Attr.
+x
+Semantic 
+Relation
+Physics-Guided Learning (Explainable and Learnable)
+√
+√
+√
+√
+SAR 
+Amplitude
+SAR 
+complex
+conv 3x3 -512
+BN + ReLU
+fc 512xTk
+64
+64
+128
+256
+512
+Physical Layer
+Physical Layer
+Fig. 10: A physics guided network was proposed [42], [43] where a novel deep learning paradigm and loss function were
+designed to associate the SAR scattering characteristics with image semantics.
+(a)
+(b)
+Fig. 11: Visualization of physics guided signals on test data by t-sne. (a) Different colors represent semantic labels of sea-
+ice. (b) The physics guided signals are grouped into eight clusters, where each color indicates samples with similar physical
+scattering properties.
+known targets. For example, a common approach for calibra-
+tion/validation of SAR data is to employ an electronic target
+(transponder) that receives a signal, applies a controllable time
+delay, and transmits the delayed signal towards the receiver
+of the bistatic/monostatic system. Such a target can be used
+to validate results related to deformation measurements (e.g.,
+atmospheric corrections) or polarimetric analysis.
+Some real world applications of SAR requires the measure-
+ment of reliability and uncertainty. One example is the sea-
+ice classification in the untraversed polar regions where the
+ice is always promptly changeable, that would result in the
+difficulty for annotation and the lack of reference data. In this
+case, the predictions in unknown polar areas obtained by AI
+model need to be trusted by humans. Strong robustness and
+plausible degree of confidence of ML system prediction are
+equally as important as its accuracy.
+Fig. 12 a indicates building orientations have a great impact
+on polarization orientation angles [58] and scattering mech-
+anisms [38]. The zoomed-in region mainly contains ortho
+buildings buildings where φ1 is close to 0◦ and orientated
+buildings with a larger φ2. The polarization orientation angles
+of ortho buildings are obviously smaller than those of oriented
+buildings. Ortho built-up areas mainly depict double scattering
+(DS) and mixed scattering (MS) where the double scattering
+dominates. The oriented buildings are with volume scattering
+(VS). Fig. 12 b shows limited robustness of recognition
+performance as the angle of test data varies when training with
+a small range of angles. A trustworthy model is expected to
+perceive SAR scattering variations with a variety of physical
+parameters and be perturbation-tolerant.
+B. Trustworthy Modeling with Uncertainty Quantification
+The development of Bayesian deep learning [59] has caught
+much attention in recent years, where the posterior distribution
+over parameters are obtained instead of the point estimation.
+A crucial property of the Bayesian method is its ability to
+quantify uncertainty, to the benefit of constructing trustworthy
+model.
+In the case of Fig. 12 b, the performance of deep neural
+networks drops dramatically when testing SAR targets of very
+different orientation angles with training data. The model is
+over-confident about some uncertain data that cannot be per-
+ceived by frequentist method. Bayesian deep neural network,
+instead, is able to calibrate the output score and measure
+the uncertainty of the prediction. Some recent studies applied
+
+Young Ice
+Glacier
+First year ice
+Water bodies
+Iceberg
+Floating ice
+Old icePhy. Scat 0
+Phy. Scat 1
+Phy. Scat 2
+Phy. Scat 3
+Phy. Scat 4
+Phy. Scat 5
+Phy. Scat 6
+Phy. Scat 71.0
+0.8
+0.6
+0.4 -
+0.2
+0.0
+0
+25
+50 
+75
+100
+125
+150
+175ACCEPTED VERSION
+10
+Fig. 12: A trustworthy model should perceive the SAR scattering variations with a variety of physical parameters and be
+perturbation-tolerant. a Differently oriented buildings reflect various polarization orientation angles and scattering mechanisms
+in a PolSAR image. b SAR targets vary violently with orientation angles. When training with a small range of angles, limited
+robustness of recognition performance is observed as the angle of test data varies.
+Bayesian deep learning for SAR sea-ice segmentation [60]–
+[62], as well as target discrimination [63]. The generated
+uncertainty map can serve as a guideline for the experts in
+annotation and improve trust between users and the model.
+Some approximation strategies of Bayesian deep neural net-
+work, such as Monte Carlo Dropout [64] and Deep ensembles
+[65], are promising for different SAR applications.
+We give an example of SAR ship detection for demonstra-
+tion. The limited labeled training data, and the interference
+of complex scattering from target itself or the inshore back-
+ground, would strongly restricted the detection performance.
+The ship detection result on some selected SAR images from
+AIR-SARShip-1.0 dataset [66], obtained by FCOS detection
+algorithm [67], are shown in the first row of Fig. 13. Compared
+with the ground truth annotation in the third row, the detection
+result appears many false alarms. It is crucial to estimate the
+model uncertainty, which is basically brought by inadequate
+training data, to evaluate the reliability of SAR ship detection
+model and provide more trustworthy predictions.
+When we apply the Monte Carlo (MC) Dropout training
+strategy to approximate the Bayesian inference [64], it captures
+the uncertainty from the existing deep model for SAR ship
+detection. The results with high uncertainty and very low
+classification scores are discarded, with only the trustworthy
+predictions preserved. The results are shown in the second row
+of Fig. 13, where the false alarms are evidently reduced. In
+the fourth and fifth SAR image, the localization uncertainty
+of two large ships visualized with circles around the corner
+of the predicted bounding box is relatively high. Intuitively,
+we can infer the reason for the weak capability of the trained
+model in detecting such kind of targets is probably the lack
+of the large size ships in the training set. The feedback from
+the uncertainty estimation should further inspire the follow-
+up studies to improve the algorithm and build trustworthier
+models.
+V. SUPPLEMENTARY EXPLANATIONS
+Beyond the hybrid and trustworthy modeling, extra expla-
+nations and other interpretable models are as well required to
+assist with developing more transparent AI model for SAR.
+The explainable artificial intelligence (XAI) techniques, such
+as gradient based, attention based, and occlusion based expla-
+nation methods, are helpful to demonstrate the effectiveness
+of integrating physical layers to achieve explainability.
+The transparent machine learning models, such as linear
+regression, decision trees, and Bayesian models, are inter-
+pretable [68]. The algorithm itself provides explanations, for
+example, Latent Dirichlet Allocation (LDA) builds a three-
+level hierarchical Bayesian model to describe the underlying
+relationship among document-topic-word. That is, the docu-
+ment can be explained with a set of topic, where each topic in
+turn, is represented by a distribution over words. Karmakar et
+al. [69] used the LDA model for SAR image data mining to
+generate the topic compositions and group them into semantic
+classes, which were fused with domain knowledge obtained
+by active learning from experts. The transparent model can
+be also integrated in a deep learning framework to approach
+the explainability. Huang et al. [42], [43] applied the LDA
+model to generate the physical attributes representation as the
+guided physics signals, rather than directly using the physical
+scattering characteristic labels to train the physics guided
+network. That is because the learned physics-aware features
+are expected to the benefit of semantic label prediction, but the
+semantic gap actually exists between the physical scattering
+characteristics and the semantic annotation. Consequently, the
+LDA model enables the guided signals to gain the abstract
+semantics and be explained with physical scattering properties.
+The other purpose for approaching the explainability lies in
+the applications of transfer learning. The manual annotation
+in SAR domain is difficult and the deficiency of labeled
+data basically restricts the development of data-driven meth-
+
+PolSAR image
+Built-up area
+Scattering mechanism
+ib
+a
+-Ss
+-DS
+HMS
+150°
+±180°
++150°
+-120°
+1VS
+0
+-30
+30
+Polarization orientation angle map
+ ZSU234
+Range
+09-
+60
+ZIL131
+d.
+口
+ D7
+BRDM2
+-90
+口
+2S1
+dt:
+90
+口
+BTR60
+T72
+120
+BTR70
+-120
+口
+BMP2
+Azimuth
+Building orientations
+-150
+150
+±180
+Test performance of different anglesACCEPTED VERSION
+11
+FCOS result
+Ground Truth
+FCOS-MC Dropout
+Fig. 13: The SAR ship detection result on AIR-SARShip-1.0 dataset [66], obtained by the detection deep learning algorithm
+FCOS [67]. Many false alarms appear in the detection result, due to the limited training data and the interference of complex
+scattering. The prediction uncertainty is estimated by MC-Dropout [64] and the uncertain results are discard to achieve a better
+performance.
+ods. Facing a wide variety of launched SAR platforms with
+various frequency bands and resolutions, as well as other
+multi-spectral, hyper-spectral, optical remote sensing sensors,
+it is of vital importance for elucidating the transferability
+of ML models among inhomogeneous data. Arrieta et al.
+[68] indicated the transferability is one of the goals toward
+reaching the explainability. Although many researches have
+explored different deep transfer learning methods in SAR
+domain [46], [70], [71], the inner transfer mechanisms of deep
+learning model still need explanation of insight. An insufficient
+understanding of the model may mislead the user toward
+inappropriate design of algorithm and fatal consequences, i.e.
+the negative transfer. Based on SAR target recognition, we
+proposed to analyze the transferability of features in DNN,
+which contributed to explaining what, where, and how to
+transfer more effectively for SAR images [72]. The inspiration
+also motivates the follow-up studies, including the SAR-
+specific pretrained model [73], the application in detection task
+[74], and the interpretability analysis of deep learning model
+in radar image [75].
+VI. CONCLUSION AND PERSPECTIVES
+In this paper, we prospect an AI paradigm shift for SAR
+applications that is explainable, physics aware and trustwor-
+thy. To ground this, SAR physical layers embedded with
+domain knowledge are introduced, which are supposed to
+be integrated and interacted with neural networks for hy-
+brid modeling. Some illustrative examples are provided to
+demonstrate the general patterns, showing algorithmic and
+scientific explainability. In addition, we emphasize the impor-
+tance and approaches of trustworthy modeling with Bayesian
+deep learning, as well as illustrating some other techniques
+such as interpretable machine learning method, explainable
+techniques, and model transferability, that would assist with
+developing more transparent AI model for SAR. In fact, this
+field belonging to interdisciplinary research is still largely
+undeveloped. To our best knowledge, such approaches have
+not been formulated in the past years. So far, only some plain
+attempts have been made. Significant questions and challenges
+remain, e.g., the feasible representation of SAR physical layer,
+the optimized form of physical constraint, and hybrid modeling
+optimization.
+Currently there are several smart sensing techniques in the
+SAR community that can be exploited as pre-processing steps
+of data fed into DNNs, e.g., multi-aperture focusing in bistatic
+configurations [76], monostatic/bistatic tomography, polari-
+metric decomposition, deformation time series. The outputs
+of these techniques can expose features that probably cannot
+be directly extracted by a DNN, especially when using a
+small training data set. The newly introduced AI paradigms
+can apply to the broad class of coherent imaging systems. A
+few examples can be enumerated: computer tomography, THz
+imaging, echographs in medicine or industrial applications,
+sonar or seismic observations in Earth sciences, or radio-
+telescope data in astrophysics.
+ACKNOWLEDGMENT
+This work was supported by the National Natural Science
+Foundation of China under Grant 62101459, China Post-
+doctoral Science Foundation under Grant BX2021248, the
+Fundamental Research Funds for the Central Universities
+under Grant G2021KY05104, and a grant of the Romanian
+Ministry of Education and Research, CNCS - UEFISCDI,
+project number PN-III-P4-ID-PCE-2020-2120, within PNCDI
+III. We would like to thank the associate editor and the
+anonymous reviewers for their great contribution to this article.
+
+baseline
+Our method
+U:0.0001
+GACCEPTED VERSION
+12
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+Huang,
+C.
+O.
+Dumitru,
+Z.
+Pan,
+B.
+Lei,
+and
+M.
+Datcu,
+“Classification of Large-Scale High-Resolution SAR Images With
+Deep Transfer Learning,” IEEE Geoscience and Remote Sensing
+Letters, vol. 18, no. 1, pp. 107–111, jan 2021. [Online]. Available:
+https://ieeexplore.ieee.org/document/8966281/
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+interpretability of deep learning model in radar image,” SCIENTIA
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+Observations and Remote Sensing, vol. 13, pp. 5823–5832, 2020.
+
+ACCEPTED VERSION
+14
+PLACE
+PHOTO
+HERE
+Mihai Datcu (DLR - German Aerospace Center
+(DLR), Oberpfaffenhofen, Wesling 82234, Germany,
+and University POLITEHNICA of Bucharest (UPB),
+[email protected]). His research interests include
+explainable and physics aware AI, smart radar sen-
+sors design, and quantum machine learning with
+applications in Earth Observation (EO). He holds
+a Professor position in AI and information theory
+with UPB, and he is Senior Scientist with DLR. He
+elaborated AI4EO programs and systems for CNES,
+DLR, EC, ESA, NASA, ROSA. He was the recipient
+of the Chaire d’excellence internationale Blaise Pascal 2017 for international
+recognition in the field of Data Science in Earth Observation. He is IEEE
+Fellow.
+PLACE
+PHOTO
+HERE
+Zhongling
+Huang
+(Northwestern
+Polytechnical
+University,
+Xi’an,
+China.
+[email protected]).
+She
+received
+the B.Sc. degree in electronic information science
+and technology from Beijing Normal University,
+Beijing, China, in 2015, and the Ph.D. degree
+from
+the
+University
+of
+Chinese
+Academy
+of
+Sciences (UCAS) and the Aerospace Information
+Research Institute, Chinese Academy of Sciences,
+Beijing, China, in 2020. She served as a visiting
+scholar in German Aerospace Center (DLR) during
+2018-2019, funded by UCAS. She is currently working in the BRain
+and Artificial INtelligence Lab (BRAIN LAB), School of Automation,
+Northwestern Polytechnical University, Xi’an, China. Her research interests
+include explainable deep learning for synthetic aperture radar (XAI4SAR),
+SAR image interpretation, deep learning, and remote sensing data mining.
+PLACE
+PHOTO
+HERE
+Andrei
+Anghel
+(University
+POLITEHNICA
+of
+Bucharest
+(UPB),
+313
+Splaiul
+Independentei,
+Bucharest
+060042,
+Romania,
+andrei.anghelatmunde.pub.ro).
+His
+current
+research
+interests
+include
+remote
+sensing,
+smart radar, microwaves and signal processing.
+Between 2012 and 2015, he worked as a PhD
+researcher with Grenoble Image Speech Signal
+Automatics
+Laboratory
+(GIPSA-lab),
+Grenoble,
+France. Presently he is Associate Professor in
+telecommunications at UPB designing bistatic SAR
+systems for ESA. He is IEEE Senior Member.
+PLACE
+PHOTO
+HERE
+Juanping Zhao (School of Electronic Information
+and Electrical Engineering, Shanghai Jiao Tong Uni-
+versity, Shanghai 200240, China, juanpingzhaoat-
+sjtu.edu.cn). Her research interests include AI for
+SAR and PolSAR image interpretation, pattern
+recognition, and machine learning. From 2018 to
+2019, she was a visiting Ph.D. student with German
+Aerospace Center (DLR), Oberpfaffenhofen, Ger-
+many.
+PLACE
+PHOTO
+HERE
+Remus
+Cacoveanu
+(University
+POLIEHNICA
+of Bucharest (UPB), 313 Splaiul Independentei,
+Bucharest 060042, Romania, rcacoveanu at ya-
+hoo.com). His main field of expertise is in the
+wireless communication systems, antennas, radar
+sensors, propagation, and microwave circuits. He
+holds an Associate Professor position in telecommu-
+nications with UPB. For more than 10 years he was
+the technical lead of the Redline Communications’
+Romanian branch and between 2011-2015 he was
+technical consultant for Blinq Networks Canada.
+One of the designed products received the “Best of WiMAX World EMEA
+2008 Industry Choice Award”, Munich Germany May 21st 2008.
+

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+CARAMALAU ET AL.: MOBYV2AL: SELF-SUPERVISED ACTIVE LEARNING
+1
+MoBYv2AL: Self-supervised Active Learning
+for Image Classification
+Razvan Caramalau1
+[email protected]
+Binod Bhattarai2
+[email protected]
+Danail Stoyanov2
+[email protected]
+Tae-Kyun Kim1,3
+[email protected]
+1 Imperial College London, UK
+2 University College London, UK
+3 School of Computing,
+KAIST, Daejeon, South Korea
+Abstract
+Active learning(AL) has recently gained popularity for deep learning(DL) models.
+This is due to efficient and informative sampling, especially when the learner requires
+large-scale labelled datasets. Commonly, the sampling and training happen in stages
+while more batches are added. One main bottleneck in this strategy is the narrow repre-
+sentation learned by the model that affects the overall AL selection.
+We present MoBYv2AL, a novel self-supervised active learning framework for im-
+age classification. Our contribution lies in lifting MoBY – one of the most successful
+self-supervised learning algorithms to the AL pipeline. Thus, we add the downstream
+task-aware objective function and optimize it jointly with contrastive loss. Further, we
+derive a data-distribution selection function from labelling the new examples. Finally,
+we test and study our pipeline robustness and performance for image classification tasks.
+We successfully achieved state-of-the-art results when compared to recent AL methods.
+Code available: https://github.com/razvancaramalau/MoBYv2AL
+1
+Introduction
+Active Learning (AL) [1, 3, 6, 13, 21, 22, 32, 42] has recently gained more popularity in
+the research community. The goal of AL is to sample the most informative and diverse
+examples from a large pool of unlabelled data to query their labels. The existing AL meth-
+ods can be grouped into two based on the selection criteria. The first group is uncertainty-
+based algorithms [12, 15, 42] that select the challenging and informative examples. Whereas
+representative-based algorithms select the most diverse examples from the data set. To select
+diverse examples, existing methods first project the images into a feature space followed by
+applying sampling techniques such as CoreSet [28]. Our work falls in the latter category.
+Prominent works on representative-based methods for AL in the past few years have tack-
+led a wide range of architectures to learn the image representations such as Convolutional
+Neural Network [42], Graph Convolutional Neural Networks [6], Bayesian Network [5],
+© 2022. The copyright of this document resides with its authors.
+It may be distributed unchanged freely in print or electronic forms.
+arXiv:2301.01531v1  [cs.CV]  4 Jan 2023
+
+2
+CARAMALAU ET AL.: MOBYV2AL: SELF-SUPERVISED ACTIVE LEARNING
+Variational Auto-Encoders [21, 32], and too few to mention. These works have proven that
+the learned features of the images have directly influenced the performance of the pipeline.
+However, these methods suffer from cold-start problem. As we know, in the early selection
+stage, we have limited annotated examples, and the above-mentioned architectures are hard
+to train with the small number of training examples. Thus, the features extracted from such
+models get biased from the beginning and continue to become sub-optimal in the subsequent
+selection stages. This problem is commonly known as cold-start problem. To address such a
+problem, recent works in AL have explored self-supervised learning methods [3, 13, 17, 20].
+Self-supervised learning methods [4, 7, 16, 19, 41] have made tremendous progress in
+generating discriminative representations of the images. Some methods have even come
+close to supervised methods in generalization [10, 16, 41]. One of the earliest works in this
+direction [13] employed consistency loss between the input image and its geometrically aug-
+mented versions along with the objective of downstream tasks. However, this method limits
+augmentation methods in the primitive form. Similarly, J. Bengar et al. [3] introduced con-
+trastive learning in AL, but the self-supervised method and end-task objective are optimised
+in multi-stage form. This makes the model sub-optimal, affecting the features’ representa-
+tiveness during selection. Simple random labelling overpasses any AL criteria. Thus, the
+existing works in this direction show explicit limitations.
+To address the issues of those methods, we introduce contrastive learning as MoBYv2
+(from its predecessor MoBY [41]) in our AL pipeline, MoBYv2AL, and jointly train the
+learner. We choose MoBY SSL because it addresses the computational complexities and
+shortcomings of other previous methods, such as SimCLR [8] or BYOL[16]. MoBY has two
+branches (as shown in Figure 1). One updates with gradient (query encoder) and another
+with momentum (key encoder). The parameters of the momentum encoder are updated in
+slow-moving averages with the query one. Moreover, the memory bank of keys from the mo-
+mentum encoder keeps long dependencies with several mini-batches. Apart from minimising
+a contrastive loss, another advantage consists in the asymmetric structure of BYOL that cap-
+tures distances from mean representation. The AL process of MoBYv2AL culminates with
+the concept-aware selection function, CoreSet.
+We state our contributions and achievements with the following:
+• a task-aware self-supervised method jointly trained with the learner - MoBYv2;
+• a quantitative evaluation with MoBYv2AL on multiple image classification bench-
+marks such as: CIFAR-10/100[24], SVHN[14] and FashionMNIST [40];
+• state-of-the-art performance over the existing AL baselines.
+2
+Related Works
+Recent Advances in Active Learning. Recent advancement in AL are either uncertainty-
+oriented [5, 12, 15, 30, 32, 42] or data representativeness [1, 28, 35]; and some of them are
+the mixture of both [2, 6, 13, 21].
+Under the pool-based setting [29], deep active learning has been initially tackled with un-
+certainty estimation. For classification tasks, this was addressed from the maximum entropy
+of the posterior or through Bayesian approximation with Monte Carlo (MC) Dropout[5, 12,
+15]. Concurrently, methods that used latent representations to sample have outperformed
+the ones that explored uncertainty. From these works, we recognise CoreSet [28] as the
+
+CARAMALAU ET AL.: MOBYV2AL: SELF-SUPERVISED ACTIVE LEARNING
+3
+most revised and competitive baseline. However, more recently, a new trend shifted the
+AL acquisition process to parameterised modules. The first work, Learning Loss [42] opti-
+mises a predictor for the loss of the learner. Still tracking uncertainty, VAAL [32] deploys
+a dedicated variational auto-encoder (VAE) to adversarial distinguish between labelled and
+unlabelled images. CoreGCN[6] and CDAL [1], on the other hand, proposed to improve
+data representativeness with graph convolutional networks and categorical contextual diver-
+sity, respectively. We test these methods in the experiments section and we further detail
+them in the Supplementary. Given the shared selection criteria with CoreSet, our MoBYv2
+AL framework falls in the representativeness-based category.
+Self-supervised Learning (SSL). For the past years, a new pillar, SSL, has arisen in unsu-
+pervised environments with linked goals to AL. Learning generalised concepts from large-
+scale data is critical for further expansion to various vision applications. We can divide
+the SSL in two approaches: consistency-based [4, 7, 33, 34] and contrastive energy-based
+[8, 10, 16, 19, 25, 41]. Consistency regularisation looks to preserve the class of unlabelled
+data even after a series of augmentations. For example, both MixMatch [4] and DINO [7]
+sharpen the averaged pseudo-labelled predictions. Conversely, contrastive learning generally
+demands pairs of positive and negative examples while optimising the similarity/contrast
+between them. Dual networks are usually deployed to evaluate these losses either within
+the batch (as in SimCLR [8]) or within a dictionary of keys (for methods like MoCo[19],
+MoBY[41]). Because contrastive learning is foundational to our proposal, we revise these
+techniques in Sec. 3.
+AL with self-supervision. In the beginning, SSL and AL evolved in parallel. Only re-
+cently, these fields have merged to further progress data sampling. Although SSL brings
+better visual constructs, there is still the question of which labelled information to allocate.
+By leveraging the unlabelled data behaviour, CSAL [13] firstly integrated MixMatch in the
+AL training and selection. We follow a similar strategy, but our end-to-end training learns
+contrastive representations. Despite this, CSAL is included in the SSL-based experiment
+as it is directly comparable. Two new works tackle contrastive learning either in acquiring
+language samples, CAL [26], or by adapting the sequential SSL SimSiam [9] in [3]. CAL is
+task-dependent on natural language processing. In [3], the multi-stage AL selection has no
+effect against random sampling. To this extent, we omit these works in our analysis.
+3
+Methodology
+In this section, we explain our pipeline in detail. First, we introduce deep active learning for
+image classification in general, followed by our contributions.
+Standard AL requires an online environment where the task learner selects and optimises
+simultaneously. We consider a large unlabelled pool of data DU from which we uniformly
+random sample and label an initial subset S0
+L << DU. Let (xL,yL) ∈ S0
+L be the available
+images and their corresponding classes. Commonly, we deploy a learner by a DL model
+comprising of a feature encoder f and a class discriminator g. The objective loss for the
+learner is the categorical cross-entropy defined as Lclassi fication = −∑S0
+L yL ·logg(f(xL)).
+Following the AL objective, we decide upon the exploration-exploitation trade-off in
+conjunction with our classification performance. Thus, we set up the exploitation rate through
+a budget b across DU/ S0
+L guided by a selection criteria. Consequently, we label the new
+sampled subset S1
+L and re-train our learner. The exploration factor is expressed by the num-
+ber of stages S0...N
+L
+we repeat this loop according to the targeted performance. While we may
+
+4
+CARAMALAU ET AL.: MOBYV2AL: SELF-SUPERVISED ACTIVE LEARNING
+Query Feature
+Encoder
+Keys Feature
+Encoder
+MLP  
+Projector
+MLP  
+Projector
+Queue
+of  
+Keys
+Queries
+Discriminator
+Contrastive
+Loss
+Task Optimiser
+Update
+Exponential moving
+average update
+Task
+Discriminator
+Cross-Entropy
+Loss
+strong/weak
+augmentation
+strong/weak
+augmentation
+Contrastive Optimiser
+Update
+Active Learning Selection
+New
+Labelled 
+Batch
+Learner
+Figure 1: SSL-AL training framework under the proposed MoBYv2AL configuration. The
+query feature encoder plays two roles: to map the features to the task discriminator for classi-
+fication; to capture contrastive visual representation with the asymmetry of the query and key
+modules. For unlabelled data, the blue lines show the back-propagation of contrastive loss
+and its exponential moving average (dashed). The green lines also include the cross-entropy
+loss during training when the annotation is available. Once training ends, the unlabelled
+samples pass through the learner for AL selection.
+limit the exploration cycles, in our proposal, we primarily focus on exploitation.
+Contrastive Semi-supervised learning framework. We tackle the contrastive unsupervised
+learning approach compared to previous semi-supervised AL techniques [13, 17, 20] that rely
+on consistency measurement. Here, we briefly re-introduce the key aspects of the previous
+SSL techniques. These are constituent to our MoBYv2AL proposal.
+The goal of self-supervised learning aligns with the AL problem, where there is plenty
+of unlabelled data and a costly annotation procedure. However, the former tends to learn
+generalised visual representation in aid of the objective task. For contrastive learning, the
+main approach to obtaining these representations is by analysing the similarity (dissimilarity)
+within the data space. From the most successful works [7, 8, 16, 19, 41], we can broadly
+form the contrastive learning process of these main parts: data augmentation with or without
+dual encoder, feature-vector projections, and similarity approximation by a dedicated loss
+function.
+We design the self-supervision framework according to MoBY [41]. This method com-
+bines two innovative prior works MoCo[19] and BYOL[16] on visual transformers [11, 38].
+MoCo[19] pioneers contrastive learning by addressing the similarity between an image and
+a specific dictionary of samples. To deploy the loss, positive examples are required through
+data augmentation of the input query together with the other negative keys from the dictio-
+nary. The self-supervision training pipeline consists of two feature encoders and two MLP
+projectors for mapping the query and the keys. Consequently, the keys are permuted in a
+large memory bank, while the positive examples are inferred through the online encoder.
+The gradient over the dictionary of keys needs a slower update. Thus, a gradual momentum
+update is implemented.
+BYOL[16], on the other hand, has a different approach for contrastive self-supervision.
+
+CARAMALAU ET AL.: MOBYV2AL: SELF-SUPERVISED ACTIVE LEARNING
+5
+It simplifies MoCo by relying only on positive examples. In this way, the memory bank can
+be discarded. The InfoNCE[27] loss is also replaced with a l2 loss given the new setting. The
+contrastive learning strategy of BYOL is indirectly obtained through batch normalisation. To
+achieve this, further modifications are proposed. Thus, the architecture of the dual encoders
+is asymmetric in regard to MoCo, and BYOL adds a prediction module to the projector of the
+online encoder. Following only positive examples, the inputs to the two networks are strong-
+augmented versions of the same image. Finally, BYOL preserves the common mode from
+the data and inherits contrastive learning when passing a slow exponential moving average
+from the online to the momentum encoder. We intuitively explore the contrastive learning
+strategies from both MoCo and BYOL and align the self-supervision with MoBY[41]. We
+further present the combined pipeline depicted in 1.
+From a design perspective, we adopt the asymmetric dual encoders from BYOL as shown
+in Fig. 1. The top branch in Figure 1 culminates with a discriminator gq to match the outputs
+from the bottom. Despite this, both branches consist of the same feature extractor archi-
+tecture followed by an MLP projector (f
+′
+q,f
+′
+k for query and key, respectively). Distinctively
+from MoBY, we tackle convolutional neural networks (CNNs) as feature encoders. More-
+over, we reduce the MLP projectors and the query discriminator to a single layer with batch
+normalisation and ReLU activation.
+The asymmetric pipeline helps to mimic the contrastive learning principle of BYOL.
+However, to include the concepts from MoCo, we minimise our objective with the InfoNCE
+loss. In this case, we will also need to keep the memory bank for the queue of keys. We
+define the contrastive loss as a sum of InfoNCE from two augmented versions of a query
+{q,q′} and of a different key {k,k′}:
+Lcontrastive = −log
+exp(q·k
+′/τ)
+∑m
+i=0 exp(q·k
+′
+i/τ) −log
+exp(q
+′ ·k/τ)
+∑m
+i=0 exp(q
+′ ·ki/τ)),
+(1)
+where m is the size of the memory bank and τ is the adjusting temperature [39]. During
+training, the online query encoder branch is updated by gradient while the key encoder takes
+the slow-moving average with momentum. We ensure with this combined design the preser-
+vation of both MoCo and BYOL representation concepts. On the one hand, the asymmetric
+structure indirectly finds discrepancies from the average image with moving average and
+batch normalisation. On the other hand, the contrastive loss with the queue of different keys
+maintains the direct distinctiveness between the images.
+The standard SSL techniques MoCo, BYOL and MoBY demand the supervision stage
+where the pre-trained models are fine-tuned for the task objective. Such multi-stage pipelines
+seem ineffective in AL [3]. In this paper, we extended the SSL pipeline of MoBY to minimise
+both the self-supervised objective and downstream task objective jointly.
+Joint Objective. A final step to clarify before presenting the joint training procedure is
+data augmentation. MoBY derives the augmentation strategy from BYOL, where the inputs
+suffer strong transformations. In our proposal, we choose an alternation between strong and
+weak augmentation, similarly to MoCov2[10]. This change boosted the performance of its
+predecessor [19]. We also observed in our experiments that using only strong augmentations
+can affect the optimisation of the task-aware branch. The weak augmentations comprise
+horizontal flips and random crops. In addition, the strong transformation includes colour
+jitter (on brightness, contrast, saturation, hue), Gaussian blur, grayscale conversion and pixel
+inversion (solarise). From equation 1, {q,k} can be referred as the weak transformations of
+query and key, and {q
+′,k
+′} their corresponding stronger versions.
+
+6
+CARAMALAU ET AL.: MOBYV2AL: SELF-SUPERVISED ACTIVE LEARNING
+With all these elements in place, we can change the learner from the existing AL frame-
+work with the modified MoBY and train jointly the pipeline. Starting from the first cycle, we
+consider the available labelled samples (xL,yL) ∈ S0
+L and the remaining unlabelled xU ∈ DU
+as queries and keys. A strong augmentation is marked as {˜xL
+q, ˜xL
+k}, while a weak is rep-
+resented with {¯xL
+q, ¯xL
+k}. When training, we alternate between batches of labelled and unla-
+belled data with every inference. Therefore, we back-propagate only the contrastive loss for
+the unlabelled to 1. In this context, given the pipeline from Figure 1 for this contrastive loss
+LU
+contrastive(q,q
+′;k,k
+′), {q,k} and {q
+′,k
+′} can be obtained so:
+{q,q
+′} = gq (f
+′
+q (fq({¯xU
+q , ˜xU
+q }))),
+(2)
+{k,k
+′} = f
+′
+k (fk ({¯xU
+k , ˜xU
+k })).
+(3)
+Similarly, we can compute LL
+contrastive, the contrastive loss for the labelled images. In
+addition, we also minimise the categorical cross-entropy, Lclassification, with the output from
+the task discriminator. Once computed, we back-propagate both the contrastive and the
+classification loss. Therefore, the combined loss, adjusted by a scaling factor λc, can be
+expressed as:
+LL
+combined = Lclassi fication +λcLL
+contrastive
+(4)
+While the contrastive loss is computed continuously regarding the classification loss, we de-
+cide to reduce its influence over the gradients with λc = 0.5. Finally, it is worth mentioning
+that the exponential moving average and the queue of keys are updated on the bottom branch
+for both labelled and unlabelled samples.
+Unlabelled samples selection. We emphasise that our proposal minimises the self-supervised
+loss inspired by MoBY. With this, the end-task objective jointly enriches the visual repre-
+sentations of the data compared to the standard AL strategy. AL selection methods that rely
+on the learner’s data distribution will perform better. CoreSet [28] has been proven to be
+effective in such scenario. To this extent, we primarily choose this selection function with
+MoBYv2AL. Briefly, CoreSet aims to find a subset of data points where a constant radius
+bounds the loss difference with the entire data space. This technique is approximated with
+k-Centre Greedy algorithm [37] in the euclidean space of our feature encoder outputs fq(x).
+A thorough visual selection of different AL selection approaches together with CoreSet in
+presented in the Supplementary.
+4
+Experiments
+Datasets.For the quantitative evaluation, we put forward four well-known image classifica-
+tion datasets: CIFAR-10, CIFAR-100 [24], SVHN[14] and FashionMNIST[40].
+Models. We mentioned in 3 that we use different CNNs for feature encoders. To show
+that MoBYv2AL is robust to architectural changes, we opt for VGG-16 [31] in the CIFAR-
+10/100 quantitative experiments and for ResNet-18 [18] in SVHN and FashionMNIST.
+Training settings. We train at every selection stage for 200 epochs, and we keep the batch
+size at 128. The dictionary size for the keys m is set up as in MoBY at 4096. We noticed
+in our experiments that the contrastive and cross-entropy loss converge together after 200
+epochs. The learning rate starts at 0.01, and it follows a schedule for the queue encoder and
+
+CARAMALAU ET AL.: MOBYV2AL: SELF-SUPERVISED ACTIVE LEARNING
+7
+Figure 2: Evaluations on CIFAR-10 (left), CIFAR-100 (right) [Zoom in for better view]
+Figure 3: Evaluations on SVHN (left), FashionMNIST (right) [Zoom in for better view]
+task discriminator that decreases ten times at 120 and 160 epochs. However, we keep the
+momentum scheduler update in the key bottom branch (gradual momentum increment from
+0.99). In the contrastive loss, for both queues, we fix the temperature parameter to 0.2.
+AL settings. We followed the AL settings of VAAL[32], CDAL[1] and CoreGCN[6]. For
+more details, please see Supplementary.
+Baselines. We compared our method MoBYv2AL with a wide range of methods in active
+learning such as: MC Dropout [15], DBAL [12], Learning Loss[42], VAAL[32], , Learning
+Loss [42], CoreGCN[6] and CDAL[1].
+4.1
+Quantitative experiments
+CIFAR10/100. To maintain a fair comparison, in Figure 2, we report the performance charts
+obtained by CDAL[1] and VAAL[32]. All methods use VGG-16 for the feature encoder.
+MoBYv2AL has a considerable advantage with the proposed SSL framework in the CIFAR-
+10/100 experiments from the first selection stage. In both scenarios, we gain 20% testing
+accuracy over standard learning (62% and 28% on CIFAR-10/100). This justifies the impor-
+tance of the joint training framework from MoBYv2AL.
+Our pipeline’s more refined visual representations direct helpful information to the Core-
+Set selection method. Thus, we notice a gradual increase in Figure 2, where after 7 cy-
+cles, with 40% labelled data, MoBYv2AL achieves 89.6% mean accuracy on CIFAR-10
+and 63.1% on CIFAR-100. Another observation in the CIFAR-10 experiment is that the AL
+performance saturates faster than in CIFAR-100. This effect occurs due to a large initial
+labelled pool in relation to the complexity of the task. MoBYv2 exploits more contrastive
+information, and it limits the exploratory potential in the next stages.
+
+Testing performance on CIFAR-10
+90
+(mean of 5 trials)
+85
+80
+75
+Test accuracy (
+70
+Random
+VAAL
+MC Dropout
+CDAL
+65
+DBAL
+MoBYv2AL
+CoreSet
+60
+10
+15
+20
+25
+30
+35
+40
+Percentage [%l of labelledTesting performance on CIFAR-100
+65
+60
+55
+50
+45
+40
+Random
+VAAL
+35
+MC Dropout
+CDAL
+DBAL
+MoBYv2AL
+30
+CoreSet
+10
+15
+20
+25
+30
+35
+40
+Percentage [%l of labelledTesting performance on SVHN
+95
+trials)
+90
+85
+80
+Random
+UncertainGCN
+75
+VAAL
+Learning Loss
+70
+CoreSet
+CoreGCN
+MoBYv2AL
+65
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+9000
+10000
+Number of labelled samplesTesting accuracy on FashionMNIST
+94
+(mean of 5 trials)
+92
+90
+88
+Test accuracy (
+86
+Random
+CoreSet
+84
+UncertainGCN
+CoreGCN
+VAAL
+MoBYv2AL
+82
+Learning Loss
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+8000
+9000
+10000
+Number of labelled samples8
+CARAMALAU ET AL.: MOBYV2AL: SELF-SUPERVISED ACTIVE LEARNING
+SVHN/FashionMNIST. We can deduct, from Figure 3 as well, that MoBYv2AL balances
+the exploration-exploitation trade-off when the initial labelled set is relatively low to the
+number of classes. The dark dashed line displays the supervised baseline training on the
+entire labelled set. While on CIFAR-10/100 and FashionMNIST, MoBYv2AL reaches com-
+parable performance, by the end of the cycles, on SVHN, it surpasses after the sixth one
+(95%). Here, we emphasise the relevance of the strong/weak augmentations in enriching the
+discrete data distribution. Furthermore, grayscale data (as in FashionMNIST) can also ben-
+efit from the proposed AL framework. In Figure 3, we keep the same results of the previous
+baselines from CoreGCN[6]. Even under these settings, we outperform the state-of-the-arts
+with a noticeable consistent margin: for SVHN and FashionMNIST a gap of at least 2% -
+3%.
+SSL-AL method vs percentages of labelled
+10%
+15%
+20%
+25%
+30%
+CSAL
+58.1
+63.76
+67.13
+69.28
+70.08
+MoBYv2AL
+67.66
+68.24
+68.49
+68.57
+70.11
+Table 1: Comparison with the SSL-AL method CSAL on CIFAR-100 with a WideResNet-28
+learner
+Comparison with other SSL-AL. MoBYv2 leverages unlabelled data for contrastive learn-
+ing in the AL framework. Previously, we chose this amount of data equal to the avail-
+able labelled samples. Therefore, at every AL cycle, this size increases with the newly
+selected data. Another recent SSL-AL baseline CSAL[13], however, deployed the consis-
+tency measurements from MixMatch[4] on the entire unlabelled data. We could identify
+that MoBYv2AL over-exploits as CSAL the captured representation under these conditions.
+We further compare the 2 methods on CIFAR-100 in Table 1 and adjust the feature encoder
+to WideResNet-28[43]. In this experiment, MoBYv2AL maintains the initial performance
+gain.
+Imbalanced dataset experiment. Apart from SVHN, all the previous experiments have
+a uniform distribution over the classes. This rarely occurs during real-world acquisition
+scenarios. Therefore, as in CoreGCN, we simulate an imbalanced CIFAR-10 unlabelled set.
+Each of the ten classes has originally 5000 training examples. We decide to reduce 5 of
+the classes to 500 images (resulting in a pool of 27500). The learner contains a ResNet-18
+encoder, and it is trained with an initial set of 1000 labelled examples. We apply MoBYv2AL
+together with the other baselines from CoreGCN[6] for 7 cycles. Figure 4(left) presents
+the ability of MoBYv2AL to outperform the previous methods even in possible real-world
+environments. Investigation of long-tail distributions is still part of our future work.
+Figure 4: CIFAR-10 imbalanced dataset experiment(left); Mitigating the distribution shift
+with MoBYv2AL(right) [Zoom in for better view]
+
+Testing performance on CIFAR-1O imbalanced
+80
+70
+60
+50
+Random
+CoreSet
+UncertainGCN
+CoreGCN
+VAAL
+MoBYv2AL
+40
+Learning Loss
+1000
+2000
+3000
+4000
+5000
+6000
+7000
+Number of labelled samplesCIFAR1O - Distribution shift - Random vs Lowest contrastive loss unlabelled during training
+First Stage 1000 labeled images
+Second Stage 2000 labeled images
+Second Stage 20o0 labeled images without Distribution Shift
+100
+90
+90
+88
+89
+84
+85
+83.85
+83
+81
+80
+80
+81
+81
+80
+78
+72
+70
+67
+67
+65
+63
+63
+61
+60
+56
+54
+51
+50
+46
+39
+40
+36
+30
+airplane
+car
+bird
+cat
+deer
+dog
+frog
+horse
+ship
+truckCARAMALAU ET AL.: MOBYV2AL: SELF-SUPERVISED ACTIVE LEARNING
+9
+4.2
+Distribution shift discussion
+In deep AL, the cyclical process of re-training the learner with the new labelled data may
+result in optimising to different local minima. Therefore, the exploration and exploitation of
+the AL method will be affected by this distribution shift at every stage. During experiments,
+this is commonly shown through jaggy curves (especially for uncertainty-based methods like
+MC Dropout[15], DBAL[12] or UncertainGCN[6]). To address this known issue [23], we
+analyse MoBYv2AL performance on the entire CIFAR-10 training set when providing 1000
+and 2000 samples.
+The dark blue bars of each class in Figure 4 (right) level the corresponding classification
+accuracy with the first 1000 random samples. Tracking the performance on the entire set
+challenges the learner to prefer certain classes. We continue to select with MoBYv2AL
+another set of images. Consequently, the resulted accuracy is displayed by the cyan bar. We
+can clearly observe that the minima shifted in a different direction where only some classes
+improved at the expense of the others. To mitigate this shift, we investigated what impact
+the unlabelled samples have in our end-to-end training. These samples play a key role in
+building up the dictionary of keys. Our insight is that the CoreSet selection on MoBYv2
+data representation targets primarily high contrastive samples. We can control this effect by
+customising the unlabelled set deployed in training our learners. To this extent, we propose
+to use the unlabelled data with the lowest contrastive loss. In Figure 4 (right), we displayed
+on green bars the performance with this mechanism. From an initial 1000 set accuracy
+(dark blue) we get an effective linear increase for all the 10 classes. This effect is consistent
+throughout all the previous quantitative experiments as well.
+4.3
+SSL modules variation and ablation study
+We continue to motivate the proposed design of MoBYv2AL with a set of ablation experi-
+ments and by varying its SSL module. On the left side of Table 2, we swap in the end-to-end
+training pipeline the original version of MoBY [41] and the preceding SSL state-of-the-arts,
+MoCov2 [10] and BYOL [16]. Apart from MoBY, the learner did not converge on any selec-
+tion cycle with the other SSL modules. Thus, the setup of large batches and specific training
+conditions (low learning rates, cosine scheduler) and learners can hardly adapt to this semi-
+supervision configuration. For MoBYv2AL, the weak-augmented inferences to the learner
+stabilise its performance in regard to the original version. Furthermore, our method distances
+by 4% class accuracy with each AL cycle. One can argue that our SSL framework comprises
+SSL model / No. of labelled
+1000
+2000
+3000
+MoCov2
+11.62±.9
+11.92±.6
+12.89±.6
+BYOL
+12.32±.7
+11.72±.4
+11.47±.2
+MoBY
+62.62±.4
+72±.5
+76.43±.1
+MoBYv2AL (Ours)
+63.06±.5
+76.04±.6
+80.63±.3
+MoBYv2AL / No. of labelled
+1000
+2000
+3000
+w/o Discriminator
+60.44±.4
+72.53±.8
+77.89±.3
+w/o MLP Projector
+58.57±.6
+71.96±.5
+77.02±.6
+w/o Strong Augmentation
+47.7±.4
+58±.5
+64.85±.5
+MoBYv2AL (Ours)
+63.06±.5
+76.04±.6
+80.63±.3
+Table 2: Variation of SSL pipeline (left) and ablation study of MoBYv2AL (right). Average
+testing performance (5 trials) on CIFAR-10 for 3 AL cycles with ResNet-18 encoder
+several building blocks, and its implementation can deter developers. While we value the
+significant dominance of MoBYv2 in AL selection, we still motivate the relevance of each
+part in Table 2 (right). In the ablation evaluation, we successfully remove the queue Discrim-
+inator and the MLP projectors. As a result, we detect a continuous accuracy drop. Projecting
+larger features and simulating the asymmetry brings the advantage of contrastive learning in
+MoBYv2. Moreover, strong augmentations also play a crucial role in the SSL pipeline.
+
+10
+CARAMALAU ET AL.: MOBYV2AL: SELF-SUPERVISED ACTIVE LEARNING
+MoBYv2AL
+1000
+2000
+3000
+Multi-stage semi supervised
+34.8±.1
+34.96±.2
+35.09±.1
+Jointly with end-task
+63.06±.5
+76.04±.6
+80.63±.3
+SSL method
+Supervised
+MoCov2
+BYOL
+DINO
+MoBYv2AL
+CIFAR-10 Test accuracy
+90.08
+76.7
+77.89
+81.2
+88.62
+Table 3: Multi-stage SSL-AL vs Jointly end-task AL (left). Semi-supervised learning com-
+parison (right). Testing performance on CIFAR-10 with ResNet-18 encoder
+4.4
+SSL results and multi-stage AL
+MoBYv2 SSL for AL strategy is designed in a joint manner with the end task. Despite this,
+the recent work [3] that proposes contrastive learning with SimSiam[9] adopts multi-stage
+learning for the learner. The pipeline proposed fails to sample better than random in the
+AL paradigm. In Table 3(left), we experiment with MoBYv2 the multi-stage training (with
+unsupervised contrastive learning and second task fine-tuning) for CIFAR-10. We observe
+that the performance suffers in context to the end-task, where limited labelled examples are
+used. Similarly to [3], we also notice a minor improvement when adding more selected data
+with CoreSet. To this extent, we decided to use the entire training set during fine-tuning. We
+re-iterated the same experiment for SSL cross-validation with MoCov2[10], DINO[7] and
+BYOL[16].
+5
+Limitations and Conclusions
+Although we can adapt MoBYv2AL to other applications, we expect further research on the
+effects of the augmentations and the momentum encoder. Another limiting factor should
+be analysed at the first AL selection stage, where developers may tune the exploration-
+exploitation ratio to avoid saturation.
+We have presented an SSL-based AL framework for image classification. The main
+contributions lie in the task-aware contrastive learning pipeline. MoBYv2AL retains the
+higher visual concepts and aligns them with the downstream task. The joint training is
+efficient and modular, allowing diverse backbones and sampling functions. We conduct
+quantitative experiments and demonstrate the state-of-the-art on four datasets. Our method
+shows robustness even in simulated class-imbalanced data pools.
+6
+Acknowledgements
+This work is in part sponsored by KAIA grant (22CTAP-C163793-02, MOLIT), NST grant
+(CRC 21011, MSIT), KOCCA grant (R2022020028, MCST) and the Samsung Display cor-
+poration. BB and DS are funded in whole, or in part, by the Wellcome/EPSRC Centre for
+Interventional and Surgical Sciences (WEISS) [203145/Z/16/Z]; the Engineering and Phys-
+ical Sciences Research Council (EPSRC) [EP/P027938/1, EP/R004080/1, EP/P012841/1];
+and the Royal Academy of Engineering Chair in Emerging Technologies Scheme; and En-
+doMapper project by Horizon 2020 FET (GA 863146).
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+14
+CARAMALAU ET AL.: MOBYV2AL: SELF-SUPERVISED ACTIVE LEARNING
+A
+Detailed settings for the AL experiments on
+MoBYv2AL
+Datasets.For the quantitative evaluation, we put forward four well-known image classifica-
+tion datasets: CIFAR-10, CIFAR-100 [24], SVHN[14] and FashionMNIST[40]. CIFAR-10
+and CIFAR-100 contain the same 50000 training examples but with different labelling sys-
+tems (10 and 100 classes). SVHN and FashionMNIST are separated into ten classes each
+as CIFAR-10. However, both datasets are larger, with 73257 coloured street numbers and
+60000 grayscale images for FashionMNIST. Although CIFAR-10/100 and FashionMNIST
+have class-balanced data, this is not the case for SVHN. From another perspective, deploying
+grayscale images from FashionMNIST challenges our contrastive learning approach, previ-
+ously customised to RGB data.
+Models. We mentioned in the Methodology that we use different CNNs for feature en-
+coders.
+To show that MoBYv2 is robust to architectural changes, we opt for VGG-16
+[31] in the CIFAR-10/100 quantitative experiments and for ResNet-18 [18] in SVHN and
+FashionMNIST. Moreover, for the SSL comparison with CSAL we align the encoder with
+WideResNet-28[43].
+AL settings. Under the exploration-exploitation trade-off, we characterise the budget to se-
+lect as an exploiting factor while the exploration is captured in the number selection cycles.
+The initial random-sampled labelled dataset varies between the CIFAR-10/100 experiments
+and SVHN/FashionMNIST. For CIFAR-10/100, we consider 10% (5000) of the entire train-
+ing set as labelled and the rest as unlabelled data. The budget is limited to 5% (2500) samples
+for selection, and we repeat this cycle seven times. In the second set of experiments, we test
+our method in a more restrictive environment with a starting set of 1000 labelled and a sim-
+ilar fixed budget. Despite this, we expanded the exploration to 10 cycles reaching 10000
+labelled data. As a performance measurement, we evaluate the average of 5 trials testing
+accuracy in the AL framework.
+B
+Selection function analysis
+Figure B.1: Quantitative evaluation with different selection functions for CIFAR-10 (left),
+CIFAR-100 (right) [Zoom in for better view]
+Our proposed pipeline, MoBYv2AL, can easily adapt to multiple selection methods.
+Here, we quantitatively motivate the choice of CoreSet from section 3. Therefore, we re-
+
+Different selection functions for MoBYv2 on CIFAR-1O
+90
+88
+86
+84
+Random
+High Contrastive Loss
+Max Entropy
+CoreSet
+82
+10
+15
+20
+25
+30
+35
+40
+Percentage
+「%l of labelledDifferent selection functions for MoBYv2 on CIFAR-1o0
+65.0
+Test accuracy (mean of 5 trials)
+62.5
+60.0
+57.5
+55.0
+52.5
+50.0
+47.5
+Max Entropy
+High Contrastive Loss
+45.0
+High Contrastive Feauture variance
+CoreSet
+10
+15
+20
+25
+30
+35
+40
+Percentage [%l of labelledCARAMALAU ET AL.: MOBYV2AL: SELF-SUPERVISED ACTIVE LEARNING
+15
+evaluate MoBYv2AL on CIFAR-10/100 benchmarks in Figure B.1. We vary the selection of
+the new budget between random, maximum class entropy and CoreSet. Intuitively, we also
+analyse the effect of selecting unlabelled examples with high contrastive loss.
+In both benchmarks, sampling with random or max entropy benefits the less MoBYv2AL
+pipeline. On the other hand, a representativeness-oriented method like CoreSet suits our
+hypothesis better. When sampling with high contrastive loss, we detected repetitive examples
+from some specific classes. This can be explained by higher contextual variance in that
+category. Specifically, on CIFAR-10, animal classes (cat, deer, dog), with stronger patterns,
+were more preferred than the vehicle ones (car, truck, ship).
+For a better visual analysis, we have simulated a toy-set experiment with the first five
+classes from SVHN. Here, we take t-SNE[36] representations of the MoBYv2AL query en-
+coder outputs of unlabelled data. In Figure B.2, the samples marked with crosses construct
+the new labelled set. The selection behaviour of the Max Entropy and CoreSet can be inter-
+Max Entropy
+Highest contrastive loss
+CoreSet
+Figure B.2: Qualitative AL selection analysis on MoBYv2. t-SNE representations at the first
+selection stage for 5 classes of SVHN. [Zoom in for better view]
+preted as expected: on the left side, the uncertainty-based technique tracks the most class-
+variant images; CoreSet, on the right side, samples both in and out-of-distribution according
+to the Euclidean space.
+
+0
+1
+2
+3
+4-
+40
+20
+0
+20
+40

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+arXiv:2301.04487v1  [stat.ME]  11 Jan 2023
+Testing separability for continuous functional data
+Holger Dette, Gauthier Dierickx, Tim Kutta
+January 12, 2023
+Abstract
+Analyzing the covariance structure of data is a fundamental task of statistics.
+While this task is
+simple for low-dimensional observations, it becomes challenging for more intricate objects, such as
+multivariate functions.
+Here, the covariance can be so complex that just saving a non-parametric
+estimate is impractical and structural assumptions are necessary to tame the model. One popular
+assumption for space-time data is separability of the covariance into purely spatial and temporal
+factors.
+In this paper, we present a new test for separability in the context of dependent functional time
+series. While most of the related work studies functional data in a Hilbert space of square integrable
+functions, we model the observations as objects in the space of continuous functions equipped with
+the supremum norm. We argue that this (mathematically challenging) setup enhances interpretability
+for users and is more in line with practical preprocessing. Our test statistic measures the maximal
+deviation between the estimated covariance kernel and a separable approximation. Critical values are
+obtained by a non-standard multiplier bootstrap for dependent data. We prove the statistical validity
+of our approach and demonstrate its practicability in a simulation study and a data example.
+Keywords: Dependent multiplier bootstrap; space-time data; separability; Banach space; functional time
+series.
+MSC: 62G10; 62R10.
+1
+Introduction
+Over the last decades, the analysis of high-dimensional space-time data has become a cornerstone of geo-
+statistics. New technologies allow the collection of high-frequency and high-resolution measurements for
+variables such as temperature, magnetic fields or pollutant concentrations (see, for example, Gromenko et al.
+(2012); Aue et al. (2018); King et al. (2018)). One way to analyze such data, is to smooth it over space
+and time, which yields time series of spatio-temporal processes. This approach of reconstructing and ana-
+lyzing random functions follows the paradigm of functional data analysis (FDA) and has recently gained
+attraction in the geostatistical community (for overviews see, e.g., Martínez-Hernández and Genton (2020)
+and the monograph of Mateu and Giraldo (2022)).
+While the analysis of such processes promises deep scientific insights, their high complexity can push
+statistical methods to their limit. For example, commonly used tools such as PCA and Kriging hinge on
+1
+
+an approximation of the processes’ covariance operator - an object that can be too massive to be stored or
+to be inverted for the purpose of prediction. For a concise overview of these computational challenges, we
+refer to Table 1 in Masak et al. (2020). To reduce complexity, many works impose structural assumptions
+on the covariance, one of them being separability.
+Roughly speaking, separability states that the covariance of a space-time process can be decomposed
+into a purely temporal and a purely spatial component. The implied elimination of space-time interactions
+cuts the number of model parameters and makes the covariance tractable again (see Genton (2007)).
+Besides, separability entails a product structure for the principal components, facilitating the construction
+of estimators and inference methods (see Gromenko et al. (2012, 2016)). To rigorously define separability,
+consider a stochastic process {X(s, t) | s ∈ K1, t ∈ K2}, with one argument in a spatial domain K1 and one
+in a temporal domain K2. Then, under suitable conditions (see, for example, Janson and Kaijser (2015)),
+its covariance operator can be defined point-wise as
+C(s, t, s′, t′) := E [(X(s, t) − EX(s, t))(X(s′, t′) − EX(s′, t′))] .
+We call C separable, if there exist two functions C1 (spatial), C2 (temporal), such that
+C(s, t, s′, t′) = C1(s, s′) · C2(t, t′)
+∀s, s′ ∈ K1, t, t′ ∈ K2.
+As pointed out before, the product structure of separability prunes model parameters, making the model
+statistically and computationally more tractable.
+However, separability is not a free lunch for space-time data. Indeed, erroneously assuming a sep-
+arable model can lead to inconsistent estimates and biased inference results. This point is crucial, as
+separability is rarely self-evident, as noticed by many authors (see Scaccia and Martin (2005); Genton
+(2007); Aston et al. (2017), among many others).
+To address this issue, statistical tests have been
+proposed to examine separability, such as for finite dimensional data by Matsuda and Yajima (2004);
+Scaccia and Martin (2005); Fuentes (2006); Crujeiras et al. (2010). More recently, nonparametric methods
+tailored to functional observations have been devised by Aston et al. (2017); Constantinou et al. (2017,
+2018); Bagchi and Dette (2020); Dette et al. (2022).
+While these latter works differ in terms of their
+inference strategies, they share the mathematical setup of modelling observations in a space of square inte-
+grable functions. This approach is standard in FDA (see the monographs of Horváth and Kokoszka (2012);
+Hsing and Eubank (2015)) and provides the most immediate extension of finite dimensional methodology
+to function spaces. Nevertheless, the choice of L2-spaces is usually more informed by mathematical con-
+venience, than by practicability. Indeed, as many undergraduate textbooks point out, the L2-distance
+is hard to geometrically interpret and often stuns the novice by its unintuitive notion of convergence.
+More specific to FDA, basing a theory on L2-spaces usually ignores structural features of the functions,
+such as continuity. Notice that the bulk of FDA relies on continuous, non-parametric curve estimation as
+preprocessing. In such cases, insisting on an L2-framework can feel disjoint from a user’s intuition and
+defy heuristic interpretations of results.
+Following recent work of Degras (2011); Cao et al. (2012); Degras (2017) and Dette et al. (2020), we
+opt instead to conduct FDA on the in our opinion more natural space of continuous functions equipped
+with the supremum norm. On this more intricate space, we study functional space-time processes and
+advance a new test for separability of the covariance, which is based on an estimate of the maximum
+2
+
+deviation between the covariance operator and an approximation by a covariance operator from a separable
+process. We combine the profound theory of weak convergence of stochastic processes (see, for example,
+van der Vaart and Wellner (1996) or Giné and Nickl (2016) among many others) with new differentiability
+results for these separability measures to study the asymptotic properties of the corresponding estimates.
+To improve the finite sample properties and to avoid the estimation of complicated nuisance parameters
+we propose a multiplier bootstrap for dependent data as a more practicable alternative. In particular, our
+results hold under weaker dependence and stationarity assumptions compared to previous works.
+The rest of this paper is organized as follows. Section 2 provides a brief introduction to random
+variables on the space of continuous functions and the notion of separability. Subsequently, we discuss
+three methods from the related literature to approximate a covariance kernel C by a separable version (we
+develop theory for all three approximations at a later point). In Section 3, we present the test statistic for
+the hypothesis of separability and prove its weak convergence. To approximate the limiting distribution,
+we discuss a non-standard multiplier bootstrap for dependent data. Section 4 is dedicated to the finite
+sample properties of our test, which we study by virtue of simulations as well as a data example. Finally,
+all proofs and technical details are deferred to the Appendix.
+2
+Mathematical concepts
+In this section, we lay the mathematical foundations of FDA in the space of continuous functions. Sec-
+tion 2.1 begins with a review of random, continuous functions, as well as basic concepts, such as functional
+expectations and covariance kernels. Subsequently, we define the model assumption of separability, which
+is the focus of our below statistical analysis. In Section 2.2, we discuss three methods to approximate a
+kernel A by a separable version Ax and show that each approximation map (A �→ Ax) is well-defined and
+differentiable (see Theorem 2.2).
+2.1
+Mathematical preliminaries
+Let
+C(K) :=
+�
+f : K → R | f continuous
+�
+,
+denote the space of continuous, real valued functions defined on a non-empty, compact set K ⊂ Rd, which
+equipped with the common supremum norm (or "sup-norm" for short)
+∥f∥ := sup
+t∈K
+|f(t)|
+(2.1)
+is a Banach space. If there is no danger of confusion, we sometimes refer to a function f ∈ C(K) by its
+evaluation f(t). Moreover, we also use the notation ∥ · ∥ to refer to the max-norm (maximum absolute
+entry) for matrices or vectors (corresponding to a finite set K in (2.1)).
+Letting (Ω, A, P) denote a complete probability space, we call a map X : (Ω, A, P) → C(K) a
+random C(K)-valued function, if X is Borel-measurable w.r.t. the sup-norm. We point out that Borel-
+measurability of X (as a Banach space valued function) is equivalent to measurability of the real valued
+marginals X(t) : (Ω, A, P) → R for all t ∈ K. Supposing that the first absolute moment of X exists, in the
+sense that E∥X∥ < ∞, the expectation of X is well-defined (in the Bochner sense), with EX ∈ C(K) and
+3
+
+equal to the point-wise expectation E[X(t)] for any t ∈ K. If the stronger moment condition E∥X∥2 < ∞
+holds, we can define the expectation of the product function {(X(s)− EX(s))(X(t)− EX(t))}s,t∈K on the
+space C(K2) := C(K × K). We call this the covariance kernel and define it point-wise as
+C(s, t) := E[X(s) − EX(s)][X(t) − EX(t)].
+For further details on defining moment of Banach space valued random variables we refer to Janson and Kaijser
+(2015). As common in the study of functional data, we sometimes consider for a continuous kernel function
+A ∈ C(K2) the corresponding integral operator {f(t)}t∈K �→ {
+�
+K A(s, t)f(s)ds}t∈K. For ease of notation,
+we usually identify the integral operator with the associated kernel (notice that the relation is one-one) and
+define the expression A[f] := {
+�
+K A(s, t)f(s)ds}t∈K. In particular, we call the integral operator associated
+with C the covariance operator. Finally, we observe that C(K) can be understood as a subspace of L2(K),
+the space of square integrable functions, equipped with the canonical semi-norm ∥f∥L2 := {
+�
+K f(s)2ds}1/2.
+Recall that the L2-topology is weaker than the uniform topology, since we consider functions with compact
+support.
+In the next step, we consider the mathematical property of separability. Let K1 ⊂ Rd, K2 ⊂ Rq be
+compact and non-empty sets. Then we say that a kernel A ∈ C((K1 × K2)2) is separable, if there exist
+kernels Ai ∈ C(K2
+i ) for i = 1, 2, s.t.
+A(s, t, s′, t′) = A1(s, s′) A2(t, t′)
+∀s, s′ ∈ K1, t, t′ ∈ K2.
+Later, we will study the case of a separable covariance kernel C of a random function X. In order to
+assess separability, we will employ an estimator ˆCN and compare it to a separable approximation. The
+construction of such separable approximations is the subject of the next section.
+2.2
+Separable approximations
+In this section, we discuss the approximation of a kernel function A ∈ C((K1 × K2)2) by the (separable)
+product of two functions B = A1 · A2 with Ai ∈ C(K2
+i ) and i = 1, 2. This approximation is key to our
+subsequent analysis, as it characterizes separability by the vanishing goodness-of-fit measure ∥A−B∥ = 0.
+How then should we construct a separable approximation? The most natural way might be to optimize
+over all separable maps w.r.t. the norm ∥ · ∥, i.e., to find
+Bopt ∈ argmin{∥A − B∥ : B separable}.
+Unfortunately, this type of approximation is computationally infeasible. To illustrate this point, let us
+consider the analogue problem of finding an optimal separable approximation for a d × d-matrix w.r.t.
+to the max-norm (also denoted by ∥ · ∥).
+This problem reflects the search for an optimal, separable
+approximation for discretized a version of A (as it would be saved on a computer in a real application).
+In the following Lemma "⊗" denotes the well-known Kronecker product for matrices.
+Remark 2.1. Let M ∈ Rd×d be a real valued matrix with d = pq and p, q ∈ N. Then the problem of
+finding a solution to the following minimization problem
+“minimize ∥M − M1 ⊗ M2∥
+with M1 ∈ Rp×p,
+M2 ∈ Rq×q”,
+is NP-complete.
+4
+
+Proof. The problem of finding optimal rank-1-approximations with regard to the max-norm is an NP-
+complete problem, according to Gillis and Shitov (2017). According to Section 2 of Genton (2007) the
+problem of finding separable approximations is equivalent to finding rank-1-approximations.
+The notion of NP-completeness in the above Remark 2.1 refers to a set of problems which are (by today’s
+knowledge) not efficiently solvable (for details, see Garey and Johnson (1979)).
+Evidently, if it is not
+feasible to find max-norm approximations for matrices, it has a fortiori to be true for continuous kernels
+(which then cannot be optimally approximated by a separable kernel even on a grid with reasonable effort).
+This insight suggests to use different, suboptimal, but computationally feasible approximation methods. In
+the following, we discuss three procedures which have been applied previously in the study of separability
+for L2-functions.
+2.2.1
+Partial trace approximations
+Let A ∈ C((K1 × K2)2) be a kernel function. We can then define the marginal kernels Atr
+1 ∈ C(K2
+1) and
+Atr
+2 ∈ C(K2
+2) point-wise as
+Atr
+1 (s, s′) :=
+�
+K2
+A(s, w, s′, w)dw
+and
+Atr
+2 (t, t′) =
+�
+K1
+A(u, t, u, t′)du
+(2.2)
+and therewith the separable approximation Atr of A as
+Atr(s, t, s′, t′) :=
+Atr
+1 (s, s′)Atr
+2 (t, t′)
+�
+K1
+�
+K2 A(u, w, u, w)du dw .
+(2.3)
+Notice that Atr is well-defined, if the denominator in (2.3) is non-zero. For a covariance kernel (sym-
+metric and positive semidefinite) this is automatically satisfied, unless the kernel is trivial. Partial trace
+approximations are well-known from quantum physics (see Bhatia (2003) and references therein) and have
+recently been applied for separability tests of space-time processes in L2-spaces (see Constantinou et al.
+(2017), Aston et al. (2017)). In a recent work of Masak et al. (2020) generalizations of partial traces have
+been investigated in the context of “almost separable matrices”.
+2.2.2
+Partial product approximations
+Closely related to the notion of partial traces are partial products. The partial product approximation
+depends on a user determined function ψ ∈ C(K2
+2), where typical choices are discussed in Bagchi and Dette
+(2020) (one being the constant 1). We then define for A ∈ C((K1 × K2)2) point-wise the marginal kernels
+Apr
+1 (s, s′) :=
+�
+K2
+2
+A(s, w, s′, w′)ψ(w, w′)dw dw′
+and
+Apr
+2 (t, t′) :=
+�
+K2
+1
+A(u, t, u′, t′)Apr
+1 (u, u′)du du′
+(2.4)
+and therewith the separable approximation Apr as
+Apr(s, t, s′, t′) :=
+Apr
+1 (s, s′)Apr
+2 (t, t′)
+�
+K2
+1 (Apr
+1 (u, u′))2 du du′ .
+(2.5)
+As for partial traces, the approximation is well-defined for a non-vanishing numerator, i.e., for
+Apr
+1 ̸= 0.
+(2.6)
+5
+
+This condition is fulfilled if A ̸= 0 for an appropriate the choice of ψ. Recently, partial products have been
+used in Dette et al. (2022) for the quantification of separability in L2-spaces and more recently for the
+efficient derivation of optimal separable approximations w.r.t. the L2-norm in Masak et al. (2020). These
+latter approximations are discussed next.
+2.2.3
+SPCA approximations
+SPCA (separable principal component analysis) provides the last approximation method that we want to
+investigate. With respect to the L2-norm, SPCA-approximations are optimal, whereas for the sup-norm
+they do not occupy this special position (see Remark 2.1). SPCA-approximations have been known in
+finite dimensions for several decades (Van Loan and Pitsianis (1993); Genton (2007)) and have recently
+been generalized to infinite-dimensional Hilbert spaces by Dette et al. (2022).
+The name “SPCA” has
+been introduced by Masak et al. (2020), who proposed an efficient algorithm for the calculation of these
+approximations by virtue of the partial product (see Section 2.2.2 in this reference). Consider the kernels
+˜APCA
+1
+(s, s′, ¯s, ¯s′) :=
+�
+K2
+2
+A(s, w, s′, w′)A(¯s, w′, ¯s′, w)dw dw′
+and
+˜APCA
+2
+(t, t′, ¯t, ¯t′) :=
+�
+K2
+1
+A(u, t, u′, t′)A(u′, ¯t, u, ¯t′)du du′,
+both of which are continuous, symmetric (w.r.t. to the first and second pair of components) and positive
+definite. According to Mercer’s theorem (Theorem 4.49 in Steinwart and Christmann (2008)) we can write
+them down as
+˜APCA
+1
+(s, s′, ¯s, ¯s′) =
+�
+i≥1
+λivi(s, s′)vi(¯s, ¯s′),
+˜APCA
+2
+(t, t′, ¯t, ¯t′) =
+�
+i≥1
+λiui(t, t′)ui(¯t, ¯t′).
+Here {λi, vi}i∈N and {λi, ui}i∈N are the eigensystems of the respective integral operators and the eigen-
+values are supposed to be in descending order. It is not difficult to show that if the strict inequality
+λ1 > λ2
+(2.7)
+holds, the eigenfunction v1, u1 are well-defined (up to sign) and continuous (see for both results Ap-
+pendix A.3). Supposing that this is true, we define the marginals
+APCA
+1
+(s, s′) =
+�
+λ1v1(s, s′),
+APCA
+2
+(t, t′) =
+�
+λ1u1(t, t′)
+and therewith the SPCA approximation
+APCA(s, t, s′, t′) = APCA
+1
+(s, s′)APCA
+2
+(t, t′).
+(2.8)
+We conclude this section with a general result regarding the approximation maps A �→ Ax, for every
+x ∈ {tr, pr, PCA}. We demonstrate that each of these maps is well-defined (in the sense that the resulting
+kernels are indeed continuous and positive definite). Moreover, we conclude that the approximation maps
+are Fréchet-differentiable, which is critical for the subsequent application of the functional Delta-method
+(see, for instance, Section 3.9 in van der Vaart and Wellner (1996)).
+6
+
+Theorem 2.2. Let K1 ⊂ Rp, K2 ⊂ Rq be compact, non-empty sets and let ˜A ∈ C((K1 × K2)2) be a
+covariance kernel with ˜A ̸= 0. Moreover, suppose that equation (2.6) and (2.7) are satisfied. Then the
+maps
+
+
+
+Fx
+i : C((K1 × K2)2) → C(K2
+i ) : A �→ Ax
+i
+Fx : C((K1 × K2)2) → C((K1 × K2)2) : A �→ Ax
+are for i = 1, 2 and x ∈ {tr, pr, PCA} well-defined in a sufficiently small, open neighborhood of ˜A and
+Fréchet differentiable in ˜A. Moreover, Fx
+i [ ˜A] and Fx[ ˜A] are again covariance kernels.
+The proof of this Theorem can be found in Appendix A, where we also state the explicit form of the
+derivatives. In the next section, we will use this result in the context of statistical inference for spatio-
+temporal data.
+3
+Testing separability for a continuous covariance kernel
+In this section, we develop a statistical test for the hypothesis of a separable covariance kernel in the space
+of continuous functions. First, we specify the statistical framework in Section 3.1 and, second, investigate
+test statistics for the hypothesis of separability in Section 3.2. Lemma 3.5 entails weak convergence of
+these test statistics to suprema of Gaussian processes (under the null hypothesis) and hence provides the
+theoretical tools for separability tests. To approximate the asymptotic quantiles of the test statistics, we
+present in Section 3.3 a multiplier bootstrap for dependent data.
+3.1
+Notations and assumptions
+Let K1 ⊂ Rp and K2 ⊂ Rq be compact, non-empty sets and (Xn)n∈Z be a time series of random functions
+in the Banach space C(K1 × K2) (for a definition, see Section 2.1). In the following, we will assume that
+(Xn)n∈Z satisfies fourth order stationarity, in the sense that for any indices (n1, . . . , n4) ��� Z4 and k ∈ Z
+the vectors (Xn1, . . . , Xn4) and (Xn1+k, . . . , Xn4+k) have the same distribution. In particular, supposing
+that E∥X1∥2 < ∞, both the mean function EXn(s, t) and the covariance
+C(s, t, s′, t′) := E [Xn(s, t)Xn(s′, t′)] − E [Xn(s, t)] E [Xn(s′, t′)]
+do not depend on n and are well-defined on the spaces C(K1 × K2) and C((K1 × K2)2) respectively. In
+the following, we want to construct a test for the hypothesis of a separable covariance operator, i.e.,
+H0 : C is separable
+vs.
+H1 : C is not separable,
+(3.1)
+where separability is defined in Section 2.1. For this purpose, suppose that we observe a sample of N
+random functions X1, . . . , XN (from the time series (Xn)n∈Z). For estimating C we use the (standard)
+empirical covariance estimator defined as:
+ˆCN(s, t, s′, t′) := 1
+N
+N
+�
+n=1
+�
+Xn(s, t) − XN(s, t)
+� �
+Xn(s′, t′) − XN(s′, t′)
+�
+,
+(3.2)
+where XN(s, t) := 1
+N
+�N
+n=1 Xn(s, t), (s, t) ∈ K1 ×K2, denotes the sample mean estimator function. In the
+next section, we will construct a test statistic for the hypothesis H0, by comparing ˆCN with a separable
+7
+
+approximation. The use of this statistic is motivated by the fact that, under suitable assumptions on the
+dependence structure, ˆCN is a consistent estimator of C by the law of large numbers (on Banach spaces).
+In order to quantify dependence, we introduce the popular concept of α-mixing sequences.
+Definition 3.1. Let (Xn)n∈Z be a sequence of random variables on some Banach space. For index sets
+I, J ⊂ Z we define the set distance
+dist(I, J ) = min{|i − j| : i ∈ I, j ∈ J }
+and use the notation FI := σ ({Xi : i ∈ I}) for the σ−algebra generated by the family of random variables
+{Xi : i ∈ I}. For r ∈ N0 the r-th α-mixing coefficient is then defined as
+α(r) = sup {|P(A ∩ B) − P(A)P(B)| : A ∈ FI, B ∈ FJ , dist(I, J ) ≥ r, I, J ⊂ Z} .
+The sequence (Xn)n∈Z is called α-mixing if α(r) → 0, as r → ∞.
+We can now state the theoretical assumptions for the separability test, developed in this section.
+Assumptions 3.2.
+i) The sequence (Xn)n∈Z consists of centered, random functions in C(K1 × K2) and is fourth order
+stationary.
+ii) There exists a non-negative random variable M, parameters β ∈ (0, 1] and J ≥ 0 with the constraint
+Jβ > ⌈2(dim(K1) + dim(K2))⌉ + 1 such that
+E(∥X1∥JM J) < ∞,
+and
+|Xn(s, t) − Xn(s′, t′)| ≤ M max{∥s − s′∥β, ∥t − t′∥β},
+(3.3)
+holds (almost surely) for all (s, t), (s′, t′) ∈ K1 × K2 and all n ∈ Z.
+iii) For some γ > max{J, 8} it holds that E∥X1∥γ < ∞.
+iv) The sequence (Xn)n∈Z is α-mixing in the sense of Definition (3.1), with α(r) ≤ κ/(1 + r)a, where
+κ > 0 is some constant and a > 2γ/(γ − 8).
+We briefly discuss each of these Assumptions.
+Remark 3.3.
+i) We require second order stationarity s.t. the empirical mean XN and the empirical covariance oper-
+ator ˆCN are consistent estimators. The stronger assumption of fourth order stationarity guarantees
+existence of the long-run variance operator of
+√
+N( ˆCN − C). An examination of our proofs shows
+that these assumptions can be further relaxed to conditions on the moments of Xn. However, for par-
+simony of presentation, we do not discuss these mathematically weaker (but harder to understand)
+adaptions. Our stationarity assumption is weaker than those in the related literature (both in L2-
+and C-spaces), where either independence or strict stationarity is considered (see Constantinou et al.
+(2017); Aston et al. (2017); Bagchi and Dette (2020); Dette and Kokot (2022)).
+8
+
+ii) To devise asymptotic tests for separability, we derive a CLT on the Banach space of continuous
+functions. Such results require the validation of tightness conditions, which depend on the geometry
+of the underlying space (reflected by the entropy rate). In the case of i.i.d. observations, sufficient
+conditions for a CLT can be found in Jain and Marcus (1975), and for the dependent case (under α-
+and φ-mixing) in Dmitrovskii et al. (1984). More specifically, Theorem 1 of Jain and Marcus (1975)
+requires a random Lipschitz condition, which together with an entropy condition entails a functional
+CLT on C(K). In this paper, we further relax this assumption by requiring only a random β-Hölder
+condition. This condition specifically includes sample paths of the Brownian motion, that are almost
+surely β-Hölder continuous, for β < 1/2. However, additional smoothness helps to reduce moment
+conditions imposed on the data (see the next assumption).
+iii) − iv) The existence of sufficiently many moments is the key to proving our CLT (and its bootstrap variant).
+As might be expected, stronger moment conditions can be traded off against weaker smoothness
+assumptions for the data functions, as well as milder conditions on temporal dependence. We quantify
+dependence by the decay rate of strong mixing coefficients, where a slower decay expresses stronger
+time-dependence. In comparison to the related literature, our mixing conditions are rather weak and
+include large classes of dependent time series.
+3.2
+Central limit theorems in C((K1 × K2)2)
+We begin our theoretical derivations by proving weak convergence of the standardized empirical covariance
+operator
+√
+N( ˆCN − C) to a Gaussian process G. This result, together with the corresponding bootstrap
+convergence, is of independent interest for the statistical investigation of the covariance on the space of
+continuous functions (notice that Theorem 3.4 is a special consequence of Theorem 3.6).
+Theorem 3.4. Suppose that Assumption 3.2 holds. Then there exists a centered Gaussian process G on
+the space C((K1 × K2)2) such that
+√
+N
+� ˆCN − C
+�
+d→ G.
+We now combine weak convergence of the empirical covariance with the Fréchet-differentiability of the
+separable approximation maps (Theorem 2.2). This yields weak convergence of the empirical separability
+measure ∥ ˆCN − ˆCx
+N∥ for any one of the approximation types x ∈ {tr, pr, PCA}, considered in Sections A.2-
+A.3.
+Lemma 3.5. Suppose our Assumption 3.2 holds, that C ̸= 0 is separable and that (2.6), (2.7) are satisfied.
+Then under the hypothesis H0 (defined in (3.1), it holds that
+√
+N
+�
+ˆCN − ˆCx
+N
+�
+d→ (Id −DCFx)G
+(3.4)
+for all x ∈ {tr, pr, PCA}. Here, DCFx is the derivative of the separable approximation map (see Section 2),
+Id the identity operator and G the Gaussian process from Theorem 3.4. In particular, it holds under H0
+(separability), that
+√
+N
+��� ˆCN − ˆCx
+N
+���
+d→ ∥(Id −DCFx)[G]∥
+(3.5)
+and under H1 (non-separability) that
+√
+N
+��� ˆCN − ˆCx
+N
+���
+d→ ∞.
+9
+
+The weak convergence in (3.4) follows by a version of the Delta-method (see for instance Section 3.9 in
+van der Vaart and Wellner (1996)) applied to the process
+√
+N( ˆCN −C). The statement in (3.5) is a direct
+consequence of this, using the continuous mapping theorem (see Theorem 1.3.6 in van der Vaart and Wellner
+(1996)). Finally, divergence under H1 is entailed by consistency of ˆCN and ˆCx
+N.
+Lemma 3.5 implies a straightforward test for the hypothesis H0: If the Gaussian process G (or equivalently
+its covariance) were known, we could easily approximate the upper (1 − α) quantile q1−α of the limiting
+distribution and reject whenever
+√
+N
+��� ˆCN − ˆCx
+N
+��� > q1−α.
+Unfortunately, the covariance CG of G is unknown and extremely difficult to approximate. Notice that it
+is defined on the product space C((K1 × K2)4), making it nearly impossible: either to calculate or save
+(recall that our discussion is motivated by the intractability of C, and CG consumes the squared amount
+of memory space). In the literature on L2-tests, Aston et al. (2017); Constantinou et al. (2018) have tried
+to evade this problem by using projection methods. While this approach is viable for functional data with
+a discrete spatial component, it seems less effective in the case of continuous spatial data (see also our
+simulations in Section 4.1). As a more practicable and fully functional alternative, we therefore discuss a
+multiplier bootstrap for dependent data in the next section.
+3.3
+A multiplier bootstrap
+In order to approximate the distribution of
+√
+N( ˆCN − C) we propose a multiplier bootstrap for dependent
+data, which is inspired by the methodology of Bücher and Kojadinovic (2013) and has recently been
+adapted to functional data by Dette et al. (2020). While various alternative bootstraps exist for functional
+data (an important example for the covariance is Paparoditis and Sapatinas (2016)), their validity is
+usually demonstrated in an L2-framework, making them inapplicable in our setup.
+We begin our discussion by fixing a number r ∈ N of bootstrap replicates. For 1 ≤ k ≤ r, we consider
+vectors of random weights (w(k)
+1,N, . . . , w(k)
+N,N) ∈ RN, which are independent of the data X1, . . . , XN and
+independent across k. We assume that each weight-vector follows a multivariate normal distribution. More
+precisely, the variables w(k)
+i,N are supposed to be centered with unit variance, and (lN − 1)-dependent with
+E[w(k)
+i,Nw(k)
+j,N] = (1 − |i − j|/lN)
+for any |i − j| ≤ lN.
+Here lN ∈ N is a bandwidth parameter, comparable to the block-length in a
+block bootstrap (see, for instance, Theorem 2.1 in Bücher and Kojadinovic (2013)). We then define the
+bootstrapped process B(k)
+N
+point-wise as
+B(k)
+N (s, t, s′, t′) := 1
+N
+N
+�
+n=1
+w(k)
+n,N
+�
+[Xn(s, t) − XN(s, t)][Xn(s′, t′) − XN(s′, t′)] − ˆCN(s, t, s′, t′)
+�
+(3.6)
+for 1 ≤ k ≤ r.
+Theorem 3.6 (Multiplier Bootstrap). Suppose that Assumption 3.2 holds and that lN satisfies both re-
+strictions lN → ∞ and lN/
+√
+N → 0. Then for any r ∈ N the weak convergence
+√
+N
+�
+ˆCN − C, B(1)
+N , . . . , B(r)
+N
+�
+d→
+�
+G, G(1), . . . , G(r)�
+(3.7)
+holds, where G is the Gaussian process from Theorem 3.4 and G(1), . . . , G(r) are i.i.d. copies of G.
+10
+
+A detailed proof of Theorem 3.6 is given in the Appendix, but let us briefly give some arguments why (3.7)
+holds. First, notice that due to independence of the weights (of the data and among themselves across
+k), we have uncorrelatedness of B(k)
+N
+with ˆCN − C and with any other B(k′)
+N
+. Hence, if these objects are
+(jointly) normal in the limit, they also have to be independent. This leaves open the questions, why B(k)
+N
+is normal and why it has the right covariance structure. On a high level, asymptotic normality of B(k)
+N
+is obvious, as it is a sum of weakly dependent random variables. A closer look reveals that dependence
+in (3.6) is governed by the bandwidth lN, which cannot increase too fast for a CLT to hold (this will
+be reflected in the assumption lN = o(
+√
+N)). On the other hand, lN has to diverge for N → ∞, s.t.
+B(k)
+N
+has the same asymptotic covariance as G. As lN grows, neighboring terms in B(k)
+N
+have (almost)
+identical weights. Consequently, their covariance is (almost) identical to that of the terms in ˆCN, without
+a multiplier.
+By Theorem 3.6 the bootstrap variable B(k)
+N
+mimics the random fluctuations of the empirical covariance
+ˆCN around C. Thus, it can help us to approximate the distribution of the separability measure ∥ ˆCN − ˆCx
+N∥;
+the actual object of interest. To make this clear, we use the representation
+∥ ˆCN − ˆCx
+N∥ = ∥{ ˆCN − C} − ([C + { ˆCN − C}]x − C)∥.
+To give a bootstrap approximation, we can replace { ˆCN − C} by B(k)
+N .
+Furthermore, since we want
+to approximate the distribution under the null, we have to replace C by the separable estimator ˆCx
+N
+everywhere else. This gives us the bootstrap version
+B∞,(k)
+N
+:= ∥B(k)
+N − ([ ˆCx
+N + B(k)
+N ]x − ˆCx
+N)∥.
+(3.8)
+Lemma 3.7. Under the assumptions of Theorem 3.6 it holds that
+√
+N(B∞,(1)
+N
+, · · · , B∞,(r)
+N
+)
+d→
+�
+∥(Id −DCFx)[G1]∥ , · · · , ∥(Id −DCFx)[Gr]∥
+�
+,
+where G is the Gaussian process from Theorem 3.4 and G(1), . . . , G(r) are i.i.d. copies of G.
+Lemma 3.7 theoretically underpins the following test decision: Let ˆq(r)
+1−α be the empirical (1 − α) quantile
+of the vector (B∞,(1)
+N
+, · · · , B∞,(r)
+N
+). Reject the hypothesis H0 (defined in (3.1)), if
+∥ ˆCN − ˆCx
+N∥ > ˆq(r)
+1−α.
+(3.9)
+Lemma 3.7 implies that the resulting test holds asymptotic level α, as r → ∞, that is
+lim
+r→∞ lim
+N→∞ PH0
+�
+∥ ˆCN − ˆCx
+N∥ > ˆq(r)
+1−α
+�
+= α,
+and is consistent as N → ∞ under the alternative H1, that is
+lim
+N→∞ PH0
+�
+∥ ˆCN − ˆCx
+N∥ > ˆq(r)
+1−α
+�
+= 1
+for any r ∈ N.
+Remark 3.8.
+11
+
+i) There are different ways to mathematically validate a bootstrap procedure. One way is proving
+convergence of the bootstrap measure (conditional on the data) to the correct limiting distribu-
+tion. Here, convergence is measured w.r.t. some metric on the probability measures, such as the
+Kolmogorov–Smirnov distance for probability measures on Rd or the bounded Lipschitz distance
+on general metric spaces (see van der Vaart and Wellner (1996) for a definition). This approach of
+studying conditional distributions is considered in many classical works, such as Hall (1992). As
+an alternative, it is possible to derive unconditional convergence results (such as Theorem 3.6 and
+Lemma 3.7), where it is shown that a number of bootstrapped statistics converge to independent
+copies of the same limiting distribution. Such results reflect the need of generating bootstrap rep-
+etitions for most practical test decisions. One merit of this approach is its clearer interpretability
+compared to the abstract notion of convergence on the spaces of measures, w.r.t. a difficult metric.
+Yet, from a theoretical standpoint, the two approaches are in many instances equivalent. In par-
+ticular, Lemma 3.7 implies for the conditional bootstrap measure PB(1)
+N |X1,··· ,XN the convergence in
+probability
+d
+�
+PB(1)
+N |X1,··· ,XN , P∥ ˆ
+CN− ˆ
+Cx
+N∥� P→ 0,
+where d is the Kolmogorov–Smirnov metric for probability measures on R and P∥ ˆ
+CN− ˆ
+Cx
+N∥ denotes
+the probability measure of the test statistic ∥ ˆCN − ˆCx
+N∥. The proof of this assertion follows directly
+from Lemma 2.3 in Bücher and Kojadinovic (2019) and a similar, conditional version of Theorem 3.6
+can be given by using the techniques in Section 3 of that paper.
+ii) The bandwidth parameter l = lN in Theorem 3.6 plays a similar role as the block length in a block
+bootstrap. In the special case of independent data it is not necessary to assume that lN → ∞ and
+a choice l = 1 yields the desired result (for multiplier bootstraps in the independent case, see also
+Section 3.6 in van der Vaart and Wellner (1996)). We also want to highlight that our assumption
+lN/
+√
+N → 0 is weaker than in previous works, where usually a polynomial decay rate of lN/
+√
+N was
+assumed (see Dette and Kokot (2022)).
+iii) The bootstrap test decision presented in this paper can be implemented without ever saving the entire
+empirical covariance operator, or any other object of comparable size. In the following, we want to
+sketch an argument why this is true. Let us therefore focus on the partial trace approximation: To
+approximate the partial trace approximation ( ˆCN)tr, it suffices to calculate the trace of the empirical
+covariance Tr[ ˆCN], as well as the partial traces ( ˆCN)tr
+1 , ( ˆCN)tr
+2 . All of these objects can be calculated
+directly from the data, as e.g.,
+Tr[ ˆCN] = 1
+N
+N
+�
+i=1
+�
+K1
+�
+K2
+X2
+i (s, t)ds dt
+(3.10)
+or
+( ˆCN)tr
+1 (s, s′) = 1
+N
+N
+�
+i=1
+�
+K2
+Xi(s, t)Xi(s′, t)dt.
+(3.11)
+In particular, we do not have to save ˆCN to calculate ( ˆCN)tr. In contrast to ˆCN, the three objects
+Tr[ ˆCN], ( ˆCN)tr
+1 , ( ˆCN)tr
+2 are small (a discretization takes about as much memory as a data function
+Xn) and hence we assume that they (and thereby ( ˆCN)tr) are tractable.
+So, from now on we
+12
+
+assume that these objects are saved. Now, calculating the test statistic ∥ ˆCN − ˆCtr
+N ∥ can be done
+by maximizing the distance | ˆCN − ˆCtr
+N | blockwise (it is easy to calculate for example ˆCN only for
+certain subsets of the arguments (s, t, s′, t′) and then evaluate the maximum over all subsets). The
+calculation of the bootstrap statistic B∞,(k)
+N
+(defined in (3.8)) is slightly more intricate. Here, we
+have to calculate [ ˆCtr
+N + B(k)
+N ]tr (the other objects in B∞,(k)
+N
+can again be calculated for subsets of
+indices in a straightforward way). Calculating [ ˆCtr
+N + B(k)
+N ]tr boils down to calculating the (partial)
+traces of ˆCtr
+N and B(k)
+N
+separately (as the partial traces are linear). For ˆCtr
+N they equal ( ˆCN)tr
+1 , ( ˆCN)tr
+2
+and we have already calculated and saved them before. For B(k)
+N , they are equal to the partial traces
+of
+1
+N
+�N
+n=1 w(k)
+n,N ˆCN (so essentially those of ˆCN) and those of
+1
+N
+�N
+n=1 w(k)
+n,N [Xn − XN] · [Xn − XN].
+Notice that we cannot save this later object (it is of the same size as ˆCN). So, as a computational
+trick, we can express it as the “covariance” of the complex valued data
+�
+sign(w(k)
+n,N)·|w(k)
+n,N|·[Xn−XN].
+These objects are again small enough to be saved, and from them, we can calculate the partial traces
+of
+1
+N
+�N
+n=1 w(k)
+n,N [Xn − XN] · [Xn − XN] directly (see (3.10) and (3.11)). Notice that in practice
+it is necessary to take the real part in the end, to eliminate complex valued remainders, due to
+computational imprecisions.
+4
+Finite sample properties
+In this section, we study the finite sample performance of our new method. We begin by a simulation
+study, where we compare the performance of the test with a benchmark procedure from Constantinou et al.
+(2018). Subsequently, we apply our test to a dataset of roman language recordings as described and used
+in Aston et al. (2017).
+4.1
+Simulations
+Following Constantinou et al. (2018), we generate spatio-temporal data by virtue of a functional MA(1)-
+process. More precisely, we generate Gaussian processes e0, . . . , eN, living on the unit square [0, 1]2, with
+covariance kernel C, point-wise defined as
+C(s, t, s′, t′) :=
+1
+(a|t − t′| + 1)1/2 exp
+�
+−
+b2|s − s′|2
+(a|t − t′| + 1)c
+�
+,
+s, s′, t, t′ ∈ [0, 1].
+(4.1)
+As in Constantinou et al. (2018), we set a = 3 and b = 2 and consider the observations
+Xn(s, t) :=
+S
+�
+s′=1
+exp
+�
+−b2(s − s′)2�
+[en(t, s′) + en−1(t, s′)]
+n = 1, · · · , N.
+Notice that for c = 0 the kernel C in (4.1) factorizes into purely spatial and temporal components. By
+implication, the covariance of Xn is separable for c = 0 (the hypothesis). On the other hand, if c > 0 the
+kernel C is not separable and this inseparability is inherited by the covariance of Xn. We hence generate
+data under H0 by setting c = 0 and under the alternative by setting c = 1. As in Constantinou et al.
+(2018), we discretize the time component t, by dividing the unit-interval into T = 50 equidistant points
+(1/T, 2/T, . . ., 50/T ). For the spatial component, Constantinou et al. (2018) use a similar discretization
+(1/S, 2/S, . . ., (S − 1)/S), but for smaller numbers of gridpoints with S = 4, 6, 8, 10, 12, 14. Considering
+13
+
+larger S is problematic for the procedure in Constantinou et al. (2018), which relies on the estimation
+of the asymptotic covariance operator of
+√
+N( ˆCN − C) - an expensive and difficult undertaking in high
+dimensions (we touched this point in our discussion at the end of Section 3.2). To implement their pro-
+cedure, Constantinou et al. (2018) rely on dimension reduction, for S ≤ 8 in time, and for S > 8 in
+both space and time. As we might expect, such projections entail deteriorating power for larger S as the
+amount of variance explained decreases. In our study, we use the same number of gridpoints (to allow
+meaningful comparisons) but also study the larger sizes of S = 20 and S = 30. As sample sizes, we
+consider N = 50, 100, 150, 200 and as corresponding block-lengths lN = 2, 2, 3, 4. The number of bootstrap
+repetitions for the generation of the empirical quantile is fixed at 400 and the nominal level at α = 5%.
+All reported results are based on 1000 simulation runs.
+In Table 1 we report empirical rejection probabilities under the hypothesis of separability (c = 0) and
+the alternative (c = 1). In brackets, we include the rejection probabilities reported in Constantinou et al.
+(2018) (whenever available), where we have chosen each time the maximum number of projection param-
+eters (in time for S ≤ 8 and in space and time for S > 8), which produced the most powerful results (see
+their Tables I-IV for a full picture of performance under variation of the projection parameters).
+rejection probability under H0
+rejection probability under H1
+S
+N = 50
+N = 100
+N = 150
+N = 200
+N = 50
+N = 100
+N = 150
+N = 200
+4
+3.3
+5.8 (5.0)
+5.2 (5.3)
+5.4 (5.5)
+36.0
+91.0 (95.1)
+99.2 (99.8)
+100.0 (100.0)
+6
+2.5
+6.0 (5.3)
+5.2 (6.6)
+4.5 (5.1)
+40.5
+92.8 (89.0)
+99.1 (99.3)
+100.0 (100.0)
+8
+3.75
+6.5 (7.5)
+4.2 (4.7)
+5.1 (5.7)
+34.8
+92.3 (85.2)
+98.0 (98.7)
+99.9 (100.0)
+10
+2.75
+7.6 (4.7)
+3.4 (5.3)
+3.7 (5.8)
+41.8
+90.6 (84.7)
+98.6 (98.5)
+99.9 (100.0)
+12
+1.5
+7.2 (5.0)
+3.1 (6.0)
+5.2 (5.7)
+37.8
+90.4(82.9)
+99.0 (97.8)
+99.9 (100.0)
+14
+2.0
+7.0 (4.6)
+5.3 (5.7)
+3.7 (5.6)
+33.5
+92.5 (78.9)
+98.6 (93.7)
+100.0 (95.7)
+20
+3.4
+5.2
+5.7
+3.80
+38.0
+90.3
+98.3
+99.9
+30
+2.7
+7.0
+5.3
+3.61
+36.1
+87.4
+98.2
+99.9
+Table 1:
+Empirical rejection probabilities of the bootstrap test (3.9) under the hypothesis (c = 0) and the
+alternative (c = 1). Benchmark values from the test of Constantinou et al. (2018) are given in brackets if
+available.
+Our results in Table 1 attest a satisfactory performance of the bootstrap test.
+The nominal level is
+reasonably approximated for N ≥ 100, while for N = 50 the test is somewhat conservative. The power of
+the bootstrap test is high in most scenarios. While for small values of S, the benchmark test fares slightly
+better, the bootstrap’s performance does not deteriorate for larger S, where it clearly outperforms the
+benchmark. Even raising S to 20 or 30 does not impinge on performance in any systematic way (exactly
+what a theory for continuous processes would suggest). Computationally, the bootstrap is particularly
+user-friendly: It allows a straightforward parallelization in the generation of bootstrap samples and is
+hence easy to implement. We have run our simulations on a standard desktop computer (3.2 GHz Apple
+14
+
+M1 Pro, Octa-core, 16 GB RAM) and any test evaluation needed less than a minute (for S ≤ 10 less than
+2 seconds), which underpins the practicability of this approach. The subsequent data example was run on
+the same machine for even larger values of T and S.
+4.2
+A data example
+We apply our test to the acoustic phonetic dataset of acoustic (log-)spectograms of Aston et al. (2017). A
+brief discussion with additional references can be found in Section 4.2 of their paper. Both the raw and pre-
+processed data together with detailed descriptions are contained in the file Acoustic_Data_And_Code.zip
+available from Series C datasets of Volume 67 (https://rss.onlinelibrary.wiley.com/hub/journal/
+14679876/series-c-datasets/67_5). The data consist of recordings of spoken words (in this case the
+numbers one to ten) in five different roman languages. For statistical use they have been transformed in
+the form of acoustic log-spectograms. For our purposes only the preprocessed data are used (namely the
+files WarpedPSD.RData and SVRF_WarpedPSD_SuppMat.RData). There are in total 219 data "functions" in
+a frequency-time domain as described in Sections 2-4 of Pigoli et al. (2018).
+As already indicated by the analysis in Pigoli et al. (2018) the null hypothesis (3.1) of separability seems to
+be violated in this instance. Before applying our test statistic we look at relative measures of separability.
+More concretely, for the separable trace approximation, the relative measure is given by
+∥ ˆCN − Ftr( ˆCN)∥
+∥ ˆCN∥
+.
+Indeed, as a preliminary analysis, we found that the relative measure of separability for the language
+covariance operators are rather high, see Table 2.
+French
+Italian
+Portuguese
+American Spanish
+Iberian Spanish
+Relative
+measure
+0.631
+0.895
+0.868
+0.947
+0.882
+Table 2: The relative measure of separability w.r.t. the trace approximation of the five roman languages.
+For every of the five languages we ran our bootstrap statistic of 1, 000 repetitions on the residual (i.e.
+centered) surface data for each language separately. As a result the hypotheses of separability for every
+of these languages is rejected with a highly significant p−value of less than 0.1%. See the first row of
+Table 3 for the p-values of our test. For the sake of completeness the second and third row contain the
+p-values obtained in Aston et al. (2017) for 2 frequency and 3 time dimensions and 8 frequency and 10
+dimensions, respectively. Our method provides several advantages, it is free of any tuning parameter, in
+contrast to the Studentized version of the empirical bootstrap of Aston et al. (2017), who have to choose
+additional parameters of dimensions of eigendirections (note that their test can only detect deviations
+from separability along those eigendirections). Moreover, when too few eigendirections are chosen (see
+Section 4.2 of Aston et al. (2017)) the hypotheses of separability cannot be rejected at the 5%-level.
+Secondly, in order to calculate the bootstrap we do only need to save the data, not the whole covariance
+operator, hence avoiding storage problems.
+15
+
+French
+Italian
+Portuguese
+American Spanish
+Iberian Spanish
+Test (3.9)
+<0.001
+<0.001
+0.001
+<0.001
+<0.001
+Emp. stud. test
+(2, 3)
+0.078
+0.197
+0.022
+0.360
+0.013
+Emp. stud. test
+(8, 10)
+0.001
+0.002
+0.001
+0.001
+<0.001
+Table 3: p−values of three different bootstrap tests of five Roman languages. First row: the test (3.9)
+proposed in this paper.
+Second and third row: the studentized version of the empirical bootstrap test
+proposed by Aston et al. (2017) with frequencey and time dimensions (2, 3) and (8, 10) respectively.
+Acknowledgements. This work was partially supported by the DFG Research unit 5381 Mathematical
+Statistics in the Information Age.
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+19
+
+A
+Proof of Theorem 2.2
+In this section, we investigate the differentiability of the approximation maps (·)tr, (·)pr, (·)PCA. We adapt
+results from Dette et al. (2022) (Theorem 3.4), where differentiability of the approximations on the space
+of L2-functions is shown. Since the continuous functions form a subspace of L2, endowed with a stronger
+norm, the Fréchet-differentials (for each map) have to coincide on both spaces (if they exist). Therefore, it
+only remains to show that the “differential quotients” converge in the space of continuous functions. Notice
+that we do not establish positive-semi definitness for any of the approximations, which is well known in
+the literature (for partial traces see Lemma 2.4 in Filipiak et al. (2018) and for the other approximation
+types the discussion in Masak et al. (2020)).
+A.1
+Differentiability of the partial trace
+Recall the definition of the partial trace kernels in (2.2) and the partial product approximation in
+(2.3).
+Lemma A.1. Let K1 ⊂ Rp, K2 ⊂ Rq be compact sets. Then the (partial) trace operators
+tr : C((K1 × K2)2) → R
+tri : C((K1 × K2)2) → C(K2
+i )
+defined in Section 2.2.1 are continuous.
+Proof. The proof follows by elementary calculations. For A ∈ C((K1 × K2)2) we can upper bound the
+integral
+|tr[A]| =
+����
+�
+K1×K2
+A(u, w, u, w)du dw
+���� ≤ |K1| · |K2| · ∥A∥.
+Here,|Ki| denotes the Lebesgue measure of Ki in the appropriate dimension. Analogous arguments can
+be used for the continuity of tr1, tr2 using their definitions in Section 2.2.1.
+Theorem A.2. Let K1 ⊂ Rp, K2 ⊂ Rq be compact sets. Then, the map Ftr defined in (2.3) is Fréchet
+differentiable in any non-zero covariance kernel C.
+Proof. The Fréchet differentiability of Ftr
+i = Htr ◦ Gtr, follows by the chain rule from the differentiability
+of the two maps
+Gtr : C((K1 × K2)2) → C(K2
+1) × C(K2
+2)
+and
+Htr : C(K2
+1) × C(K2
+2) → C((K1 × K2)2)
+defined (point-wise) as
+
+
+
+Gtr
+1 [T ](s, s′)
+Gtr
+2 [T ](t, t′)
+
+
+ =
+
+
+
+
+�
+K2 T (s, w, s′, w)dw
+�
+K1×K2 T (u, w, u, w)du dw
+�
+K1 T (u, t, u, t′)du
+
+
+
+
+Htr
+
+
+
+
+
+F
+G
+
+
+
+
+ (s, t, s′, t′) = F(s, s′)G(t, t′).
+20
+
+Using (the proof of) Theorem 3.4 in Dette et al. (2022), we claim that the Fréchet differential of Gtr in
+C, i.e., DCGtr[T ](s, t, s′, t′), can be point-wise expressed as
+
+
+
+
+
+�
+K2 T (s, w, s′, w)dw
+�
+K1×K2 C(u, w, u, w)dudw −
+�
+K1×K2 T (u, w, u, w)dudw
+�
+K2 C(s, w, s′, w)dw
+��
+K1×K2 C(u, w, u, w)dudw
+�2
+�
+K1 T (u, t, u, t′)du
+
+
+
+
+
+Boundedness of the map DCGtr follows directly from the continuity of tr, tr1, tr2 (see Lemma A.1). Now,
+we verify that DCGtr is indeed the differential. Since the second component is linear and continuous in
+the sup-norm it must be its own differential. Hence it is enough to consider the first component. For this
+purpose, let H ∈ C((K1 × K2)2) such that ∥H∥ → 0. Notice that for H sufficiently small, all (partial)
+traces in the subsequent objects are well-defined. A simple calculation yields the representation
+Gtr
+1 [C + H](s, t, s′, t′) − Gtr
+1 [C](s, t, s′, t′) − DCGtr
+1 [H](s, t, s′, t′)
+=
+−
+��
+K2 H(s, w, s′, w)dw
+� ��
+K1×K2 H(u, w, u, w)du dw
+�
+��
+K1×K2(C + H)(u, w, u, w)du dw
+� ��
+K1×K2 C(u, w, u, w)du dw
+�
++
+��
+K1×K2 H(u, w, u, w)du dw
+�2 �
+K2 C(s, w, s′, w)dw
+��
+K1×K2(C + H)(u, w, u, w)du dw
+� ��
+K1×K2 C(u, w, u, w)du dw
+�2 .
+Arguing as in the proof of Lemma A.1 above one easily sees that both terms on the right are uniformly
+of order O
+�
+∥H∥2�
+, which shows the desired property of Fréchet differentiability. Next we turn to the
+differentiability of the map Htr, where again (the proof of) Theorem 3.4 in Dette et al. (2022) suggests
+the following candidate for a differential
+D(L1,L2)Htr
+
+
+
+
+
+F
+G
+
+
+
+
+ (s, t, s′, t′) = L1(s, s′)G(t, t′) + F(s, s′)L2(t, t′)
+with (L1, L2)t := Gtr[C]. Proving that D(L1,L2)Htr is indeed the Fréchet differential follows by similar,
+but simpler calculations as in the proof of the differentiability of Hpr in Theorem A.4 (below) and is
+therefore omitted.
+A.2
+Differentiability of the partial product
+Recall the definition of the partial product kernels in (2.4) and the partial product approximation in
+(2.5).
+Lemma A.3. For any ψ ∈ C(K2
+2), the partial product operators as defined in Section 2.2.2 are continuous.
+Proof. The proof is trivial for the linear operator Fpr
+1 , which is evidently bounded. In order to show
+21
+
+continuity of Fpr
+2 , let A, H ∈ C(K2
+2) with ∥H∥ → 0. Then, by definition of its kernel we have
+�
+K2
+1
+H(u, t, u′, t′)(A + H)pr
+1 (u, u′)du du′ −
+�
+K2
+1
+A(u, t, u′, t′) ((A + H)pr
+1 (u, u′) − Apr
+1 (u, u′)) du du′
+(A.1)
+=
+�
+K2
+1
+(H(u, t, u′, t′)Hpr
+1 (s, s′)(u, u′) + H(u, t, u′, t′)Apr
+1 (u, u′) + A(u, t, u′, t′)Hpr
+1 (u, u′)) du du′
+where we used linearity of Fpr
+1
+in the second equality. The first integral is of order O(∥H∥2) where we
+have used the (Lipschitz) continuity of the linear map Fpr
+1 . The second and third integral are evidently
+(bounded) linear maps in H corresponding to the derivative.
+Theorem A.4. Let K1 ⊂ Rp, K2 ⊂ Rq be compact sets and ψ ∈ C(K2
+2) be chosen such that Cpr
+1
+(defined
+in (2.4)) is not identically 0. Then, the map Fpr defined in (2.5) is Fréchet differentiable in C.
+Proof. As in the proof of Theorem A.2 above, we can again decompose Fpr into two simpler ones
+Gpr : C((K1 × K2)2) → C(K2
+1) × C(K2
+2)
+and
+Hpr : C(K2
+1) × C(K2
+2) → C((K1 × K2)2)
+point-wise defined as
+Gpr[T ](s, t, s′, t′) =
+
+
+
+�
+K2
+2 T (s, w, s′, w′)ψ(w, w′)dw dw′
+�
+K2
+1 T (u, t, u′, t′)
+��
+K2
+2 T (u, w, u′, w′)ψ(w, w′)dw dw′�
+du du′
+
+
+
+Hpr
+
+
+
+
+
+
+F
+G
+
+
+
+
+
+ (s, t, s′, t′) =
+F(s, s′)G(t, t′)
+�
+K2
+1 (F(u, u′))2du du′ .
+The derivative in C of the first component of Gpr is by linearity and boundedness (w.r.t. the sup
+norm) the map itself. The second component is differentiable, which follows by the decomposition (A.1),
+where the last two integrals are linear maps of H (the derivative) and the third one is of order O(∥H∥2).
+Indeed defining the map (which in fact only depends actually on the variables (t, t′))
+DCGpr
+2 [H](s, t, s′, t′) =
+�
+K2
+1
+C(u, t, u′, t′)
+��
+K2
+2
+H(u, w, u′, w′)∆(w, w′)dw dw′
+�
+du du′
++
+�
+K2
+1
+H(u, t, u′, t′)
+��
+K2
+2
+C(u, w, u′, w′)∆(w, w′)dw dw′
+�
+du du′.
+and subtracting it from (Gpr
+2 [C + H] − Gpr
+2 [C])(s, t, s′, t′), we infer as in (A.1) that this difference is of the
+order O(∥H∥2). Next we calculate the derivative of Hpr for a generic pair (L1, L2)t, Li ∈ C(K2
+i ), i = 1, 2
+22
+
+with L1 ̸= 0. It is point-wise given by
+D(L1,L2)tHpr
+
+
+
+
+
+
+F
+G
+
+
+
+
+
+ (s, t, s′, t′) =F(s, s′)L2(t, t′) + L1(s, s′)G(t, t′)
+�
+K2
+1(L1(u, u′))2du du′
+−
+2
+��
+K2
+1 F(u, u′)L1(u, u′)du du′�
+L1(s, s′)L2(t, t′)
+��
+K2
+1 (L1(u, u′))2du du′
+�2
+Boundedness of the derivative follows directly from boundedness of the kernels F, G (as well as the bound-
+edness away from 0 of the denominator). Now consider Hi ∈ C(K2
+i ) with max(∥H1∥, ∥H2∥) → 0 differen-
+tiability of the map Hpr follows by decomposing the difference
+
+
+
+Hpr
+
+
+
+
+
+
+L1 + H1
+L2 + H2
+
+
+
+
+
+ − Hpr
+
+
+
+
+
+
+L1
+L2
+
+
+
+
+
+ − D(L1,L2)tHpr
+
+
+
+
+
+
+H1
+H2
+
+
+
+
+
+
+
+
+
+ (s, t, s′, t′)
+=
+
+H1(s, s′)L2(t, t′) + L1(s, s′)H2(t, t′) − 2L1(s, s′)L2(t, t′)
+�
+K2
+1 L1(u, u′)H1(u, u′)du du′
+�
+K2
+1 (L1(u, u′))2 du du′
+
+
+×
+
+
+1
+�
+K2
+1 ((L1 + H1)(u, u′))2 du du′ −
+1
+�
+K2
+1 (L1(u, u′))2 du du′
+
+
+Evidently, the first factor is of order O(max{∥H1∥, ∥H2∥}). Furthermore, a small calculation reveals the
+same rate for the second one. This completes the proof of the differentiability for the maps Gpr and Hpr
+and the differentiability of the partial product approximation follows by the chain rule and the identity
+Fpr = Hpr ◦ Gpr.
+A.3
+Differentiability of the SPCA
+In order to make the following derivations easier to read, we define for a compact set K the space of
+continuous, symmetric kernels C(K2)Sym as
+C(K2)Sym := {A ∈ C(K2) : A(x, y) = A(y, x) ∀x, y ∈ K}.
+Any kernel A represents the corresponding Hilbert–Schmidt (integral) operator, which (according to the
+spectral theorem for normal operators acting on L2[0, 1]) can be decomposed as
+A(x, y) =
+�
+i∈N
+vA
+i (x)vA
+i (y)λA
+i ,
+(A.2)
+where {vA
+i }i∈N are the eigenfunctions and {λA
+i }i∈N the corresponding eigenvalues. Without loss of gen-
+erality, we assume in the following that |λA
+1 | ≥ |λA
+2 | ≥ |λA
+i | for any i ≥ 3. Notice that a priori, the
+identity (A.2) only holds true in an L2-sense, but it can be shown that it remains true in the space of
+23
+
+continuous functions w.r.t. the sup-norm by Mercer’s Theorem, see, for instance, Theorem 3.a.1 in König
+(1986). Moreover, it can be shown that the eigenfunctions allow the choice of a continuous representative
+(see Lemma C.1). Therefore, we will subsequently assume that vA
+i
+is this representative in C(K). Let
+us now assume that A0 ∈ C(K2)Sym is a kernel which satisfies |λA0
+1 | > |λA0
+2 |. This means that the first
+eigenvalue of A0 is unique, and the first eigenfunction (corresponding to λA0
+1 ) as well. Since eigenfunctions
+are generally only determined up to sign, we suppose here that some choice of sign for vA0
+1
+has been fixed.
+Then, it follows that there exists a sufficiently small δ = δ(A0) > 0, such that for any A ∈ C(K2)Sym with
+∥A − A0∥ < δ it holds that |λA
+1 | > |λA
+2 | making the first eigenvalue of A unique. This also (up to a sign)
+identifies the first eigenfunction vA
+1 of A and we may fix that choice of sign, which minimizes the distance
+�
+K(vA0
+1 (t) − vA
+1 (t))2dt (see Lemma C.3 for details). Now, the “eigen-maps”
+Λ :
+
+
+
+Uδ(A0) → R,
+A �→ λA
+1
+V :
+
+
+
+Uδ(A0) → C(K),
+A �→ vA
+1
+(A.3)
+are well-defined. Since the separable approximation is (essentially) a rank-1-approximation of an operator,
+the key step in proving differentiability of the SPCA-approximation is proving differentiability of the
+eigenfunction-map and eigenvalue-map. Since the eigen-maps are known to be differentiable in an L2-sense
+(again see the proof of Theorem 3.4 in Dette et al. (2022)) we only have to validate that the differentials
+are still the limit of the “differential quotients” (as in the above proofs). In the case of the eigenfunction-
+maps this requires a non-standard representation of the differential, to still guarantee that it maps into
+the space of continuous functions.
+Lemma A.5. Suppose that A0 ∈ C(K2)Sym satisfies |λA0
+1 | > |λA0
+2 |. Then for δ = δ(A0) sufficiently small,
+it holds that the eigen-maps (defined in (A.3)) are Fréchet-differentiable.
+Proof. Adapting the proof from Theorem 3.4 in Dette et al. (2022) is trivial in the case of the eigenvalue-
+map, where the differential is given by
+DA0Λ[T ] =
+�
+K2 T (x, y)vA0
+1 (x)vA0
+1 (y)dx dy.
+In contrast, establishing the result for the eigenfunction-map is more intricate. Therefore, it warrants a
+detailed discussion. This time we do not start with the differential, but find it easier to derive it in a
+step-by-step process. For this purpose, consider H ∈ C(K2)Sym with ∥H∥ → 0 (in particular, we may
+assume that A0 + H ∈ Uδ(A0)). We now investigate the decomposition
+vA0+H
+1
+− vA0
+1
+= (A0 + H)
+λA0+H
+1
+�
+vA0+H
+1
+�
+− A0
+λA0
+1
+�
+vA0
+1
+�
+=
+�
+1
+λA0+H
+1
+−
+1
+λA0
+1
+�
+(A0 + H)
+�
+vA0+H
+1
+�
++ (A0 + H)
+λA0
+1
+�
+vA0+H
+1
+− vA0
+1
+�
++ H
+λA0
+1
+�
+vA0
+1
+�
+=: T1 + T2 + T3
+Next, we will find the differential of each term separately.
+We start with the first term T1 and note
+that T1 = −(λA0+H
+1
+− λA0
+1 )vA0+H
+1
+/λA0+H
+1
+, since (A0 + H)
+�
+vA0+H
+1
+�
+= λA0+H
+1
+vA0+H
+1
+. We claim that the
+24
+
+differential is given by H �→ −vA0
+1 /λA0
+1
+·
+�
+K2 H(x, y)vA0
+1 (x)vA0
+1 (y)dx dy, which is obviously linear and
+continuous w.r.t. the sup-norm. Adding and subtracting cross-terms yields the following bound
+�����T1 + vA0
+1
+λA0
+1
+�
+K2 H(x, y)vA0
+1 (x)vA0
+1 (y)dx dy
+����� ≤
+�
+K2 H(x, y)vA0
+1 (x)vA0
+1 (y)dx dy ∥vA0+H
+1
+− vA0
+1 ∥
+λA0
+1
++ ∥vA0
+1 ∥
+λA0
+1
+����
+�
+K2 H(x, y)vA0
+1 (x)vA0
+1 (y)dx dy − (λA0+H
+1
+− λA0
+1 )
+����
+Both terms are of order O
+�
+∥H∥2�
+. For the first one we observe that the integral is obviously of order
+O
+�
+∥H∥2�
+and the bound of O
+�
+∥H∥2�
+for the difference of eigenfunctions follows by Lemma C.2 part ii).
+The second term on the right is of order O
+�
+∥H∥2�
+, due to the differentiability of the eigenvalue-map (see
+the beginning of this proof), as
+DA0Λ[H] =
+�
+K2 H(x, y)vA0
+1 (x)vA0
+1 (y)dx dy
+(for details we refer to Dette et al. (2022)).
+We continue by analyzing T2. Notice that by the (second) inequality of Lemma 2 in Kokoszka and Reimherr
+(2013), one has
+∥H[vA0+H
+1
+− vA0
+1 ]/λA0
+1 ∥ ≤ ∥H∥
+|λA0
+1 |
+��
+K
+�
+vA0+H
+1
+− vA0
+1
+�2
+(x)dx
+�1/2
+< κ∥H∥2.
+This implies that T2 = A0[vA0+H
+1
+− vA0
+1 ]/λA0
+1
++ O(∥H∥2). We can rewrite the non-negligible term as
+A0
+λA0
+1
+�
+vA0+H
+1
+− vA0
+1
+�
+= A0
+λA0
+1
+
+�
+i≥1
+��
+K
+�
+vA0+H
+1
+(x) − vA0
+1 (x)
+�
+vA0
+i
+(x)dx
+�
+vA0
+i
+
+
+(A.4)
+Notice that the RHS is nothing else than the L2-basis expansion of vA0+H
+1
+−vA0
+1
+w.r.t. the ONB {vA0
+i }i∈N.
+This relation definitely holds w.r.t. the L2-norm, since A0 is an integral operator, but it needs not hold
+w.r.t. the sup norm. However, we claim it does. First, note that both side are well-defined continuous
+functions. Moreover, notice that
+sup
+x
+������
+�
+K
+A0(x, y)
+
+vA0+H
+1
+(y) − vA0
+1 (y) −
+�
+i≥1
+��
+K
+�
+vA0+H
+1
+(u) − vA0
+1 (u)
+�
+vA0
+i (u)du
+�
+vA0
+i
+(y)
+
+ dy
+������
+≤∥A0∥
+�
+K
+������
+vA0+H
+1
+(y) − vA0
+1 (y) −
+�
+i≥1
+��
+K
+�
+vA0+H
+1
+(u) − vA0
+1 (u)
+�
+vA0
+i
+(u)du
+�
+vA0
+i (y)dy
+������
+= 0
+where a0 is the continuous kernel of A0. Using the first identity and the second identity of Lemma 1 in
+Kokoszka and Reimherr (2013) on
+��
+K
+�
+vA0+H
+1
+(x) − vA0
+1 (x)
+�
+vA0
+1 (x)dx
+�
+vA0
+1
+and
+�
+i≥2
+��
+K
+�
+vA0+H
+1
+(x) − vA0
+1 (x)
+�
+vA0
+i (x)dx
+�
+vA0
+i
+respectively, one sees that the RHS of (A.4) equals
+− A0
+2λA0
+1
+�
+vA0
+1
+� �
+K
+(vA0+H
+1
+(x) − vA0
+1 (x))2dx + A0
+λA0
+1
+��
+i>1
+vA0
+i
+λA0+H
+1
+− λA0
+i
+�
+K2 H(x, y)vA0+H
+1
+(x)vA0
+i
+(y)dx dy
+�
+25
+
+where equality holds w.r.t. sup-norm by the same argument as above. From the proof of Lemma 2 in
+Kokoszka and Reimherr (2013), it follows easily that
+�
+K(vA0+H
+1
+(x) − vA0
+1 (x))2dx = O(∥H∥2) so that the first term is negligible. Using a similar reasoning
+as before (together with the bounds of Lemma C.2), we can then show that
+A0
+λA0
+1
+��
+i>1
+vA0
+i
+λA0+H
+1
+− λA0
+i
+�
+K2 H(x, y)vA0+H
+1
+(x)vA0
+i
+(y)dx dy
+�
+= A0
+λA0
+1
+��
+i>1
+vA0
+i
+λA0
+1
+− λA0
+i
+�
+K2 H(x, y)vA0
+1 (x)vA0
+i
+(y)dx dy
+�
++ O(∥H∥2),
+where we omit the precise calculations to avoid redundancy. The remaining series in the second line is
+linear in H. A simple calculation shows that it is also bounded. Finally, we notice that the term T3 already
+is a linear, bounded map in H. As a consequence, we can rewrite the decomposition
+vA0+H
+1
+− vA0
+1
+= T1 + T2 + T3 = −vA0
+1 /λA0
+1
+�
+K2 H(x, y)vA0
+1 (x)vA0
+1 (y)dx dy
++ A0
+λA0
+1
+��
+i>1
+vA0
+i
+λA0
+1
+− λA0
+i
+�
+K2 H(x, y)vA0
+1 (x)vA0
+i
+(y)dx dy
+�
++ H
+λA0
+1
+�
+vA0
+1
+�
++ O(∥H∥2).
+The non-vanishing part is the Fréchet-differential. Notice that in L2 this differential can be further sim-
+plified to the more common expression ���
+i>1
+vA0
+i
+λA0
+1
+−λA0
+i
+�
+K2 H(x, y)vA0
+1 (x)vA0
+i
+(y)dx dy (where it has been
+used that A0[vA0
+i
+] = λA0
+i vA0
+i
+in an L2-sense), for instances, see Kokoszka and Reimherr (2013). However,
+this expression is not necessarily an element of the continuous functions anymore, since even the conti-
+nuity of the eigenfunctions may not hold for all i ∈ N. These considerations conclude our proof of the
+differentiability of the eigen-maps.
+Theorem A.6. Let K1 ⊂ Rp, K2 ⊂ Rq be compact sets and C ∈ C((K1 × K2)2) be a separable covariance
+operator. Then the map FPCA as defined in (2.8) is Fréchet differentiable in C w.r.t. the sup-norm.
+Proof. Having established differentiability of the first eigenvalue and eigenfunction of a symmetric operator,
+the proof now follows step by step as that of Theorem 3.4 of Dette et al. (2022).
+B
+Proof of Theorem 3.6
+We show weak convergence in the space of continuous functions, relying on the theory of weak conver-
+gence for spaces of bounded functions as described in van der Vaart and Wellner (1996). For this purpose
+two conditions need to be verified, tightness and convergence of the marginals, see their Theorem 1.5.4.
+First, we demonstrate weak convergence of the marginals of the covariance operator, using the Cramér–
+Wold device (see Theorem 29.4 in Billingsley (2012)). We use classical blocking technique (to handle the
+bootstrapped part) together with moment inequalities from Yoshihara (1978) for strongly mixing random
+variables. Second, we prove tightness, by establishing asymptotic equicontinuity (see, for instance, The-
+orem 1.5.7 from van der Vaart and Wellner (1996)). The latter is done in turn by controlling entropy
+bounds. We define the packing numbers and entropy of sets.
+26
+
+For the sake of brevity, we introduce the following point-wise notations for the objects of interest.
+So for (s, t), (s′, t′) ∈ K1 × K2 we set
+˜Xi(s, t) := Xi(s, t) − EXi(s, t) ;
+˜CN(s, t, s′, t′) := 1
+N
+N
+�
+i=1
+˜Xi(s, t) ˜Xi(s′, t′),
+and
+˜B(k)
+N (s, t, s′, t′) := 1
+N
+N
+�
+i=1
+( ˜Xi(s, t) ˜Xi(s′, t′) − C(s, t, s′, t′)) wi,N.
+The second equality follows by change of summation, where we set Z(k)
+j
+= 0 (deterministic) for any j ≤ 0.
+In the last equality, we have defined the (random) weight wi,N in the obvious way.
+In order to enhance the clarity of our proof we make two simplifications: First, instead of proving con-
+vergence of the vector
+√
+N( ˆCN − C, B(1)
+N , · · · , B(r)
+N ) we confine ourselves to convergence of their centered
+versions, i.e., of
+√
+N( ˜CN − C, ˜B(1)
+N , · · · , ˜B(r)
+N ). Proving that the difference of these vectors is of order oP(1)
+follows by similar, but simpler techniques as used in the below proof and is therefore omitted. (It is a
+consequence of the fact that ˜Xi, i ≥ 1 also satisfies a CLT). Second, w.l.o.g. we set r = 1, since adjusting
+for r > 1 is straightforward and a notational burden.
+Step 1:
+We begin, proving weak convergence of the finite dimensional distributions, by means of the
+Cramér–Wold device (see, for instance, Theorem 29.4 in Billingsley (2012)). For this purpose, let p ∈ N be
+fixed but arbitrary and consider arbitrary tuples (sm, tm, s′
+m, t′
+m) ∈ (K1 × K2)2 and numbers am, a′
+m ∈ R,
+for 1 ≤ j ≤ p. Recall that to apply the Cramér–Wold device, we have to establish weak convergence of
+the real-valued, random variables
+CWN :=
+√
+N
+p
+�
+m=1
+�
+am
+�
+˜CN(sm, tm, s′
+m, t′
+m) − C(sm, tm, s′
+m, t′
+m)
+�
++ a′
+m ˜B(1)
+N (sm, tm, s′
+m, t′
+m)
+�
+to
+p
+�
+m=1
+�
+amG(sm, tm, s′
+m, t′
+m) + a′
+mG(1)(sm, tm, s′
+m, t′
+m)
+�
+,
+where G, G(1) are two independent identically distributed Gaussian processes. We proceed by a blocking
+technique. It follows, by definition, that we can rewrite CWN as
+CWN =
+1
+√
+N
+N
+�
+i=1
+p
+�
+m=1
+�
+˜Xi(sm, tm) ˜Xi(s′
+m, t′
+m) − C(sm, tm, s′
+m, t′
+m)
+�
+(am + a′
+mwi,N)
+=
+�
+bN
+N
+⌊N/bN⌋
+�
+j=1
+jbN
+�
+i=(j−1)bN +1
+p
+�
+m=1
+�
+˜Xi(sm, tm) ˜Xi(s′
+m, t′
+m) − C(sm, tm, s′
+m, t′
+m)
+� (am + a′
+mwi,N)
+√bN
++ Rem1 .
+Here (bN)N∈N is a sequence of natural numbers, such that lN/bN → 0 and bN/
+√
+N → 0 and Rem1 a
+remainder capturing all “overhanging terms” with indices between bN⌊N/bN⌋ and N. Using the triangle
+inequality (and counting terms), it is straightforward to see that E| Rem1 | = O(bN/
+√
+N) = o(1) and hence
+27
+
+Rem1 = oP(1). For ease of reference, we now define the random variables
+Yj :=
+jbN
+�
+i=(j−1)bN +1
+p
+�
+m=1
+�
+˜Xi(sm, tm) ˜Xi(s′
+m, t′
+m) − C(sm, tm, s′
+m, t′
+m)
+� (am + a′
+mwi,N)
+√bN
+(B.1)
+=
+jbN
+�
+i=(j−1)bN +1
+p
+�
+m=1
+˘Yi,m
+(am + a′
+mwi,N)
+√bN
+,
+where ˘Yi,m is defined in the obvious way.
+This allows us to write CWN =
+�
+bN
+N
+�
+j Yj + oP(1).
+In
+the following, we demonstrate weak convergence of the (non-negligible) sum, by virtue of the central
+limit theorem of Wooldridge–White (Theorem 5.20 in White (2001)). Therefore, we have to check three
+conditions: Sufficiently fast decay of the mixing coefficients of (Yj)j=1,··· ,⌊N/bN⌋, uniform boundedness of
+fourth moments and existence of the asymptotic variance.
+Mixing: The variables (Yj)j=1,··· ,⌊N/bN⌋ form a triangular array of α-mixing random variables, with
+mixing coefficients satisfying (for all N ≥ N0 and some sufficiently large N0 ∈ N) for |j − j′| ≥ 2
+α(Yj, Yj′) ≤ κ ((|j − j′| − 1) (bN − lN + 2))−a .
+(B.2)
+Here we have used Assumption 3.2 iv) together with the definition of the random variables Yj, Yj′.
+More specifically, we have used that Yj only depends on the functions ˜X(j−1)bN +1, · · · , ˜XjbN and on
+w(j−1)bN +1,N, · · · , wjbN ,N. In particular, the mixing coefficients for |j − j′| ≥ 2 converge (uniformly) to 0
+and hence the decay condition in the theorem of Wooldridge–White is trivially satisfied.
+Bounded fourth moments: We now want to show that E[Y 4
+j ] ≤ κ < ∞ uniformly in j and begin with
+the following calculation
+E[Y 4
+j ] ≤κ
+p
+�
+m=1
+E
+
+
+������
+jbN
+�
+i=(j−1)bN +1
+˘Yi,m
+(am + a′
+mwi,N)
+√bN
+������
+4
+
+(B.3)
+=κ
+p
+�
+m=1
+E
+
+E
+
+
+������
+jbN
+�
+i=(j−1)bN +1
+˘Yi,m
+(am + a′
+mwi,N)
+√bN
+������
+4 ����w1,N, . . . , wN,N
+
+
+
+
+≤κ
+p
+�
+m=1
+E
+
+
+
+
+jbN
+�
+i=(j−1)bN +1
+�(am + a′
+mwi,N)
+√bN
+�2
+
+
+4
+ ≤ κ.
+In the first inequality the constant κ only depends on p. In the first equality, we condition on the wi,N,
+1 ≤ i ≤ N (using the tower property). Hence, we can apply Theorem 3 from Yoshihara (1978), to get
+the second inequality. Notice that we assume ˘Yi,m to have a finite γ/2-moment, which holds since by iii)
+of Assumptions 3.2 Xi have finite γ-moments. In the above cited theorem, we have made the parameter
+choices m = 4 and δ = γ/2 − 4 and its summability condition (ii) is met since a > 2γ/(γ − 8) according
+to iv) of our Assumptions 3.2. The constant κ in this inequality only depends on the summability of the
+mixing coefficients (see the proof of Theorem 3 of Yoshihara (1978) for more details) and moments (of ˘Yi),
+but not on anything else. For the third inequality, we notice that (am + a′
+mwi,N)2 has bounded moments
+of any order (wi,N is normal with variance 1) and hence by counting terms the last inequality follows, with
+28
+
+κ now also depending on am, a′
+m, 1 ≤ m ≤ p.
+Convergence of the variance: Next, we have to prove the long-run variance exists. In order to
+achieve this, we decompose
+E
+
+
+
+
+�
+bN
+N
+�
+j
+Yj
+
+
+2
+ = bN
+N
+⌊N/bN⌋
+�
+j=1
+EY 2
+j + bN
+N
+�
+|j−j′|=1
+|EYjYj′| + 2
+�
+h≥2
+sup
+j
+|EYjYj+h|.
+(B.4)
+Using the mixing condition (B.2) we see that the last sum can be bounded (for sufficiently large N) by
+κb−a(γ−4)/γ
+N
+sup
+j
+(E|Yj|γ/2)4/γ �
+h≥2
+h−a(γ−4)/γ = o(1).
+Here we have used a standard covariance inequality for α-mixing (see Lemma 3.11 in Dehling et al. (2002))
+to bound |EYjYj+h| and for the small o-rate, that the sum on the right converges for a > γ/(γ − 4) (see
+Assumption 3.2 iv)). Similarly, we can upper bound the other sum of covariances by
+bN
+N
+�
+|j−j′|=1
+|EYjYj′| ≤ κ sup
+j
+|EYjYj+1|.
+We further bound the covariance |EYjYj+1| (and show that it converges to 0 uniformly in j). To this end,
+let (sN)N∈N denote an increasing sequence of natural numbers with lN/sN → 0 and sN/bN → 0. Recalling
+(B.1) we can now split up
+Yj =
+(j+1)bN −sN
+�
+i=jbN +1
+p
+�
+m=1
+˘Yi,m
+(am + a′
+mwi,N)
+√bN
++
+(j+1)bN
+�
+i=(j+1)bN −sN+1
+p
+�
+m=1
+˘Yi,m
+(am + a′
+mwi,N)
+√bN
+=: Yj,1 + Yj,2,
+Yj+1 =
+(j+2)bN −sN
+�
+i=(j+1)bN +1
+p
+�
+m=1
+˘Yi,m
+(am + a′
+mwi,N)
+√bN
++
+(j+2)bN
+�
+i=(j+2)bN −sN+1
+p
+�
+m=1
+˘Yi,m
+(am + a′
+mwi,N)
+√bN
+=: Yj+1,1 + Yj+1,2.
+Here the variables Yj,1, Yj,2, Yj+1,1, Yj+1,2 are defined in the obvious way. Using the same techniques as
+in (B.3), it follows that E[(Yj,1)4], E[(Yj+1,1)4] < ∞ and (counting terms) E[(Yj,2)4], E[(Yj+1,2)4] = o(1).
+The Cauchy–Schwarz inequality thus implies E[Yj,1Yj+1,2], E[Yj,2Yj+1,2], E[Yj,2Yj+1,1] = o(1). Moreover,
+using the covariance inequality for α-mixing E[Yj,1Yj+1,2] ≤ {E[(Yj,1)4]E[(Yj+1,1)4]}1/2α(Yj,1, Yj+1,1)1/2.
+Now α(Yj,1, Yj+1,1) ≤ κ(sN − lN)−a = o(1), where we have used Assumption 3.2 iv) together with the
+definition of the random variables Yj,1, Yj+1,1. Recall therefore, that Yj,1 only depends on the functions
+˜XjbN +1, . . . , ˜X(j+1)bN −sN and on the weights wjbN +1,N, . . . , w(j+1)bN −sN ,N. These considerations imply
+that the second and third term, on the right of (B.4), i.e., the mixed terms, are of order o(1) and hence the
+variance is equal to bN
+N
+�⌊N/bN⌋
+j=1
+EY 2
+j . Finally, we have to show that this variance convergences. Therefore,
+let us consider, for 1 < j ≤ ⌊N/bN⌋, EY 2
+j
+=E
+
+E
+
+
+
+
+p
+�
+m=1
+(j+1)bN
+�
+i=jbN +1
+˘Yi,m
+(am + a′
+mwi,N)
+√bN
+
+
+2 ���w1,N . . . , wN,N
+
+
+
+
+(B.5)
+=
+p
+�
+m,l=1
+1
+bN
+(j+1)bN
+�
+i,k=jbN +1
+E[ ˘Yi,m ˘Yi,l] E[(am + a′
+mwi,N)(al + a′
+lwk,N)]
+29
+
+where we used independence of ˘Yi,m and wi,N, 1 ≤ i ≤ N. Due to our stationarity Assumption 3.2, we
+have E[ ˘Yi,m ˘Yk,l] = τ m,l
+Y
+(|i − k|) for a function τm,l
+Y
+: N → R. Moreover, by construction the covariance of
+wi,N and wk,N also only depends on |i − k| and N. Hence, we can consistently define
+τ m,l
+w,N(|i − k|) := E[(am + a′
+mwi,N)(al + a′
+lwk,N)] = amal + a′
+ma′
+lE[wi,Nwk,N]
+which converges as N → ∞ and E[wi,Nwk,N] → 1 (this follows by definition of the weights; see the very
+beginning of this proof). We can hence rewrite (B.5) as
+p
+�
+m,l=1
+�
+|h|<bN
+τ m,l
+Y
+(|h|)τ m,l
+w,N(|h|)(1 − |h|/bN).
+Notice that this object does not depend on j and converges to the long-run variance �p
+m,l=1
+�
+h∈Z τY (|h|)m,l.
+This latter convergence can be established directly by the dominated convergence theorem. Indeed, first
+observe that |τm,l
+w,N(|h|)| ≤ κ, for (1 − |h|/bN) ≤ 1. Secondly, the terms τ m,l
+Y
+(|h|) are summable for any
+m, l, which again follows by the covariance inequality for α-mixing random variables. Let |i− j| = |h| ≥ 1,
+then
+τ m,l
+Y
+(|h|) = E[ ˘Yi,m ˘Yk,l] ≤ κ{E[| ˘Yi,m|γ/2]}2/γ{E[| ˘Yk,l|γ/2]}2/γ(|h| + 1)−a(γ−4)/γ,
+which is summable since a > γ/(γ − 4) and due to uniform boundedness of the moments, see iii) of our
+Assumptions 3.2. It follows by straightforward modifications that also EY 2
+1 converges to the same vari-
+ance. As a consequence of the above considerations, the variance in (B.4) converges and we can apply the
+central limit theorem of Wooldridge–White (Theorem 5.20 in White (2001)), which entails convergence of
+the marginal distributions.
+It remains to show that
+√
+N �N
+p=1 am( ˜CN − C)(sm, tm, s′
+m, t′
+m) and
+√
+N �N
+p=1 a′
+m ˜B(1)
+N (sm, tm, s′
+m, t′
+m) are
+asymptotically independent and have the same (asymptotic) variance. The asymptotic independence fol-
+lows readily from the uncorraletedness of the sequences ( ˜Xi)i∈Z and the weights together with the Gaussian
+limit. As for the equivalence of their variance, this follows by a quick calculation using similar techniques
+as in the first part of our proof.
+Before proceeding to the proof of tightness by asymptotic equicontinuity we state the definition of
+packing numbers for the sake of completeness.
+Definition B.1. Let (X, d) be a metric space and B(x, r) a ball of radius r > 0 centered around x ∈ X.
+Then for ε > 0, we define the ε-packing number D(ε, d) as
+sup
+�
+n ∈ N |
+n�
+i=1
+B(xi, ε) ⊃ X where d(xi, xj) > ε, xi ∈ X, 1 ≤ i ≤ n
+�
+.
+Note that, clearly, the packing number becomes bigger for smaller ε > 0 and remains finite for any totally
+bounded sets.
+In the subsequent part of our proof, for K ⊂ Rp we set ρK(x, y) := 1 ∧ maxp
+i=1 |xi − yi|.
+Step 2:
+We show that the process
+√
+N( ˜CN − C, ˜B(1)
+N ) is asymptotically uniformly ˜ρ-equicontinuous in
+probability, where
+˜ρ((s, t, u, v), (s′, t′, u′, v′)) := max{ρK1×K2((s, t), (s′, t′)), ρK1×K2((u, v), (u′, v′))}
+30
+
+is our metric on (K1 × K2)2. Moreover, recall that by Theorem 1.5.7 in van der Vaart and Wellner (1996)
+asymptotic equicontinuity of the process is equivalent to tightness. For ζ > 0, define the set of pairs
+Aζ := {((s, t, u, v), (s′, t′, u′, v′)) ∈ (K1 × K2)4 | ˜ρ((s, t, u, v), (s′, t′, u′, v′)) < ζ}.
+We will now bound, for ǫ > 0,
+lim sup
+N→∞
+P
+�
+sup
+(x,x′)∈Aζ
+���
+√
+N( ˜CN − C, ˜B(1)
+N )(x) −
+√
+N( ˜CN − C, ˜B(1)
+N )(x′)
+��� > ε
+�
+≤ lim sup
+N→∞
+E
+�
+sup
+(x,x′)∈Aζ
+���
+√
+N( ˜CN − C, ˜B(1)
+N )(x) −
+√
+N( ˜CN − C, ˜B(1)
+N )(x′)
+���
+�J
+/εJ.
+(B.6)
+Using Theorem 2.2.4. in van der Vaart and Wellner (1996) it is enough to bound the J-th moment of the
+difference in two locations. More precisely, for a J to be specified below, we upperbound
+E
+���
+√
+N( ˜CN − C)(s, t, u, v) −
+√
+N( ˜CN − C)(s′, t′, u′, v′)
+���
+J
+(B.7)
+and
+E
+���
+√
+N ˜B(1)
+N (s, t, u, v) −
+√
+N ˜B(1)
+N (s′, t′, u′, v′)
+���
+J
+.
+(B.8)
+Applying Theorem 3 in Yoshihara (1978) on the α-mixing random variables:
+˜X1(s, t) ˜X1(u, v) − E
+�
+˜X1(s, t) ˜X1(u, v)
+�
+− ˜X1(s′, t′) ˜X1(u′, v′) + E
+�
+˜X1(s′, t′) ˜
+X1(u′, v′)
+�
+with weights ai = 1/
+√
+N and (arbitrary, but small) δ > 0 we see (B.7) is less than
+κE
+���� ˜X1(s, t) ˜X1(u, v) − E
+�
+˜X1(s, t) ˜X1(u, v)
+�
+− ˜X1(s′, t′) ˜X1(u′, v′) + E
+�
+˜X1(s′, t′) ˜X1(u′, v′)
+����
+�J
+(B.9)
+whenever �
+i≥1(i + 1)J/2+1α(i)δ/(J+δ) < ∞.
+The latter term will be bound using continuity properties of our random variables as assumed in
+Assumption 3.2(ii). Indeed, taking the expectation of relation 3.3, note we have that
+| ˜Xi(s, t) − ˜Xi(s′, t′)| ≤ (M + EM)ρK1×K2((s, t), (s′, t′))β
+so the centered random variables are Hölder continuous with as new random constant ˜
+M := M + EM. A
+quick calculation then shows that
+��� ˜X1(s, t) ˜X1(u, v) − ˜X1(s′, t′) ˜X1(u′, v′)
+��� ≤ 2∥ ˜X1∥ ˜
+M ˜ρβ((s, t, u, v), (s′, t′, u′, v′)),
+and similarly as above
+���E
+�
+˜X1(s, t) ˜
+X1(u, v)
+�
+− E
+�
+˜X1(s, t) ˜X1(u, v)
+���� ≤ 2E| ˜
+M|˜ρβ((s, t, u, v), (s′, t′, u′, v′)).
+Using the two above bounds, we see that (B.9) can be upperbounded by
+κE
+�
+˜
+M J ∥X1∥J�
+˜ρJβ((s, t, u, v), (s′, t′, u′, v′)),
+where κ may depend on β, J but not on N.
+31
+
+To bound (B.8) we first condition on the weights w(1)
+i,N, 1 ≤ i ≤ N. Then the argument runs along the
+same lines as the one for bounding (B.7), namely a straightforward application of Theorem 3 in Yoshihara
+(1978). This gives the bound
+κE
+
+
+
+
+
+
+
+
+
+
+N
+�
+i=1
+�
+w(1)
+i,N
+�2
+N
+
+
+
+J
+E
+��
+( ˜X1 · ˜X1 − C)(s, t, u, v) − ( ˜X1 · ˜X1 − C)(s′, t′, u′, v′)
+�J ��w(1)
+i,N, 1 ≤ i ≤ N
+�
+
+
+
+
+
+
+
+≤κE
+
+
+
+N
+�
+i=1
+�
+w(1)
+i,N
+�2
+N
+
+
+
+J
+E
+�
+˜
+M J ∥X1∥J�
+˜ρJβ((s, t, u, v), (s′, t′, u′, v′))
+where we also used independence of ˜Xi, w(1)
+i,N.
+Another application of Yoshihara’s Theorem 3 on the
+lN−dependent sequence wi,N allows us to bound their J-th moment, which remains finite since lN/N → 0,
+as N → ∞. Recall that ˜ρ = max{ρK1, ρK2} and all norms are equivalent on finite dimensional spaces.
+Since in general finite dimensional spaces the following bound holds (see, for instance, Ex. 6 of Section 2.1
+in van der Vaart and Wellner (1996))
+D(ε, ˜ρβ) <
+κ
+ε2(d1+d2)/β
+(here κ depends on di := dim(Ki), i = 1, 2 as well as the diameter of Ki). Choosing the parameter J =
+⌈2(d1+d2)/β⌉+1 and using Markov’s inequality together with Theorem 2.2.4 in van der Vaart and Wellner
+(1996) the expression (B.6) for any arbitrary ν > 0, is less than (κ/εJ)(η−2(d1+d2)/(Jβ)+1+ζη−4(d1+d2)2/(Jβ2))
+which can be made arbitrarily small picking ζ small and then ν small. Consequently, our process is ˜ρ-
+equicontinuous in probability.
+32
+
+C
+Additional results
+Throughout this section, we always assume that the eigenvalues of an operator A satisfy |λA
+1 | ≥ |λA
+2 | ≥ |λA
+i |
+for any i ≥ 3.
+Lemma C.1. Let A ∈ C(K2)Sym be a kernel, with eigenvalues |λA
+1 | > |λA
+2 |, then we can find a continuous
+representative of the first eigenfunctions vA
+1 ∈ C(K).
+Proof. We first notice that in an L2-sense the equality vA
+1 = A[vA
+1 ]/λA
+1 holds. Now defining vA
+1 point-wise
+by the expression A[vA
+1 ]/λA
+1 , we see that it is already continuous, as
+vA
+1 (t + h) − vA
+1 (t) = (λA
+1 )−1
+�
+K
+[A(s, t + h) − A(s, t)]vA
+1 (s)ds
+≤(λA
+1 )−1� �
+K
+[A(s, t + h) − A(s, t)]2ds
+�1/2
+≤ κ sup
+t |A(s, t + h) − A(s, t)| = o(1).
+Here we have used Cauchy–Schwarz in the first inequality. The small-o refers to convergence as |h| → 0
+and follows because the continuous kernel A is uniformly continuous on the compact set K2.
+Lemma C.2. Suppose that A0 ∈ C(K2)Sym with A0 satisfying |λA0
+1 | > |λA0
+2 |. Furthermore, consider
+for some δ > 0 the operator A ∈ C(K2)Sym with ∥A − A0∥ ≤ δ and a choice of eigenfunction s.t.
+�
+K vA0
+1 (t)vA
+1 (t)dt ≥ 0. Then it holds with a constant κ := κ(A0, δ)
+i)
+For j = 1, 2
+���|λA
+j | − |λA0
+j |
+��� ≤ κ∥A − A0∥.
+ii)
+��
+K
+(vA
+1 (t) − vA0
+1 (t))2dt
+�1/2
+≤ κ∥A − A0∥.
+The identities follow directly from i) in Horváth and Kokoszka (2012) (Lemmas 2.2-3), applied to the
+operators A and A0. Notice that we have here exploited that the sup-norm is stronger than the L2-norm.
+Lemma C.3. Let A0 ∈ C(K2)Sym be a kernel, with eigenvalues |λA0
+1 | > |λA0
+2 | and eigenfunction vA0
+1
+(where some choice of sign for the eigenfunction is fixed). Then there exists a δ = δ(A0) > 0 sufficiently
+small, s.t. for any A ∈ C(K2)Sym with ∥A − A0∥ ≤ δ it holds that |λA
+1 | > |λA
+2 | and some choice of sign
+exists for vA
+1 s.t.
+�
+K(vA0
+1 (t) − vA
+1 (t))2dt <
+�
+K(vA0
+1 (t) + vA
+1 (t))2dt.
+Proof. The proof is a direct consequence of the preceding Lemma C.2. Notice that
+|λA
+1 | = (|λA
+1 | − |λA0
+1 |) + (|λA0
+1 | − |λA0
+2 |) + (|λA0
+2
+− |λA
+2 |) + |λA
+2 | ≥ |λA
+2 | − κδ,
+where κ comes from Lemma C.2. For sufficiently small δ > 0, the inequality |λA
+1 | > |λA
+2 | holds. Finally,
+the inequality
+�
+K(vA0
+1 (t) − vA
+1 (t))2dt <
+�
+K(vA0
+1 (t) + vA
+1 (t))2dt follows directly from part ii) of Lemma C.2,
+again for small enough δ.
+33
+

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+CMS Tracker Alignment Activities during LHC Long
+Shutdown 2
+Sandra Consuegra Rodríguez푎,∗
+푎Deutsches Elektronen-Synchrotron,
+Notkestraße 85, 22607 Hamburg, Germany
+E-mail: [email protected]
+The strategies for and the performance of the CMS tracker alignment during the 2021-2022 LHC
+commissioning preceding the Run 3 data-taking period are described. The results of the very
+first tracker alignment after the pixel reinstallation, performed with cosmic ray muons recorded
+with the solenoid magnet off are presented. Also, the performance of the first alignment of the
+commissioning period with collision data events, collected at center-of-mass energy of 900 GeV,
+is presented. Finally, the tracker alignment effort during the final countdown to LHC Run 3 is
+discussed.
+41st International Conference on High Energy physics - ICHEP2022
+6-13 July, 2022
+Bologna, Italy
+1On behalf of the CMS Collaboration
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+
+CMS Tracker Alignment Activities during LHC Long Shutdown 2
+Sandra Consuegra Rodríguez
+1. CMS tracker detector
+The innermost tracking system of the CMS experiment, called the tracker, consists of two de-
+vices, the Silicon Pixel and Silicon Strip detectors, within a length of 5.8 m and a diameter of 2.5
+m [1]. The closest detector to the interaction point, the Silicon Pixel detector, consists of the barrel
+pixel sub-detector (BPIX) composed of 4 layers and the forward pixel sub-detector (FPIX) with two
+forward endcaps, each composed of 3 disks, for a total of 1856 modules in the current configuration
+(Phase 1). Because of its proximity to the interaction point, the pixel detector is exposed to the
+highest density of tracks from the proton-proton collisions and, therefore, suffers more extensively
+the effects of the radiation damage. To tackle these effects, the pixel tracker was extracted from the
+CMS experimental cavern, underwent extensive repairs, was provided with a new innermost layer,
+and was reinstalled during the LHC Long Shutdown 2 (LS2). The silicon strip detector consists
+of 15 148 modules and is composed of the Tracker Inner Barrel (TIB), Tracker Inner Disks (TID),
+Tracker Outer Barrel (TOB), and Tracker EndCaps (TEC).
+2. Track-based alignment
+The tracker was specifically designed to allow for a very accurate determination of the trajec-
+tory of charged particles by ensuring an intrinsic spatial resolution of up to 10-30 microns. From
+the positions at a number of points registered in the detector in the form of hits, the trajectory of a
+charged particle can be reconstructed. Once the hits belonging to each trajectory are associated, a
+fit is performed and the track parameters such as the track curvature radius, impact parameter 푑푥푦
+in the xy plane, impact parameter 푑푧 along the beam direction, as well as the angles 휃 and 휙, are
+obtained. The tracking efficiency, a measure of how efficiently a charged particle passing through
+the tracker is reconstructed, provides a quantitative estimate of the tracking performance. Given
+the deflection of charged particles in a magnetic field, their transverse momentum can be computed
+as the product of the electric charge, the magnetic field, and the curvature radius. Thus, the track
+parameter uncertainties are propagated to the momentum measurement. Furthermore, the resolu-
+tion of a reconstructed primary vertex position depends strongly on the number of tracks used to fit
+the vertex and the transverse momentum of those tracks. Hence, high-quality track reconstruction
+paves the way for accurate primary and secondary vertex reconstruction.
+To ensure good tracking efficiency, as well as a precise momentum measurement and primary ver-
+tex reconstruction, the uncertainty on the track parameters needs to be reduced as much as possible.
+One of the main contributions to this uncertainty comes from the limited precision of the hit position
+measurements entering the track fit from which the track parameters are determined, which in turn
+is related to the overall limited knowledge of the position of the detector modules. The latter has
+two main contributions: the intrinsic spatial resolution of the sensors and the uncertainty related
+to the limited knowledge of the displacements, rotations, and surface deformations of the tracker
+modules. For an accurate determination of the track parameters, this second source of uncertainty
+in the position of the detector modules needs to be reduced to at least the intrinsic spatial resolution
+of the sensors. The correction of the position, orientation, and curvature of the tracker modules to
+reach a precision better than the intrinsic spatial resolution is the task of tracker alignment.
+The so-called track-based alignment consists of fitting a set of tracks with an appropriate track
+2
+
+CMS Tracker Alignment Activities during LHC Long Shutdown 2
+Sandra Consuegra Rodríguez
+model, and computing track-hit residuals, i.e., the difference between the measured hit position and
+the corresponding prediction obtained from the track fit. Geometry corrections can be derived from
+the 휒2 minimization of these track-hit residuals. The Millepede and HipPy alignment algorithms
+are used by CMS to solve the 휒2 minimization problem [2, 3]. The alignment parameters are de-
+termined with Millepede in a simultaneous fit of all tracks, involving two types of parameters: the
+local parameters that characterize the tracks used for the alignment, and nine global parameters that
+describe the position, orientation, and surface deformations of the modules. The local parameters
+of a single track are only connected to the subset of global parameters that are related to that track,
+and they are not directly connected to the local parameters of other tracks. The global parameters
+of each of the single modules of the detector can be corrected in a single alignment fit if enough
+tracks are available. On the other hand, when using the HipPy algorithm, the 휒2 of each sensor is
+minimized with respect to a change in the local alignment of that sensor only, keeping the parame-
+ters of every other sensor fixed, in an iterative procedure.
+Once the set of alignment constants is obtained, the improvement of post-alignment track-hits resid-
+uals is reviewed. Furthermore, before the new detector geometry is updated online for the data taking
+or used for the data reprocessing, the impact of the new set of alignment constants in the tracking
+performance, vertexing performance, and physics observables such as the mass of the Z boson res-
+onance as function of the pseudorapidity and azimuthal angle is checked.
+A simplified version of the offline alignment described above also runs online as part of the Prompt
+Calibration Loop (PCL). The PCL alignment uses the MillePede algorithm and performs the align-
+ment of the pixel high-level structures at the level of ladders and panels, which ensures the consid-
+eration of radiation effects of the innermost layer of the barrel pixel detector already during data
+taking. The obtained constants are then used for the reconstruction of the next run if movements are
+above certain thresholds. Thus, the online and offline alignment are complementary components
+of the tracker alignment within CMS, one providing automated online correction of the pixel high-
+level structures and the other refining the alignment calibration with the possibility to reach each of
+the single modules of the detector in a single alignment fit.
+3. Tracker alignment effort prior to the beginning of Run 3
+The first data-taking exercise upon the restart of operations in 2021 consisted of recording cos-
+mic ray muons with the magnetic field off, “cosmic run at zero Tesla” (CRUZET). The alignment
+with cosmic ray data has the advantage of allowing the update of the tracker alignment constants
+before the start of collision data taking. Major shifts in the pixel and strip sub-detectors (e.g., due
+to magnet cycles and temperature changes) can be identified and the geometry corrected accord-
+ingly before beams are injected into the collider and collision data becomes available. The very first
+alignment of the pixel detector after reinstallation in the experimental cavern was performed using
+2.9M cosmic ray tracks recorded during the CRUZET period, at the level of single modules for the
+pixel detector and the outer barrel of the strip detector, and of half-barrels and half-cylinders for the
+rest of the strip partitions. This period was followed by cosmic data-taking with magnetic field at
+nominal value (3.8T), “cosmic run at four Tesla” (CRAFT). In this case, the geometry was derived
+using 765k cosmic ray tracks with the alignment corrections derived at the level of single modules
+for the barrel pixel and at the level of half-barrels and half-cylinders for the forward pixel and all of
+3
+
+CMS Tracker Alignment Activities during LHC Long Shutdown 2
+Sandra Consuegra Rodríguez
+the strip sub-detectors. While the geometries derived with CRUZET and CRAFT data constituted
+relevant updates of the alignment constants starting from a potentially large misalignment, the re-
+sults are statistically limited by the available number of cosmic ray tracks, especially in the forward
+pixel endcaps, and systematically limited by the lack of kinematic variety of the tracks sample.
+For a further improvement of the alignment calibration, a sample of 255.2M pp collision tracks,
+accumulated at a center-of-mass energy of 900 GeV and 3.8T magnetic field, was used. Finally,
+shortly before the start of pp collisions in 2022, the alignment was updated using 6.3M cosmic ray
+tracks recorded at 3.8 T. Alignment corrections were derived at the level of single modules for the
+pixel detector and at the level of half-barrels and half-cylinders for the different strip sub-detectors.
+A comparison of the performance of the different sets of alignment constants obtained with cos-
+mic rays at 0T, cosmic rays at 3.8T, and pp collision tracks during 2021, as well as the alignment
+performance in 2022 prior to pp collisions at √푠=13.6 TeV, are presented in this section.
+3.1 Offline alignment using cosmic-ray and collision tracks (2021)
+The distribution of the median of the track-hit residuals per module (DMRs) constitutes a mea-
+sure of the tracking performance. The DMRs are monitored for all the tracker substructures, as
+shown for the barrel and forward pixel sub-detectors in Figure 1. A significant improvement on the
+track-hit residuals for the alignment with collision data over the alignments with cosmic ray muons
+only is observed. In the barrel region, DMR distributions can be obtained separately for the pixel
+barrel modules pointing radially inwards or outwards. The difference of their mean values Δ휇 in
+the local-x (x’) direction shown in Figure 2 as a function of the delivered integrated luminosity
+constitutes a measure of the reduction of Lorentz drift angle effects with the alignment procedure.
+60
+−
+40
+−
+20
+−
+0
+20
+40
+60
+m]
+µ
+)[
+hit
+-x'
+pred
+median(x'
+0
+100
+200
+300
+400
+500
+600
+700
+800
+m
+µ
+number of modules / 2.4 
+Preliminary
+ 
+CMS
+pp collisions (2021) 0.9 TeV
+BPIX
+Alignment with:
+0T cosmic rays
+m
+µ
+ = 19.5 
+σ
+m,  
+µ
+ = -2.1 
+µ
+ 
+3.8T cosmic rays
+m
+µ
+ = 9.0 
+σ
+m,  
+µ
+ = -1.9 
+µ
+ 
+cosmic rays + collisions
+m
+µ
+ = 1.1 
+σ
+m,  
+µ
+ = -0.1 
+µ
+ 
+40
+−
+30
+−
+20
+−
+10
+−
+0
+10
+20
+30
+40
+m]
+µ
+)[
+hit
+-x'
+pred
+median(x'
+0
+50
+100
+150
+200
+250
+300
+350
+400
+450
+m
+µ
+number of modules / 1.6 
+Preliminary
+ 
+CMS
+pp collisions (2021) 0.9 TeV
+FPIX
+Alignment with:
+0T cosmic rays
+m
+µ
+ = 9.3 
+σ
+m,  
+µ
+ = -1.9 
+µ
+ 
+3.8T cosmic rays
+m
+µ
+ = 9.3 
+σ
+m,  
+µ
+ = -1.9 
+µ
+ 
+cosmic rays + collisions
+m
+µ
+ = 0.8 
+σ
+m,  
+µ
+ = -0.2 
+µ
+ 
+Figure 1: The distribution of median residuals is shown for the local-x coordinate in the barrel pixel (left)
+and forward pixel (right). The alignment constants used for the reprocessing of the pp collision data (red) are
+compared with the ones derived using cosmic rays only, recorded at 0T (green) and 3.8T (blue). The quoted
+means 휇 and standard deviations 휎 correspond to parameters of a Gaussian fit to the distributions [4].
+The effect of the alignment calibration on the reconstruction of physics objects is also stud-
+ied. The distance between tracks and the unbiased track-vertex residuals is studied, searching for
+4
+
+CMS Tracker Alignment Activities during LHC Long Shutdown 2
+Sandra Consuegra Rodríguez
+0
+200
+400
+600
+800
+1000
+ ]
+-1
+b
+µ
+Delivered integrated luminosity  [
+5
+−
+0
+5
+10
+15
+20
+m]
+µ
+ [
+µ
+∆
+BPIX (x')
+0T cosmic rays
+3.8T cosmic rays
+cosmic rays + collisions
+Preliminary
+ 
+CMS
+pp collisions (2021) 0.9 TeV
+Alignment with
+Figure 2: Difference between the mean values Δ휇 obtained separately for the modules with the electric field
+pointing radially inwards or outwards. After alignment with cosmic and collision tracks, the mean difference
+Δ휇 is consistently closer to zero [4].
+potential biases in the primary vertex reconstruction. The mean value of the unbiased track-vertex
+residuals is shown in Figure 3 for the longitudinal and transverse planes, with a significant reduction
+of the bias when collision tracks are included in the alignment procedure.
+ [rad]
+φ
+track 
+1000
+−
+800
+−
+600
+−
+400
+−
+200
+−
+0
+200
+400
+600
+800
+1000
+m]
+µ
+ [
+〉 
+xy
+ d
+〈
+3
+−
+2
+−
+1
+−
+0
+1
+2
+3
+Alignment with:
+0T cosmic rays
+3.8T cosmic rays
+cosmic rays + collisions
+pp collisions (2021) 0.9 TeV
+CMS
+         Preliminary
+ [rad]
+φ
+track 
+600
+−
+400
+−
+200
+−
+0
+200
+400
+600
+m]
+µ
+ [
+〉 
+z
+ d
+〈
+3
+−
+2
+−
+1
+−
+0
+1
+2
+3
+Alignment with:
+0T cosmic rays
+3.8T cosmic rays
+cosmic rays + collisions
+pp collisions (2021) 0.9 TeV
+CMS
+         Preliminary
+Figure 3: The mean track-vertex impact parameter in the transverse 푑푥푦 plane (left) and longitudinal 푑푧
+plane (right) in bins of the track azimuthal angle 휙 is shown [4].
+4. Offline alignment using cosmic-ray tracks (2022)
+The alignment constants were updated before the start of Run 3 using cosmic ray muons recorded
+at 3.8 T, to correct for movements caused by the magnet cycle during the 2021-2022 winter break
+and repeated temperature cycles due to strip detector maintenance. After the geometry update, the
+bias on the distribution of median residuals for the forward pixel detector was corrected, as shown
+in Figure 4, left. Furthermore, the difference in the track impact parameters in the transverse plane
+5
+
+CMS Tracker Alignment Activities during LHC Long Shutdown 2
+Sandra Consuegra Rodríguez
+푑푥푦 for cosmic ray tracks passing through the pixel detector and split into two halves at their point
+of closest approach to the interaction region was also reduced, as shown in Figure 4, right.
+40
+−
+30
+−
+20
+−
+10
+−
+0
+10
+20
+30
+40
+m]
+µ
+)[
+hit
+-x'
+pred
+median(x'
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+m
+µ
+number of modules / 1.6 
+Preliminary
+ 
+CMS
+3.8T cosmic rays (2022)
+FPIX
+2021 geometry
+m
+µ
+m,  rms = 9.5 
+µ
+ = -4.9 
+µ
+ 
+alignment with 3.8T cosmic rays
+m
+µ
+m,  rms = 3.4 
+µ
+ = -0.0 
+µ
+ 
+100
+−
+50
+−
+0
+50
+100
+m)
+µ
+ (
+2
+ / 
+xy
+d
+∆
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+m
+µ
+fraction of tracks / 5 
+2021 geometry
+m
+µ
+m, rms = 38.2 
+µ
+ = 4.5 
+µ
+alignment with 3.8T cosmic rays
+m
+µ
+m, rms = 36.2 
+µ
+ = 0.7 
+µ
+Preliminary
+ 
+CMS
+3.8T cosmic rays (2022)
+Figure 4: Distribution of median residuals for the local-x coordinate in the forward pixel (left) and difference
+in track impact parameters in the transverse plane 푑푥푦 (right) [5].
+5. Summary
+The tracker alignment effort during the Run 3 commissioning period has been presented. The
+online alignment in the Prompt Calibration Loop and the strategy followed for the alignment calibra-
+tion considering the availability of tracks with certain topologies have been discussed. Finally, the
+data-driven methods used to derive the alignment parameters and the set of validations that monitor
+the physics performance after the update of the tracker alignment constants have been presented.
+References
+[1] CMS Collaboration, The CMS Experiment at the CERN LHC, 2008 JINST 3 S08004,
+doi:10.1088/1748-0221/3/08/S08004.
+[2] V. Blobel and C. Kleinwort, A new method for the high-precision alignment of track detectors,
+Proceedings of Conference on Advanced Statistical Techniques in Particle Physics, Durham,
+UK, 2002, https://inspirehep.net/literature/589639.
+[3] CMS Collaboration, Strategies and performance of the CMS silicon tracker alignment during
+LHC Run 2, 2022 Nucl. Instrum. Methods A 1037 166795, doi:10.1016/j.nima.2022.166795.
+[4] CMS Collaboration, CMS Tracker Alignment Performance Results CRAFT 2022, CMS Status
+Report, 150th LHCC Meeting - OPEN Session, 2022, https://indico.cern.ch/event/1156732/.
+[5] CMS Collaboration, Tracker Alignment Performance in 2021, CMS-DP-2022/017, 2022,
+https://cds.cern.ch/record/2813999/.
+6
+

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+page_content='CMS Tracker Alignment Activities during LHC Long  Shutdown 2  Sandra Consuegra Rodríguez푎,∗  푎Deutsches Elektronen-Synchrotron,  Notkestraße 85, 22607 Hamburg, Germany  E-mail: sandra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='consuegra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='rodriguez@desy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='de  The strategies for and the performance of the CMS tracker alignment during the 2021-2022 LHC  commissioning preceding the Run 3 data-taking period are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The results of the very  first tracker alignment after the pixel reinstallation, performed with cosmic ray muons recorded  with the solenoid magnet off are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Also, the performance of the first alignment of the  commissioning period with collision data events, collected at center-of-mass energy of 900 GeV,  is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Finally, the tracker alignment effort during the final countdown to LHC Run 3 is  discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  41st International Conference on High Energy physics - ICHEP2022  6-13 July, 2022  Bologna, Italy  1On behalf of the CMS Collaboration  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='it/  CMS Tracker Alignment Activities during LHC Long Shutdown 2  Sandra Consuegra Rodríguez  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' CMS tracker detector  The innermost tracking system of the CMS experiment, called the tracker, consists of two de-  vices, the Silicon Pixel and Silicon Strip detectors, within a length of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8 m and a diameter of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='5  m [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The closest detector to the interaction point, the Silicon Pixel detector, consists of the barrel  pixel sub-detector (BPIX) composed of 4 layers and the forward pixel sub-detector (FPIX) with two  forward endcaps, each composed of 3 disks, for a total of 1856 modules in the current configuration  (Phase 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Because of its proximity to the interaction point, the pixel detector is exposed to the  highest density of tracks from the proton-proton collisions and, therefore, suffers more extensively  the effects of the radiation damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' To tackle these effects, the pixel tracker was extracted from the  CMS experimental cavern, underwent extensive repairs, was provided with a new innermost layer,  and was reinstalled during the LHC Long Shutdown 2 (LS2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The silicon strip detector consists  of 15 148 modules and is composed of the Tracker Inner Barrel (TIB), Tracker Inner Disks (TID),  Tracker Outer Barrel (TOB), and Tracker EndCaps (TEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Track-based alignment  The tracker was specifically designed to allow for a very accurate determination of the trajec-  tory of charged particles by ensuring an intrinsic spatial resolution of up to 10-30 microns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' From  the positions at a number of points registered in the detector in the form of hits, the trajectory of a  charged particle can be reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Once the hits belonging to each trajectory are associated, a  fit is performed and the track parameters such as the track curvature radius, impact parameter 푑푥푦  in the xy plane, impact parameter 푑푧 along the beam direction, as well as the angles 휃 and 휙, are  obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The tracking efficiency, a measure of how efficiently a charged particle passing through  the tracker is reconstructed, provides a quantitative estimate of the tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Given  the deflection of charged particles in a magnetic field, their transverse momentum can be computed  as the product of the electric charge, the magnetic field, and the curvature radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Thus, the track  parameter uncertainties are propagated to the momentum measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Furthermore, the resolu-  tion of a reconstructed primary vertex position depends strongly on the number of tracks used to fit  the vertex and the transverse momentum of those tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Hence, high-quality track reconstruction  paves the way for accurate primary and secondary vertex reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  To ensure good tracking efficiency, as well as a precise momentum measurement and primary ver-  tex reconstruction, the uncertainty on the track parameters needs to be reduced as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  One of the main contributions to this uncertainty comes from the limited precision of the hit position  measurements entering the track fit from which the track parameters are determined, which in turn  is related to the overall limited knowledge of the position of the detector modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The latter has  two main contributions: the intrinsic spatial resolution of the sensors and the uncertainty related  to the limited knowledge of the displacements, rotations, and surface deformations of the tracker  modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' For an accurate determination of the track parameters, this second source of uncertainty  in the position of the detector modules needs to be reduced to at least the intrinsic spatial resolution  of the sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The correction of the position, orientation, and curvature of the tracker modules to  reach a precision better than the intrinsic spatial resolution is the task of tracker alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  The so-called track-based alignment consists of fitting a set of tracks with an appropriate track  2  CMS Tracker Alignment Activities during LHC Long Shutdown 2  Sandra Consuegra Rodríguez  model, and computing track-hit residuals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=', the difference between the measured hit position and  the corresponding prediction obtained from the track fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Geometry corrections can be derived from  the 휒2 minimization of these track-hit residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The Millepede and HipPy alignment algorithms  are used by CMS to solve the 휒2 minimization problem [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The alignment parameters are de-  termined with Millepede in a simultaneous fit of all tracks, involving two types of parameters: the  local parameters that characterize the tracks used for the alignment, and nine global parameters that  describe the position, orientation, and surface deformations of the modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The local parameters  of a single track are only connected to the subset of global parameters that are related to that track,  and they are not directly connected to the local parameters of other tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The global parameters  of each of the single modules of the detector can be corrected in a single alignment fit if enough  tracks are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' On the other hand, when using the HipPy algorithm, the 휒2 of each sensor is  minimized with respect to a change in the local alignment of that sensor only, keeping the parame-  ters of every other sensor fixed, in an iterative procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  Once the set of alignment constants is obtained, the improvement of post-alignment track-hits resid-  uals is reviewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Furthermore, before the new detector geometry is updated online for the data taking  or used for the data reprocessing, the impact of the new set of alignment constants in the tracking  performance, vertexing performance, and physics observables such as the mass of the Z boson res-  onance as function of the pseudorapidity and azimuthal angle is checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  A simplified version of the offline alignment described above also runs online as part of the Prompt  Calibration Loop (PCL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The PCL alignment uses the MillePede algorithm and performs the align-  ment of the pixel high-level structures at the level of ladders and panels, which ensures the consid-  eration of radiation effects of the innermost layer of the barrel pixel detector already during data  taking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The obtained constants are then used for the reconstruction of the next run if movements are  above certain thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Thus, the online and offline alignment are complementary components  of the tracker alignment within CMS, one providing automated online correction of the pixel high-  level structures and the other refining the alignment calibration with the possibility to reach each of  the single modules of the detector in a single alignment fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Tracker alignment effort prior to the beginning of Run 3  The first data-taking exercise upon the restart of operations in 2021 consisted of recording cos-  mic ray muons with the magnetic field off, “cosmic run at zero Tesla” (CRUZET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The alignment  with cosmic ray data has the advantage of allowing the update of the tracker alignment constants  before the start of collision data taking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Major shifts in the pixel and strip sub-detectors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=', due  to magnet cycles and temperature changes) can be identified and the geometry corrected accord-  ingly before beams are injected into the collider and collision data becomes available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The very first  alignment of the pixel detector after reinstallation in the experimental cavern was performed using  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='9M cosmic ray tracks recorded during the CRUZET period, at the level of single modules for the  pixel detector and the outer barrel of the strip detector, and of half-barrels and half-cylinders for the  rest of the strip partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' This period was followed by cosmic data-taking with magnetic field at  nominal value (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8T), “cosmic run at four Tesla” (CRAFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' In this case, the geometry was derived  using 765k cosmic ray tracks with the alignment corrections derived at the level of single modules  for the barrel pixel and at the level of half-barrels and half-cylinders for the forward pixel and all of  3  CMS Tracker Alignment Activities during LHC Long Shutdown 2  Sandra Consuegra Rodríguez  the strip sub-detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' While the geometries derived with CRUZET and CRAFT data constituted  relevant updates of the alignment constants starting from a potentially large misalignment, the re-  sults are statistically limited by the available number of cosmic ray tracks, especially in the forward  pixel endcaps, and systematically limited by the lack of kinematic variety of the tracks sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  For a further improvement of the alignment calibration, a sample of 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='2M pp collision tracks,  accumulated at a center-of-mass energy of 900 GeV and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8T magnetic field, was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Finally,  shortly before the start of pp collisions in 2022, the alignment was updated using 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='3M cosmic ray  tracks recorded at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Alignment corrections were derived at the level of single modules for the  pixel detector and at the level of half-barrels and half-cylinders for the different strip sub-detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  A comparison of the performance of the different sets of alignment constants obtained with cos-  mic rays at 0T, cosmic rays at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8T, and pp collision tracks during 2021, as well as the alignment  performance in 2022 prior to pp collisions at √푠=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='6 TeV, are presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='1 Offline alignment using cosmic-ray and collision tracks (2021)  The distribution of the median of the track-hit residuals per module (DMRs) constitutes a mea-  sure of the tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The DMRs are monitored for all the tracker substructures, as  shown for the barrel and forward pixel sub-detectors in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' A significant improvement on the  track-hit residuals for the alignment with collision data over the alignments with cosmic ray muons  only is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' In the barrel region, DMR distributions can be obtained separately for the pixel  barrel modules pointing radially inwards or outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The difference of their mean values Δ휇 in  the local-x (x’) direction shown in Figure 2 as a function of the delivered integrated luminosity  constitutes a measure of the reduction of Lorentz drift angle effects with the alignment procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content="  60  −  40  −  20  −  0  20  40  60  m]  µ  )[  hit  x'  pred  median(x'  0  100  200  300  400  500  600  700  800  m  µ  number of modules / 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='4  Preliminary  CMS  pp collisions (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='9 TeV  BPIX  Alignment with:  0T cosmic rays  m  µ  = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='5  σ  m,  µ  = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='1  µ  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8T cosmic rays  m  µ  = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='0  σ  m,  µ  = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='9  µ  cosmic rays + collisions  m  µ  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='1  σ  m,  µ  = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content="1  µ  40  −  30  −  20  −  10  −  0  10  20  30  40  m]  µ  )[  hit  x'  pred  median(x'  0  50  100  150  200  250  300  350  400  450  m  µ  number of modules / 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='6  Preliminary  CMS  pp collisions (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='9 TeV  FPIX  Alignment with:  0T cosmic rays  m  µ  = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='3  σ  m,  µ  = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='9  µ  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8T cosmic rays  m  µ  = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='3  σ  m,  µ  = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='9  µ  cosmic rays + collisions  m  µ  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8  σ  m,  µ  = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='2  µ  Figure 1: The distribution of median residuals is shown for the local-x coordinate in the barrel pixel (left)  and forward pixel (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The alignment constants used for the reprocessing of the pp collision data (red) are  compared with the ones derived using cosmic rays only, recorded at 0T (green) and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8T (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The quoted  means 휇 and standard deviations 휎 correspond to parameters of a Gaussian fit to the distributions [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  The effect of the alignment calibration on the reconstruction of physics objects is also stud-  ied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=" The distance between tracks and the unbiased track-vertex residuals is studied, searching for  4  CMS Tracker Alignment Activities during LHC Long Shutdown 2  Sandra Consuegra Rodríguez  0  200  400  600  800  1000  ]  1  b  µ  Delivered integrated luminosity  [  5  −  0  5  10  15  20  m]  µ  [  µ  ∆  BPIX (x')  0T cosmic rays  3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8T cosmic rays  cosmic rays + collisions  Preliminary  CMS  pp collisions (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='9 TeV  Alignment with  Figure 2: Difference between the mean values Δ휇 obtained separately for the modules with the electric field  pointing radially inwards or outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' After alignment with cosmic and collision tracks, the mean difference  Δ휇 is consistently closer to zero [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  potential biases in the primary vertex reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The mean value of the unbiased track-vertex  residuals is shown in Figure 3 for the longitudinal and transverse planes, with a significant reduction  of the bias when collision tracks are included in the alignment procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  [rad]  φ  track  1000  −  800  −  600  −  400  −  200  −  0  200  400  600  800  1000  m]  µ  [  〉  xy  d  〈  3  −  2  −  1  −  0  1  2  3  Alignment with:  0T cosmic rays  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8T cosmic rays  cosmic rays + collisions  pp collisions (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='9 TeV  CMS  Preliminary  [rad]  φ  track  600  −  400  −  200  −  0  200  400  600  m]  µ  [  〉  z  d  〈  3  −  2  −  1  −  0  1  2  3  Alignment with:  0T cosmic rays  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8T cosmic rays  cosmic rays + collisions  pp collisions (2021) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='9 TeV  CMS  Preliminary  Figure 3: The mean track-vertex impact parameter in the transverse 푑푥푦 plane (left) and longitudinal 푑푧  plane (right) in bins of the track azimuthal angle 휙 is shown [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Offline alignment using cosmic-ray tracks (2022)  The alignment constants were updated before the start of Run 3 using cosmic ray muons recorded  at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8 T, to correct for movements caused by the magnet cycle during the 2021-2022 winter break  and repeated temperature cycles due to strip detector maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' After the geometry update, the  bias on the distribution of median residuals for the forward pixel detector was corrected, as shown  in Figure 4, left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Furthermore, the difference in the track impact parameters in the transverse plane  5  CMS Tracker Alignment Activities during LHC Long Shutdown 2  Sandra Consuegra Rodríguez  푑푥푦 for cosmic ray tracks passing through the pixel detector and split into two halves at their point  of closest approach to the interaction region was also reduced, as shown in Figure 4, right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content="  40  −  30  −  20  −  10  −  0  10  20  30  40  m]  µ  )[  hit  x'  pred  median(x'  0  20  40  60  80  100  120  140  160  180  200  m  µ  number of modules / 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='6  Preliminary  CMS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8T cosmic rays (2022)  FPIX  2021 geometry  m  µ  m,  rms = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='5  µ  = -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='9  µ  alignment with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8T cosmic rays  m  µ  m,  rms = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='4  µ  = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='0  µ  100  −  50  −  0  50  100  m)  µ  (  2  /  xy  d  ∆  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='12  m  µ  fraction of tracks / 5  2021 geometry  m  µ  m, rms = 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='2  µ  = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='5  µ  alignment with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8T cosmic rays  m  µ  m, rms = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='2  µ  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='7  µ  Preliminary  CMS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='8T cosmic rays (2022)  Figure 4: Distribution of median residuals for the local-x coordinate in the forward pixel (left) and difference  in track impact parameters in the transverse plane 푑푥푦 (right) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Summary  The tracker alignment effort during the Run 3 commissioning period has been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' The  online alignment in the Prompt Calibration Loop and the strategy followed for the alignment calibra-  tion considering the availability of tracks with certain topologies have been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content=' Finally, the  data-driven methods used to derive the alignment parameters and the set of validations that monitor  the physics performance after the update of the tracker alignment constants have been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
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+page_content='1088/1748-0221/3/08/S08004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  [2] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
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+page_content='ch/event/1156732/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
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+page_content='cern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='ch/record/2813999/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}
+page_content='  6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MNFQT4oBgHgl3EQfVDaQ/content/2301.13299v1.pdf'}

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+An Empirical Study on Software Bill of Materials:
+Where We Stand and the Road Ahead
+Boming Xia∗†, Tingting Bi†‡, Zhenchang Xing†§, Qinghua Lu†, Liming Zhu†∗
+∗University of New South Wales, Sydney, Australia
+†CSIRO’s Data61, Sydney, Australia
+‡Monash University, Melbourne, Australia
+§Australian National University, Canberra, Australia
+Abstract—The rapid growth of software supply chain attacks
+has attracted considerable attention to software bill of materials
+(SBOM). SBOMs are a crucial building block to ensure the trans-
+parency of software supply chains that helps improve software
+supply chain security. Although there are significant efforts from
+academia and industry to facilitate SBOM development, it is
+still unclear how practitioners perceive SBOMs and what are the
+challenges of adopting SBOMs in practice. Furthermore, existing
+SBOM-related studies tend to be ad-hoc and lack software
+engineering focuses. To bridge this gap, we conducted the first
+empirical study to interview and survey SBOM practitioners. We
+applied a mixed qualitative and quantitative method for gathering
+data from 17 interviewees and 65 survey respondents from 15
+countries across five continents to understand how practitioners
+perceive the SBOM field. We summarized 26 statements and
+grouped them into four topics on SBOM’s states of practice.
+Based on the study results, we derived a goal model and
+highlighted future directions where practitioners can put in their
+effort.
+Index Terms—software bill of materials, SBOM, bill of mate-
+rials, responsible AI, empirical study.
+I. INTRODUCTION
+Modern software products are assembled through intricate
+and dynamic supply chains [1], while recent attacks against
+software supply chains (SSC) have increased significantly
+(e.g., SolarWinds attack [2]). According to Sonatype’s report
+[3], there was a 650% year-over-year increase in SSC attacks
+in 2021, and the number was 430% in 2020. SSC attacks
+mainly aim at the upstream open source software/components
+(OSS) [4], and the majority of the software organizations use
+OSS [5]. The reliance on OSS leads to additional risks, such
+as the lack of reliable maintenance and support compared to
+proprietary software/components [6]. For example, in March
+2022, the primary maintainer of a popular OSS package,
+node-ipc, intentionally injected malware into an update that
+overwrote file systems in specific geographical locations [7].
+The security risks of software and its supply chain call for
+improved visibility into the SSC, with which timely and
+accurate identification of the impacted software/components
+could be carried out in case of a vulnerability or an SSC attack.
+A software bill of materials (SBOM) is a formal machine-
+readable inventory of the components (and their dependency
+relationships) used for producing a software product [8].
+SBOMs enhance the security of both the proprietary and
+open source components in SSCs [9] through improved trans-
+parency. According to Linux Foundation’s SBOM and Cyber-
+security Readiness report (SBOM readiness report for short)
+[5], SBOMs are critical for enhancing SSC security. 90%
+of the surveyed organizations have started or are planning
+their SBOM journey, with 54% already addressing SBOMs.
+The report also estimated 66% and 13% growth in SBOM
+production or consumption in 2022 and 2023, respectively.
+Nonetheless, some organizations are still concerned about how
+SBOM adoption and application will evolve (e.g., 40% are
+uncertain about industrial SBOM commitment and 39% seek
+consensus on SBOM data fields).
+Despite SBOMs’ essentiality for software and SSC security,
+there remain questions to answer and problems to solve.
+Motivated by the value of SBOMs and the existing gaps, with
+the overarching goal of investigating the SBOM status quo
+from practitioners’ perspectives, this paper aims to answer the
+following research questions (RQs).
+RQ1: What is the current state of SBOM practice?
+Despite the benefits of SBOMs and the SBOM readiness
+report showing an overall 90% of SBOM readiness, how
+practitioners perceive SBOMs and how SBOMs are being
+addressed in practice is unclear. To answer this question, we
+analyzed the SBOM practice status from SBOM generation,
+distribution and sharing, validation and verification, and vul-
+nerability and exploitability management. We summarized the
+current SBOM practices and what practitioners expect.
+RQ2: What is the current state of SBOM tooling
+support?
+Despite the proliferation of the SBOM tooling market, this
+RQ focuses on the SBOM tooling status from the practition-
+ers’ perspective. We investigated the practitioners’ attitudes
+towards existing tools from the following aspects: the ne-
+cessity/availability/usability/integrity of SBOM tools. While
+exploring the current SBOM tooling state, we also looked into
+practitioners’ expectations of SBOM tools.
+RQ3: What are the main concerns for SBOM?
+This RQ investigates the most outstanding concerns SBOM
+practitioners have. Although the prospect of SBOMs is promis-
+ing, there are still challenges to resolve. With this RQ, we aim
+to provide a reference for the most imminent issues for future
+research and development on SBOMs.
+Our research aims to unveil the state of the SBOM field,
+investigating what practitioners have and how they are ad-
+arXiv:2301.05362v1  [cs.SE]  13 Jan 2023
+
+dressing SBOMs, versus what they expect. Our work on
+SBOMs has four main distinctions and contributions compared
+to existing work represented by the SBOM readiness report:
+1) Timeliness: The SBOM readiness report was published
+in January 2022, with the survey launched in June 2021.
+Google Trends shows a significant increase in SBOM
+interest since July 2021. This study provides a more
+updated view of SBOMs.
+2) Software engineering (SE) angles: The SBOM readi-
+ness report is industry-oriented and security-focused,
+whereas this research broadens and deepens the report
+and supplements SE angles, evidenced by considerations
+such as SBOM generation/update throughout software
+development lifecycle (Finding 5) and AIBOM (Section
+IV-A8). The detailed comparison between this research
+and the SBOM readiness report is presented in Table IV.
+3) Different objectives: The SBOM readiness report aims
+to investigate whether and to what extent organizations
+are prepared for SBOM production and consumption
+(i.e., readiness). In contrast, this study focuses on current
+SBOM practices and expectations from practitioners’
+perspectives. We provide a set of implications, including
+a goal model, for future endeavors towards further
+operationalizing SBOMs.
+4) Systematic methodology: We conducted the first em-
+pirical study on the SBOM status from practitioners’
+perspectives, using a mixed methodology. Instead of
+using predefined questions as in SBOM readiness re-
+port, we qualitatively and quantitatively coded in-depth
+opinions from 17 interviewees into 26 representative
+statements, which were validated in a survey with 65
+valid respondents from 15 countries.
+The remainder of this paper is organized as follows. Section
+II describes the context of the SBOM field. Section III presents
+the methodology of our study. In section IV we present the
+study results. Section V discusses the implications of this
+study. Section VI discusses related work, and section VII
+draws conclusions and outlines avenues for future work.
+II. BACKGROUND: WHAT IS SBOM?
+A bill of materials (BOM) was initially used in the manu-
+facturing industry as an inventory list of all the sub-assemblies
+and components in a parent assembly [10]. Sharing the same
+origin, an SBOM as the building block to enhanced software
+supply chain security is a software “BOM” (see Fig. 1).
+There are three main SBOM standard formats: a) Software
+Package Data eXchange (SPDX), b) CycloneDX, and c)
+Software Identification (SWID) Tagging, while the first two
+are most adopted [12]. SPDX is an open-source international
+standard hosted by the Linux Foundation, emphasizing licence
+compliance. CycloneDX was designed by OWASP in 2017
+whose primary focus is security. SWID Tagging is also an in-
+ternational standard maintained by the US National Institute of
+Standards and Technology, focusing on providing a transparent
+software/components identification mechanism.
+Fig. 1. (AI) Software supply chain and SBOM [11].
+The above three formats all have the corresponding tooling
+to help operationalize the formats into practice that are listed
+on their respective websites. It is worth mentioning that the
+SBOM formats and tooling working group under the US
+National Telecommunications and Information Administration
+(NTIA) had an effort to summarize all tools supporting dif-
+ferent standards (i.e., SPDX list, CycloneDX list, SWID list).
+In terms of government-side SBOM efforts, with incidents
+such as the SolarWinds attack ringing the alarm of SSC
+security, the US government issued an executive order on en-
+hancing cybersecurity [13] in May 2021, explicitly mandating
+all companies trading with the US government to provide
+SBOMs. NTIA has published a series of documents and
+guidelines (e.g., SBOM minimum elements [8]) to facilitate
+SBOM development. The Cybersecurity and Infrastructure
+Security Agency (CISA) is also actively working on SBOM
+facilitation by regularly hosting listening sessions with the
+SBOM industrial community.
+Notably, the Linux Foundation published the SBOM readi-
+ness report [5] in January 2022, which surveyed 412 organi-
+zations across the globe. According to the report, 98% of the
+surveyed organizations are concerned about software security,
+over 80% are aware of the US executive order, and 90%
+have started their SBOM journey. Although the evolvement of
+SBOM adoption and application remains a concern, the report
+predicts that the SBOM tool market is expected to explode
+in 2022 and 2023. As of January 2022, there were about
+20 SBOM tool vendors in the market, and some were from
+adjacent markets like Software Composition Analysis (SCA)
+which dates back to 2002 [14]. Meanwhile, various open
+source tools are available with a focus on SBOM generation.
+However, while the SBOM readiness report comprehen-
+sively covers a broad category, their reports were based on
+multi-choice questions with limited choices, which constrained
+the possible answers. The guidelines and documents published
+by NTIA provide a reference for understanding SBOM and
+SBOM practice but lack the actual perception from the SBOM
+practitioners. To fill the gaps, we interviewed 17 SBOM
+practitioners (see Table I) with open-ended questions on how
+they perceive current SBOM practice and SBOM tooling.
+We then organized an online survey containing statements
+
+SBOM
+Component 1
+SBOM
+SBOM
+Component 2
+(AIBOM)
+SBOM
+Component 3
+Software
+Data
+Model
+AIBOM
+(AI) Component 4
+Data/model co-versioing registryfrom the interviews, allowing more practitioners to validate
+the statements. Compared with the SBOM readiness report
+and NTIA’s documents, we focus on how actual SBOM
+practitioners think about SBOMs and how they are addressing
+SBOMs in production without limiting the possible answers.
+III. RESEARCH METHODOLOGY
+This study presents an exploratory empirical study on
+SBOM status in practice. Fig. 2 shows the overall methodology
+adopted in this paper that consists of three stages, following a
+mixed qualitative and quantitative approach [15]. We described
+the planning and preparation stage in Section III-A; the data
+collection and analysis processes of the interviews and the
+online survey are presented in Sections III-B and III-C.
+Fig. 2. Overall research methodology.
+A. Stage Zero: Planning and Preparation
+At the planning stage, we prepared a research protocol1 and
+drafted two types of interview questions: demographics and
+open-ended. For demographics, we ask about the participants’
+background information, such as job roles and experiences.
+For the open-ended questions, we asked how the participants
+perceive SBOMs. We obtained ethics approvals for this study.
+B. Stage One: Interview
+Pilot interview and protocol refinement. Before the formal
+interviews, we conducted a small-scale pilot interview with 3
+participants from our connections. Based on their feedback
+and suggestions, we adjusted some interview questions.
+Participant recruitment. We recruited 17 SBOM prac-
+titioners from 13 organizations (e.g., CISA, Oracle) across
+7 countries (see Table I). Interviewees were recruited by:
+a) emailing our contacts who helped further disseminate the
+invitation emails to their colleagues; b) emailing developers
+on GitHub working on SBOM-related projects whose email
+addresses are public; c) advertising on Twitter and LinkedIn
+and interested people can contact the first author. The 17
+interviewees have worked in software-related fields for around
+14 years on average (min 4 years and max 30 years), while
+they have been working actively in SBOM-related fields for
+1The detailed research protocol is provided via the submission system as
+supplementary materials. A link will be attached in the formal version.
+TABLE I
+INTERVIEWEE DEMOGRAPHICS*
+ID
+Field of work
+Country
+Work exp.
+SBOM exp.
+I1
+Dev.
+China
+10
+0.5
+I2
+Dev.
+China
+10
+0.2
+I3
+Dev.
+China
+10
+0.2
+I4
+Dev. & Sec.
+Australia
+20
+3.0
+I5
+Dev.
+US
+18
+0.5
+I6
+Dev.
+US
+20
+0.5
+I7
+Sec.
+Brazil
+4
+0.5
+I8
+Dev. & Cnslt.
+Ireland
+15
+1.5
+I9
+Sec.
+India
+5.5
+1.0
+I10
+Cnslt. & Adv.
+US
+20
+3.0
+I11
+Dev. & Sec. (SBOM tool)
+Israel
+12
+1.5
+I12
+Cnslt. & Adv.
+Israel
+15
+0.3
+I13
+Dev.&Sec.&Res.
+Australia
+30
+2.0
+I14
+Dev. & Sec. & Res.
+US
+10
+3.0
+I15
+Dev.
+India
+8
+1.5
+I16
+Dev. (SBOM tool)
+US
+20
+1.0
+I17
+Cnslt. & Adv.
+US
+15
+5.0
+*Dev./Sec.: Software Development/Security; Cnslt.: Consultant; Adv.:
+Advisor; Res.: Researcher. Experiences are listed in years, as of July 2022
+around 1.4 years on average (min 2 months and max 5 years).
+We will refer to the 17 interviewees as I1 to I17.
+Transcribing and coding. 1) Transcribing. The interviews
+were audio-recorded. The first author transcribed the audio
+recordings, and the second author double-checked the tran-
+scripts. 2) Pilot coding. The first two authors (i.e., coders)
+conducted a pilot coding of the 3 pilot interview transcripts.
+They discussed the initial coding results and reached a certain
+level of preliminary agreement on the granularity of thematic
+coding. 3) Code generation. The coders then performed the-
+matic coding to qualitatively analyze the interview transcripts
+[16], [17] of 17 interviewees using MAXQDA2022 tool. The
+first coder generated 574 codes under 86 cards (i.e., repetitive
+and similar codes classified into the same category). The
+second coder generated 364 codes under 41 cards. After
+discussing the coding results with a third author, the coders
+further cleared the coding granularity, combined similar cards,
+and disposed of cards with limited value. Finally, a total of 54
+unique cards were generated.
+Data analysis and open card sorting. The coders sep-
+arately sorted the 54 generated cards into potential themes
+(not predefined) given thematic similarity. After the sorting
+process, the coders calculated Cohen’s Kappa value [18] to
+assess their agreement level. The overall value was 0.77,
+indicating substantial agreement. The coders discussed their
+disagreements to reach a common ground. he coders reviewed
+and agreed on the final themes to reduce card sorting bias.
+Eventually, we derived 26 statements (see Table III) under
+3 themes: State of SBOMs Practice (T1), SBOM Tooling
+Support (T2), and SBOM Issues and Concerns (T3). All the
+authors have double-checked our coding results to ensure the
+reported results are accurate and consistent.
+C. Stage Two: Online Survey
+We conducted an online survey to confirm or refute the ex-
+tracted statements. We designed the survey following Kitchen-
+
+Stage 0
+Stage 1
+Stage 2
+Planning & Preparation
+Interview
+Online survey
+Overarching goal
+Pilot study
+Pilot study
+Practitioners' perception on
+SBOM
+Protocol refinement
+Protocol refinement
+Participants
+Research questions
+recruitment
+Participants
+1. State of SBOM practice
+recruitment
+Interviews
+2. State of SBOM tooling
+Online survey
+3. SBOM concerns
+Transcribing & coding
+Data analysis
+Protocol
+Ethics approval
+Statement extractionTABLE II
+SURVEY RESPONDENTS DEMOGRAPHICS*
+Field of work
+Project team
+Work exp.
+SBOM exp.
+Dev. (38.6%)
+<10 ppl. (16)
+<1 year (1)
+<6 months (17)
+Sec. (30.1%)
+10-20 ppl. (20)
+1-3 years (11)
+0.5-1 year (17)
+Cnslt./Adv. (10.8%)
+20-50 ppl. (15)
+3-5 years (13)
+1-2 years (13)
+Mgmt (9.7%)
+>50 ppl. (15)
+5-10 years (9)
+>2 years (19)
+Res. (9.7%)
+-
+>10 years (32)
+-
+Other (1.1%)
+-
+-
+-
+*Mgmt.: Management; ppl.: people. Since multiple answers are supported
+for respondents’ work, field of work is listed in percentages while the
+others are listed with response numbers.
+ham and Pfleeger’s guideline [19]. The survey was anonymous,
+and all information collected was non-identifiable.
+Survey design and pilot study. The survey was published
+via Qualtrics. Different types of questions were included in the
+survey (e.g., multiple choice and free text). The statements are
+scored on a 5-point Likert scale (Strongly disagree, Disagree,
+Neutral, Agree, Strongly agree), with an additional “Not sure”.
+We piloted the survey with six participants from Australia
+and Singapore and then refined the survey. The pilot study
+results were excluded from the final results. The formal
+survey consists of 7 sections: demographics, SBOM status quo,
+generation, distribution, tooling, benefits, and concerns.
+Participants recruitment. To increase the number of par-
+ticipants, we adopted the following strategy for recruitment:
+● We contacted industrial practitioners from several compa-
+nies worldwide and asked for their help in disseminating
+the survey invitation emails.
+● We sent invitation emails to over 2000 developers from
+GitHub whose email addresses are publicly available.
+● We posted the recruitment advertisement on social media
+platforms (i.e., Twitter and LinkedIn).
+We received a total of 129 responses, including 27 with
+respondents selecting “(Very) unfamiliar with SBOM”. After
+removing them and the incomplete responses and responses
+completed within 2 minutes, we had 65 valid responses. We
+acknowledge that the number of responses is not as ideal
+as similar empirical studies (e.g., [17], [20]). However, we
+believe this is consistent with our findings on the lack of
+SBOM adoption and education (i.e., Findings 1 and 10). The
+65 participants come from 15 countries across 5 continents.
+The top 3 countries where the participants reside are Australia,
+China, and the US. An overview of the survey respondents’
+demographics is presented in Table II. It is worth noting
+that although nearly half (47.7%) of the respondents have
+worked in the software field for over 10 years, only one
+quarter (27.7%) have worked on SBOMs for over 2 years,
+indicating that SBOM is still a relatively fresh concept to
+software practitioners.
+Data analysis. Apart from the demographics, SBOM fa-
+miliarity questions, and a final optional free-text question, all
+statements are presented as Likert-scale questions (see the bar
+charts in Table III) for the evaluation of the agreement degree.
+IV. STUDY RESULTS
+This section reports the study results. In Table III, we drew
+the Linkert distribution graphs for 26 statements that were re-
+organized into four topics and calculated the overall scores
+based on: Strongly Disagree (1), Disagree (2), Neutral (3),
+Agree (4), Strongly Agree (5), and Not Sure (0). We calculated
+the percentages of “agrees” (strongly agree and agree) and
+“disagrees” (strongly disagree and disagree) of each statement.
+Following the SBOM background in Section II, this section
+starts by introducing the current SBOM practices (Section
+IV-A). Then Section IV-B investigates the current tooling
+status of SBOMs. Finally, Section IV-C presents practitioners’
+primary concerns for SBOMs.
+A. RQ1: What is the current state of SBOM practice?
+To answer RQ1, we discussed 16 statements in this section
+based on T1 (see Table III). Our results suggest that SBOMs
+are not widely adopted. SBOM generation and distribution re-
+quire further standardization and maturer mechanisms. SBOM
+data validation is generally neglected. For the typical SBOM
+use case of vulnerability management, the exploitability status
+classification should be more than binary.
+1) SBOM benefits: We summarized 3 statements (i.e., S1-
+S3) on SBOM benefits based on the interviews.
+15 of 17 interviewees mentioned that the enhanced trans-
+parency of the software supply chain is one of the exceptional
+advantages of SBOMs [S1, 90.8% agree, 1.5% disagree].
+Transparency brings a lot of favorable consequences, such
+as end-of-life software management, vulnerability tracking,
+and license compliance checking [5]. “The biggest benefit is
+knowing exactly what is being bundled in your software, right?
+So it is to assure our customers that, if there’s a vulnerability
+reported, you immediately know (whether) you are impacted
+or not, if the SBOM is accurate.” (I13-Dev.&Sec.&Res.)
+Another benefit originates from the SBOM data. The uni-
+fication of software composition details provided by SBOMs
+is beneficial as the standardized SBOM data has the potential
+to be further built upon. “The SBOM itself isn’t the valuable
+part. The valuable part is, how do we turn that data into
+intelligence, into action.” (I17-Cnslt.&Adv.) Based on the
+SBOM data, it is promising that there will be SBOM-centric
+ecosystems emerging [S2, 86.2% agree, 1.5% disagree]. How-
+ever, due to the lack of SBOM adoption (see Section IV-A2),
+such ecosystems are long-term goals not to be achieved soon.
+Thirdly, although adopting SBOMs requires extra efforts
+(e.g., additional tools and processes, education to related
+personnel), the benefits of SBOMs outweigh the costs [S3,
+86.2% agree, 7.7% disagree]. � Although some organizations
+are “worried whether SBOMs would increase the cost of a
+software product” (I1-Dev.), � the majority favor the benefits
+brought by SBOMs as the potential loss without SBOMs can
+be devastating. “What I think about is, what is the cost when
+a vulnerability is exploited? ...if you look at SolarWinds event,
+that cost was $800 million.” (I6-Dev.)
+
+TABLE III
+INTERVIEW AND SURVEY RESULTS ON SBOM STATEMENTS
+Likert distribution
+Statement
+Graph
+Score
+T1. State of SBOM practice
+S1. Improving transparency and visibility into the software products is the biggest benefit of SBOMs.
+4.42
+S2. SBOM data form the foundation of a potential SBOM-centric ecosystem.
+4.05
+S3. The benefits brought by SBOMs outweigh the costs of SBOMs (e.g., extra learning and management of SBOMs & tools).
+Section IV-A1
+SBOM benefits
+4.34
+S4. Currently third-party (open source or proprietary) components are not equipped with SBOMs.
+4.11
+S5. SBOMs are not generated for all software products (produced/used) within an organization.
+Section IV-A2
+SBOM adoption
+4.25
+S6. SBOMs can be generated at different stages of the software development lifecycle.
+4.14
+S7. Currently, a new SBOM is not always re-generated when there’s any change to software artifacts.
+Section IV-A3
+SBOM generation points
+3.31
+S8. SBOMs are currently generated in a non-standardized format (e.g., not SPDX nor CycloneDX nor SWID).
+2.86
+S9. Despite the 7 minimum data fields recommended by NTIA, the minimum fields are not necessarily all included in SBOMs.
+3.57
+S10. In practice, SBOMs are extended with more useful data fields (other than the 7 minimum data fields) whenever possible.
+Section IV-A4
+SBOM data fields and
+standarization
+4.34
+S11. SBOMs are currently only generated for internal consumption.
+2.97
+S12. Access control should be required for the distribution of SBOMs for proprietary software/components.
+4.14
+S13. Content tailoring (sharing partial SBOMs) should be required for SBOM distribution of proprietary software/components.
+Section IV-A5
+SBOM distribution
+3.57
+S14. SBOM producer’s (i.e., software vendor) reputation is important for assessing SBOM integrity (e.g., completeness).
+3.4
+S15. Currently there are no validation mechanisms to ensure SBOM integrity (accuracy, completeness etc).
+Section IV-A6
+SBOM validation
+3.28
+S16. Current vulnerability management with SBOMs doesn’t focus on the actual exploitability of the vulnerability.
+Section IV-A7
+Vul. & exploitability
+3.72
+S17. SBOMs for AI software are different from SBOMs for traditional software.
+Section IV-A8
+AIBOM
+3.26
+T2. State of SBOM tooling support
+S18. Although existing sources (e.g., package manager, POM.xml) are already there, it is still necessary to parse and feed
+metadata from these sources into a standard format via SBOM tools.
+Section IV-B1
+Necessity of SBOM tools
+4.08
+S19. There are significantly limited tools for SBOM consumption.
+4
+S20. SBOM consumption should be integrated with existing tools (e.g., vulnerability/configuration management tools).
+Section IV-B2
+Availability of SBOM tools
+4.03
+S21. Existing SBOM tools can be hard to use (e.g., lack of usability, complexity).
+3.57
+S22. SBOM tools lack of interoperability and standardization (e.g., the hash of one component generated by different tools
+can be different).
+Section IV-B3
+Usability of SBOM tools
+3.92
+S23. End users can’t validate the integrity (e.g., accuracy and completeness) of the generated SBOMs by existing tools.
+Section IV-B4
+Validation of SBOM tools
+3.69
+T3. SBOM issues & concerns
+S24. Existing SBOM standards don’t meet current market demands (e.g., the standards support only limited fields).
+2.85
+S25. Attackers can take advantage of the information contained in SBOMs.
+3.63
+S26. There is hesitation in adopting SBOMs due to various concerns (e.g., lack of basic IT asset management).
+Section IV-C
+SBOM concerns
+3.98
+Finding 1: The transparency brought by SBOMs can
+enable accountability, traceability and security, but
+there is a lack of systematic consumption-scenario-
+driven design of SBOM features.
+2) SBOM adoption: We summarized 2 statements (i.e., S4-
+S5) on SBOM adoption status (see Table III).
+Despite the benefits of SBOMs and the SBOM readiness
+report’s [5] optimistic results that 90% of the 412 sampled
+organizations have started or are planning their SBOM journey,
+the adoption of SBOM is not as optimistic according to our
+interviews and survey. For example, most existing third-party
+software or components, either open source or proprietary, are
+not equipped with SBOMs [S4, 83.1% agree, 13.8% disagree].
+As I2 stated, “when introducing third-party components to
+our organization, we need to try to generate SBOMs for
+them because not all of them have SBOMs.” (I2-Dev.) How-
+ever, considering the prevalence of OSS, the unavailability
+of SBOMs for (open-source) software/components also holds
+software vendors back from SBOM adoption as they may
+wonder whether SBOM adoption is an industrial consensus.
+In addition, SBOMs are not generated for all software
+even inside a software vendor organization [S5, 87.7% agree,
+7.7% disagree]. As stated by I10, “software vendors may be
+producing SBOMs for some customer products. But I bet most
+of them don’t generate SBOMs for the financial software they
+are using.” (I10-Cnslt.& Adv.)
+Finding 2: A large portion of widely used software,
+especially OSS, does not have SBOMs. The incentives
+for generating SBOMs for OSS and proprietary
+software need to be propagated.
+3) SBOM generation point: We summarized 2 statements
+(i.e., S6-S7 in Table III) on SBOM (re-)generation.
+SBOMs can be generated at a set of different stages in the
+software development life cycle (e.g., build time, run time,
+before delivery, after third-party components introduction, etc.)
+[S6, 84.6% agree, 12.3% disagree], which differs from case to
+case as “the challenge here sort of builds on the sheer diversity
+of software” (I17-Cnslt.&Adv.) (e.g., modern/legacy software,
+container images, cloud-based software). For example, “for
+legacy software that’s already out there in the wild, maybe
+
+you don’t have reproducible builds. You only have that built
+artifacts... it’s like you can’t do it (generate SBOMs) at build
+time anymore... it’s just too difficult... But you could probably
+do a run time (SBOM) generator.” (I8-Dev.&Cnslt)
+As stated by I8, “I don’t really think there is a perfect
+time (for SBOM generation)”. (I8-Dev.&Cnslt) Since software
+development goes through a life cycle, ideally an SBOM
+should be generated at the early stages and then gradually
+enriched with more information from the latter stages,
+which was supported by I12 and I14. “I think the way to
+actually include the most full SBOM is to have visibility
+towards the whole cycles, from the report to the build to the
+factory.” (I12-Cnslt.&Adv.) “I do not think we should produce
+an SBOM as a one-shot process, but rather we should be
+carrying evidence and partial SBOMs and enriching them in
+every single operation.” (I14-Dev.&Sec.&Res.)
+As for SBOM re-generation, it is evident that whenever any
+change happens to any software artifact, the corresponding
+SBOMs should be timely re-generated to reflect this change,
+which is hardly the current practice followed by every SBOM
+producer [S7, 53.8% agree, 35.4% disagree]. As stated by I10,
+“that is more of an aspirational goal - it’s not something that
+will be realized right away. Because right now it would be just
+great if they put out a new SBOM whenever they did a new ma-
+jor version, and that is better than nothing.”(I10-Cnslt.&Adv.)
+However, since some organizations are re-generating SBOMs
+upon each change, and if SBOM re-generation is to be a
+standard practice, solutions like SBOM version control are
+needed for managing the SBOMs. As stated by I8, with
+different versions of SBOMs, “you need to version control
+your SBOMs, and figure out how to distribute that information
+to your customers.” (I8-Dev.&Cnslt)
+Finding 3: SBOM generation is belated and not dy-
+namic, while ideally SBOMs are expected to be gener-
+ated during early software development stages and
+continuously enriched/updated.
+4) SBOM data fields standardization: In this subsection, we
+investigated what data fields generated SBOMs contain, based
+on S8-S10 in Table III, since not knowing what to include in
+an SBOM is the second biggest concern for producing SBOMs
+according to the SBOM readiness report [5].
+First, despite the existence and relative prevalence of two
+major SBOM standards (i.e., SPDX and CycloneDX), some
+organizations generate SBOMs based on their customized non-
+standard formats [S8, 27.7% agree, 32.3% disagree]. Based
+on the survey results, although more respondents agree with
+standardized SBOM formats, over one quarter agree with
+customized SBOM generation. As I4 stated, “the people I
+spoke to (in some organizations) might have been doing this
+(generating SBOMs), but they might not have been doing
+a standard-format SBOM. They would just be keeping an
+inventory of all their software components.” (I4-Dev.&Sec.)
+Second, although there are 7 minimum SBOM data fields
+recommended by NTIA [8], in practice, generated SBOMs
+don’t always meet the minimum bar [S9, 63.1% agree, 24.6%
+disagree] for two main reasons (i.e., software vendor cus-
+tomization, data availability). Some software vendors choose
+only to include a subset of the minimum data fields, or
+customize their minimum requirements to meet their respective
+needs. According to I3, his/her organization “has its own
+minimum requirements (different from NTIA’s)” (I3, Dev.).
+Software vendors sometimes do not include all the minimum
+data fields as the relevant data is not always attainable. “The
+truth is, I tried to put in as much as what you said (7
+minimum data fields). I don’t have access all the time to all
+the details.” (I11-Dev.&Sec.(SBOM tool)) As a result, when
+relevant information of certain data fields is unavailable, the
+generated SBOMs can be “full of non-assertion elements... In
+practice, this means that, yes, the standards themselves can
+support the NTIA recommendation; No, the tools are omitting
+those fields or leaving them blank.” (I14-Dev.&Sec.&Res.)
+Third, for some organizations producing SBOMs or building
+SBOM tools to produce SBOMs, they want to include/sup-
+port as much “useful” information in SBOMs [S10, 87.7%
+agree, 4.6% disagree]. For the former, since an SBOM can
+effectively help with internal software supply chain manage-
+ment, they prefer to generate more comprehensive SBOMs
+with additional information such as vulnerability. As I13
+mentioned, “we want to produce the best information... then
+the developers can look at it and then do their work” (I13-
+Dev.&Sec.&Res.) For the latter, the more comprehensive in-
+formation their SBOM tools support, the more competitive
+they are in the market. As stated by I11, “I also have a
+big scope of metadata depending on the target other than
+the base.” (I11-Dev.&Sec.(SBOM tool)) Furthermore, as I14
+pointed out, “something relatively worse happens... to create
+business value... they are trying to extend it (an SBOM) with
+things that may or may not be relevant to the problem.” (I14-
+Dev.&Sec.&Res.)
+Finding 4: Despite official recommendations on mini-
+mum SBOM data fields, there is still a lack of consen-
+sus on what to include in SBOMs.
+5) SBOM distribution: In this subsection we discussed 3
+statements (i.e., S11-S13 in Table III).
+A considerable portion of the respondents agree that “or-
+ganizations generate SBOMs for internal consumption, rather
+than giving them to customers” (I11-Dev.&Sec.(SBOM tool)
+[S11, 40% agree, 38.5% disagree]. The authors notice a lack of
+consensus on this statement from the survey respondents. We
+believe this is consistent with the general “lack of consensus”
+status of the SBOM as in the SBOM readiness report [5],
+resulting from the relative recentness and lack of adoption.
+However, since SBOMs are now being distributed in prac-
+tice, proper distribution mechanisms are needed. According to
+
+the SBOM readiness report [5], one of the leading concerns for
+SBOM production is that some information inside an SBOM
+is too sensitive and risky to be public. Since the source code
+is already publicly available for OSS, their SBOMs should be
+public. For proprietary software/components, although some
+of the SBOMs can be public, “authenticated (access control)
+is going to be the norm” (I4-Dev.&Sec) [S12, 76.9% agree,
+13.8% disagree], depending on the software vendors’ policies.
+“At least for some segments of the market, I think access
+management is part of it. You need to be able to share your
+SBOMs with whomever you want to share, and not have all
+of the world get access to it.” (I12-Cnslt.&Adv.)
+Apart from access control, content tailoring (selective shar-
+ing) is also helpful for mitigating the above concern. There can
+be a negotiated compromise between the software vendor and
+its downstream procurers on what to include in the distributed
+SBOMs, instead of sharing the complete SBOMs [S13, 60%
+agree, 24.6% disagree]. “There’s got to be a mechanism... like
+a router in the middle, that takes the SBOMs or VEXs produced
+by the suppliers and routes them down to each end user, exactly
+what they need.” (I0-Cnslt.&Adv.), so that “only the right
+people can see the right information” (I13–Dev.&Sec.&Res.).
+Finding 5: Proprietary and sensitive information in
+SBOMs introduces barriers to SBOM distribution. Se-
+lective sharing (content tailoring) and access control
+mechanisms need to be considered.
+6) SBOM validation: This subsection is based on S14 and
+S15 (see Table III).
+The lack of SBOM integrity validation is a shared problem
+mentioned by 13 out of 17 interviewees. Without reliable vali-
+dation measures [S15, 49.3% agree, 26.2% disagree], software
+procurers may only roughly assess the quality of an SBOM
+by referring to the SBOM producer’s reputation [S14, 50.8%
+agree, 13.8% disagree]. SBOM integrity is two-fold: a) SBOM
+data integrity (whether the SBOM has been tampered with),
+and b) SBOM tooling integrity (tooling capability as to the
+competence to generate complete and accurate SBOMs; and
+tooling security as to whether the SBOM generation tools are
+hacked). We discuss SBOM tooling integrity in Section IV-B4.
+SBOM data tampering can come from outside and inside
+an organization (i.e., external/internal tampering). External
+tampering is more straightforward as an SBOM can be “easy
+to tamper (with) and easy to fake” (I14-Dev.&Sec.&Res.)
+without reliable validation methods. Thus, proper validation
+mechanisms are needed (e.g., signing using sigstore’s Cosign).
+Inside tampering means a software vendor may change the
+SBOM data considering customer acceptance and security
+issues. For example, I14 and I15 mentioned instances of
+internal tampering based on their experiences:
+a) “You could have a release engineer at the last minute,
+realizing that they wanted to change the SBOM just because
+otherwise, the customer wouldn’t take it. It’s not that the whole
+organization lied. But it does mean that they got to tamper with
+the SBOM that doesn’t again faithfully represent the product
+that they weren’t given.” (I14-Dev.&Sec.&Res.)
+b) “Whenever we are going to use an open source project,
+there has to be a security check... if some kind of (vulnerable)
+code is there, we just need to remove it... we are not actually
+passing those kinds of changes to the public.” (I15-Dev.)
+Finding 6: Trust in SBOM data needs to be assured
+considering tampering threats. SBOM data valida-
+tion/verification mechanisms and integrity services
+are needed.
+7) Vulnerability and exploitability:
+In this section we
+discuss SBOMs for vulnerability management and the ex-
+ploitability of vulnerabilities (i.e., S16 in Table III).
+Although vulnerability management is a representative
+SBOM use case [21], vulnerability management currently
+barely considers the actual exploitability [S16, 73.8% agree,
+13.8% disagree]. Nevertheless, the exploitability of a vulner-
+ability should be taken seriously, as a vulnerability may not
+necessarily be exploitable [22].
+“I think it’s a very, very interesting point, and it’s a very
+legit issue. Vulnerability and exploitability are totally different.
+You can’t simply send a long report to the developers and ask
+them to update each and every vulnerable dependency that
+has been flagged. I know the development team might end up
+ignoring your report, or come back at you saying, ‘are you
+able to exploit this vulnerability? No? Then why should I go
+and update it if you are unable to exploit it?’” (I9-Sec.)
+As mitigation, vulnerabilities are often selectively fixed
+based on the criticality (e.g., The Common Vulnerability Scor-
+ing System (CVSS) score). As stated by I9, “if it’s a critical
+or a high vulnerability... then make sure it is updated. But
+when it comes to (a) medium or low (criticality vulnerability),
+then ignore it.” (I9-Sec.) Although there are efforts towards
+exploitability, such as CISA’s Known Exploited Vulnerabilities
+Catalog that serves as a “must patch list”, there are only
+limited records (around 800 as of August 2022) in this catalog.
+Vulnerability Exploitability eXchange (VEX) has emerged
+as a tailored method to cope with such problems. � A VEX
+is a security advisory produced by a software vendor that
+allows the assertions of the vulnerability status of a software
+product [23]. As companion artifacts to SBOMs [24], VEXs
+provide SBOM operators with clearer understanding of the
+vulnerabilities and suggested remediation. � However, cur-
+rent exploitability evaluation is manual and subjective to the
+domain knowledge of the security experts [25], [26]. Also,
+“the ability to differentiate between whether it’s exploitable or
+not is a hard thing to do itself” (I11-Dev.&Sec.(SBOM tool)),
+“especially if you want to automate it”. (I13–Dev.&Sec.&Res.)
+What is more, “there is no way to confirm this (VEX), and it’s
+actually very, very hard to prove a negative. So if you see a
+VEX entry that says this (vulnerability) doesn’t hit me, the only
+
+way to prove it wrong is to make an exploit yourself. Again,
+if you are able to do that, then you’re almost making things
+worse, right? (I14-Dev.&Sec.& Res.)”
+� To further complicate this problem, there is hardly guar-
+anteed unexploitability. As I11, an SBOM tool developer
+with security (hacker) experience, stated, “I agree the more
+valuable these exploitable vulnerabilities are, but I don’t agree
+that the ones defined less exploitable are not valuable. I used
+to be on the attacker’s side... Hackers can take their time,
+and they can find a way to put together a lot of things
+that look very not exploitable, and at the end of the day,
+find themselves with very easy and exploitable access.” (I11-
+Dev.&Sec.(SBOM tool)) A possible solution is there should be
+“potential exploitability” (I14-Dev.&Sec.& Res.).
+Finding 7: It is unclear what to do with vulnerabilities
+with limited exploitability exposed by SBOMs/VEXs.
+8) AIBOM: This subsection is based on S16 (see Table III).
+AI software is software with AI components. Compared
+with SBOMs for traditional software, SBOMs for AI software
+(i.e., AIBOMs) are different [S16, 47.7% agree, 24.6% dis-
+agree]. Although some interviewees thought an AIBOM “con-
+tains only additional AI package information” (I1-Dev.), the
+AI artifacts (e.g., data, code, model, configuration) also need
+provenance and co-versioning [27], [28]. An AIBOM (see
+Fig. 1) records not only the software composition information
+as a traditional SBOM, but also contains information about
+the data/model/code/configuration co-versioning registries, al-
+lowing transparency and accountability into the AI artifacts
+for AI model training and evaluation. Considering the AI
+software deployment is continuous progress (e.g., continuous
+training in case of data/concept drift), these AI artifacts’ co-
+versioning registries are more dynamic and subject to change,
+while the component inventory information is relatively static.
+To reduce frequent re-generation of the AIBOM, the co-
+versioning registries can be independent of the AIBOMs,
+instead of being embedded in the AIBOMs.
+B. RQ2: What is the current state of SBOM tooling support?
+This section discusses 6 statements (see T2 in Table III).
+Although some practitioners argue that SBOM tools were not
+necessary, the importance and necessity of SBOM tools are
+recognized by most participants. However, the existing tools
+still lack maturity in general and require further development.
+1) Necessity of SBOM tools: This subsection is based on
+S18 in Table III.
+The most interesting argument about SBOM tooling is the
+necessity of using SBOM tools to generate SBOMs. Since
+currently few organizations (e.g., the US government agencies)
+are actually requiring SBOMs to be provided upon software
+delivery, � some interviewees think they do not have to
+generate SBOMs, especially when most SBOM tools serve
+like a “proxy”: they merely feed the existing metadata (e.g.,
+package manager) into a standard format, but the data is
+already there with or without SBOMs.
+“We do not use SBOMs internally... because we have tools,
+which is where the SBOM information is coming anyway:
+package manager. Those tools generally just take a look at
+the files in the system and the configuration of the system,
+whereas SBOM tools just essentially parse those and then try
+to use that in a format. So, if we don’t have a lot of end users...
+why would we introduce this (SBOM)? It’s like a middleman
+that really doesn’t produce much.” (I14-Dev.&Sec.& Res.)
+� That being said, most survey respondents think generat-
+ing SBOMs using SBOM tools is necessary [S18, 83.1%
+agree, 10.8% disagree], which is consistent with the benefit
+of standardization and unification of the software composition
+data enabled by SBOMs discussed in Section IV-A1.
+2) Availability of SBOM tools:
+In this subsection, we
+discussed statements S19-S20 in Table III.
+Tooling is an integral part of SBOM, as SBOMs are not
+manually generated nor intended for direct human consump-
+tion. The generation and consumption of SBOMs rely on
+SBOM tools. “Shift left” originates from DevSecOps [29],
+which means shifting the security work to earlier stages of
+the software development life cycle so that security issues can
+be identified and fixed earlier. Considering there is a “lack of
+more developer-oriented (SBOM) tools that are more familiar
+by developers” (I7-Sec.), and SBOMs are tightly coupled with
+security tasks, SBOM tools should also consider “shift left”.
+Despite there is a lot of existing SBOM tools as mentioned
+in Section II, contrary to the finding in the SBOM readiness
+report [5] that “SBOM consumption mirrors SBOM produc-
+tion”, our finding shows that currently, the “SBOM genera-
+tion is ahead of SBOM consumption” (I17-Cnslt.&Adv.), and
+there are significantly limited tools for SBOM consumption
+[S19, 75.4% agree, 12.3% disagree]. As stated by I17, “the
+large bucket of what we don’t have today in 2022 is SBOM
+consumption.” (I17-Cnslt.&Adv.) Without SBOM consumption
+tools, even if an SBOM was provided to a software procurer,
+the procurer would wonder, “what do I do with the SBOMs?
+How do I process them? How do I analyze them?” (I12-
+Cnslt.&Adv.) Besides dedicated SBOM consumption tools, a
+possible solution is to feed SBOMs into existing IT asset
+management tools [S20, 41.5% agree, 12.3% disagree], which
+requires functional extensions.
+3) Usability of SBOM tools: This section discusses S21-
+S22 in Table III.
+Although the SBOM tools market is proliferating with the
+expectation to “explode” in 2022 and 2023 [5], the usability
+of existing tools remains an issue. SBOM tools can be hard to
+use due to various reasons. (e.g., complexity, aggressivity, lack
+of generalization) [S21, 64.6% agree, 18.5% disagree]. For
+example, “to use the CycloneDX Maven plugin, it’s required
+to import this plugin in the POM.xml file. For open source
+software, it is all right. But for proprietary software, this
+introduces invasion, which can be a problem”. (I3-Dev.)
+Although four interviewees (i.e., I9-I12) mentioned that
+there were user-friendly tools such as Dependency-track, the
+
+interviewees also acknowledged that most SBOM tools were
+open source and not enterprise-ready. A problem with open
+source tools is, “an organization needs to have the capability
+of knowing open source projects, running them, fine-tuning
+them towards its needs, maintaining them” (I12-Cnslt.&Adv.),
+which can be a considerable problem for smaller-scale orga-
+nizations and start-ups.
+Tooling interoperability and standardization also hinder the
+usability of SBOM tools [S22, 73.8% agree, 10.8% disagree].
+As mentioned by I17, SBOM tooling is also “an area where
+we need further harmonization and standardization.” (I17-
+Cnslt.& Adv.) For instance, the SBOM data (e.g., component
+hash) of the same software/components generated by different
+tools can be different, while “the whole point of a hash is
+that it should be the same (for the same component), so
+that...downstream users can validate it”. (I17-Cnslt.& Adv.)
+4) Integrity of SBOM tools: In this subsection, we discuss
+SBOM tooling integrity based on S23 in Table III.
+As mentioned in Section IV-A6, SBOM integrity consists
+of SBOM data integrity and SBOM tooling integrity. SBOM
+tooling integrity also includes two aspects: a) tooling compe-
+tence: the completeness and accuracy of the accuracy caused
+by SBOM tooling capability; and b) tooling security: whether
+the SBOM generation toolchain has been maliciously altered.
+Most respondents agree that the integrity of SBOMs gen-
+erated by existing tools cannot be validated [S23, 69.2%
+agree, 18.5% disagree]. The accuracy and completeness of
+the generated SBOMs caused by tooling competence is a
+common concern for generating SBOMs. To the best of our
+knowledge, there is no comprehensive measure or validation
+against such unintentional mistakes from the end users’ point
+of view. However, the intentional tampering resulting from
+compromised toolchains is another story. One possible solution
+is to evaluate the SBOM tools’ assurance based on Automated
+Rapid Certification Of Software (ARCOS) [30] though it is
+still a work in progress.
+Finding 8: There is a lack of maturity in SBOM tool-
+ing. More reliable, user-friendly, standard-conformable,
+and interoperable enterprise-level SBOM tools, espe-
+cially SBOM consumption tools, are needed.
+C. RQ3: What are the main concerns for SBOM?
+This section investigates practitioners’ main concerns for
+SBOMs based on T3 in Table III. During the interviews,
+SBOM tool developers’ common concern lay with the SBOM
+standard formats. Most respondents remained concerned about
+SBOMs being “roadmaps for attackers” [31]. The most fun-
+damental issue is the lack of SBOM adoption and education.
+1) SBOM formats’ lack of extensibility: In this subsection
+we discussed S24 in Table III.
+There are mainly two competing SBOM standard formats
+(i.e., SPDX and CycloneDX), and neither can fully meet
+current market needs [S24, 35.4% agree, 33.8% disagree].
+Notably, this statement was mentioned by both interviewees
+(i.e., I11, I16) working on SBOM tool development. Although
+the respondents show a discrepancy and lack of consensus on
+this statement, it is consistent with the interview results.
+Interestingly, I11 considered the SBOM format standardiza-
+tion to be one of the most significant benefits, while agreed
+the existing standards remain to be developed. � On the one
+hand, “the big advantage (of SBOMs) is standardization.
+The formats allow a lot of people to understand the same
+language” (I11-Dev.&Sec.(SBOM tool)). It offers “a unified
+framework to communicate software composition information”
+(I14-Dev.&Sec.& Res.). � On the other hand, some think the
+formats are not extensible enough. For example, ��current for-
+mats only support one dependency relationship, DependsOn”
+(I11-Dev.&Sec.(SBOM tool)). We summarized possible format
+extension points in Section V-A2 based on the interviews.
+a) “My biggest concern is the dynamicity and the ability to
+use the standard formats of SBOMs... to define the things I
+want to do with these SBOMs.” (I11-Dev.&Sec.(SBOM tool))
+b) “My biggest concern... is that the standardization is really
+kind of not good... competing standards... different properties,
+and different kinds of purposes. But consolidating down until
+reasonable sets of things are the same between the competing
+formats would be great.” (I16-Dev.(SBOM tool))
+Finding 9: Although there is a set of standard formats,
+they require further consensus, standardization as
+well as additional extension points.
+2) SBOM information sensitivity: In this section, we dis-
+cussed S25 in Table III.
+There are two types of opinions among the participants
+about the SBOM information sensitivity issue: � Some think
+certain information inside the SBOM is too risky and sensitive
+to be public, and the information inside an SBOM may serve
+the attackers as a ”roadmap” of the software and supply
+chain [S25, 63.1% agree, 20% disagree]. � Meanwhile, others
+believe that there is no need to worry as the attackers do not
+need SBOMs because they already have tools to easily get
+the software composition information. SBOMs are actually
+roadmaps for defenders to help level the playing field [31].
+Just as I4 stated, “on the surface, it seems like a valid concern.
+But... if you’re highly sophisticated cooperation, you probably
+don’t necessarily need this information. As a start-up, you
+don’t get to be a target in an attack... I think the benefit of that
+visibility is certainly on the defender’s side.” (I4-Dev.&Sec.)
+In addition, access control and content tailoring (see Section
+IV-A5) can also play a role in mitigating this concern.
+3) SBOM adoption and education deficiency: In this sec-
+tion, we discussed S26 in Table III.
+The most fundamental and imminent concern is limited
+SBOM adoption [S26, 80% agree, 7.7% disagree]. Organiza-
+tions may have various reasons for their hesitation to SBOM
+adoption. For example, they may worry about the industrial
+
+consensus or SBOMs’ value to customers, or they may lack
+even the most basic IT asset management.
+a) “I’m actually a bit worried about the people producing and
+consuming SBOMs because I think the market is not really
+ready. They need to (be) educate(d), and many people don’t
+know what SBOMs are.” (I11-Dev.&Sec.(SBOM tool))
+b) “For SBOMs to become valuable to consume, many people
+need to produce (SBOMs)... So one of the concerns I have
+about SBOMs is, is everybody going to follow this pattern (to
+produce SBOMs)? Because there’re obviously people (saying)
+it’s not accurate so they don’t want to produce it.” (I6-Dev.)
+However, despite being concerned about SBOM adoption,
+I6 also agreed that “even the less accurate ones are better
+than adding no visibility” (I6-Dev.). Because “there’s always
+going to be a maturity problem. The idea that because some
+people can’t use SBOM so others shouldn’t... That’s not how
+we do security.” (I17-Cnslt.&Adv.)
+SBOM education is needed not only to educate the public
+for increased SBOM adoption, but the SBOM practitioners
+also need to be educated and realize SBOMs have unaddressed
+issues before they rush into generating SBOMs. “The problem
+right now is that we are almost putting the cart before the
+horse - we’re expecting the SBOMs to fix the problems rather
+than fixing the problems with an SBOM.” (I14-Dev.&Sec.)
+Finding 10: There is a lack of market awareness and
+good value propositions for SBOM adoption. SBOM
+advocators need to: a) leverage relevant regulation
+and use cases such as procurement evaluation and
+supply chain risk management to improve SBOM
+awareness; and b) promote more SBOM consump-
+tion tools with clear benefits.
+V. DISCUSSION AND IMPLICATIONS
+A. Implications
+This section discusses below key implications for future
+SBOM research and development.
+1) Goal model: Based on the study results, we present
+a goal model for future SBOM endeavors (see Fig. 3). As
+mentioned in Sections IV-A2 and IV-C3, the lack of SBOM
+adoption causes a substantial obstacle to SBOM progress.
+To achieve increased SBOM adoption and more SBOM-
+enabled benefits, there are three goals to be satisfied:
+a) Higher-quality SBOM generation (findings 3, 4, 6,
+8, and 9): maturer tooling support for the generation of
+more standardized tamper-proof “dynamic” SBOMs [32]. For
+instance, further standardization on SBOM-included data fields
+is needed. SBOM industry should strictly conform to an agreed
+minimum data fields, while considering different industry and
+business sectors when adding optional data fields.
+b) Clearer benefits and use cases for SBOM consumption
+(findings 1, 2, 10): SBOM education (e.g., on SBOM-enabled
+benefits) results in increased SBOM adoption (including con-
+sumption); Increased SBOM adoption, in turn, leads to more
+developed SBOM-centric ecosystems with favorable use cases.
+c) Lower barriers in SBOM sharing and distribution
+(findings 5 and 7): The distribution and sharing of SBOMs and
+the vulnerability status (e.g., VEXs) need to be more flexible
+with proper mechanisms that meet both the software ven-
+dors’ and procurers’ needs. Technologies such as Blockchain,
+confidential computing (e.g., zero-knowledge proofs, secure
+multiparty computation for sharing without access) can poten-
+tially be leveraged to communicate SBOM data. During the
+distribution of SBOM data, there also need to be risk-based
+flexible policies to communicate unfixed vulnerabilities.
+Fig. 3. SBOM goal model.
+2) Format extension points: As discussed in Section IV-C,
+the interviewees mentioned several potential extensions to
+the existing SBOM formats. Other than supporting more de-
+pendency relationship types, another possible extension point
+is component types. “For example, the components of a
+Git is the commits, and CycloneDX does not have such a
+component (type) defined.” (I11-Dev.&Sec.(SBOM tool)) The
+third extension point is file location. “My organization has its
+own SBOM format... which has... things like where does it find
+something in a file system... However, formats like SPDX might
+not have a place for that.” (I16-Dev.(SBOM tool)) Apart from
+the three points mentioned by interviewees, another essential
+point is verifiable credentials [27] embedded in or linked to an
+SBOM. For traditional software, such credentials can prove the
+validity of software/components. For AI software, responsible
+AI-related information such as conformance to certain AI
+ethics principles can be included.
+B. Positioning with respect to SBOM readiness report
+This section compares the key differences between our
+findings and the SBOM readiness report’s results. This study
+broadens/deepens the SBOM readiness report mainly from 9
+SBOM aspects: benefits, adoption, generation, distribution, in-
+tegrity, vulnerability management (with SBOMs/VEXs), tool-
+ing, concerns, and AIBOM (See Table IV).
+C. Threats to validity
+The number of survey participants may pose a threat to
+validity. However, considering the relative novelty of SBOM
+and the lack of SBOM adoption, we believe this threat can be
+justified. It is possible that some of the survey respondents
+
+Increased SBOM adoption and more
+SBOM-enabled benefits
+Higher-quality
+Clearer benefits & use cases
+Lower barriers in SBOM
+SBOM generation
+for SBOM consumption
+sharing & distribution
+10
+Tooling
+Dynamic
+Validation/
+Consumption
+SBOM
+Adoption
+(Vulnerability)
+Standardization
+verification
+-driven design incentives promotion
+Distribution
+generation
+maturityTABLE IV
+SBOM READINESS REPORT V.S. THIS PAPER
+Topics
+SBOM readiness report
+This paper
+Benefits
+16 specific benefits of SBOMs (10 for producing and 6 for
+consuming SBOMs), all enabled by transparency.
+Transparency, and subsequently enabled accountability, traceability, and
+security. (Finding 1)
+Adoption
+90% surveyed organizations have started SBOM journey;
+47% are using (i.e., producing/consuming) SBOMs.
+SBOM adoption is worrying: limited generation & more limited consumption.
+(Findings 1, 2, 10)
+Generation
+a) SBOMs can be generated at different SDLC stages.
+b) More organizations favor including more than baseline
+SBOM information.
+a) SBOMs can be generated at different SDLC stages but practitioners expect
+“dynamic” SBOM generation throughout SDLC (Finding 3).
+b) SBOM-included data fields need further standardization (Finding 4).
+Distribution
+N/A
+Secure yet flexible SBOM distribution mechanisms are needed. (Finding 5)
+Integrity
+N/A
+SBOM integrity assurances are needed against tampering threats. (Finding 6)
+Vulnerability
+SBOMs should reflect vulnerability information.
+a) Organizations may not want to share sensitive (vulnerability) data
+(Finding 5).
+b) Mechanisms are needed to communicate vulnerabilities with limited/
+undetermined exploitability (Finding 7).
+Tooling
+Limited availability of SBOM tooling
+Affirmed necessity but limited availability, usability, and integrity of SBOM
+tooling. (Finding 8)
+Concerns
+4 shared concerns for production & consumption: industry
+commitment, data fields consensus, value of SBOMs,
+tooling availability. 2 additional concerns for production:
+information privacy, correctness.
+Explicitly identifies 3 major concerns but covers more throughout
+a) Standard formats’ lack of extensibility (Finding 9).
+b) SBOM information sensitivity & privacy (Finding 5).
+c) Adoption and education deficiency (Finding 10).
+AIBOM
+N/A
+AIBOM should also include AI/ML-specific data. (Section IV-A8)
+do not have a comprehensive understanding on SBOMs,
+which may introduce noise to the collected data. To mitigate
+this threat, we let the respondents choose whether they are
+familiar with SBOMs. If their answer is no, the survey will
+automatically end. Still, we cannot fully ascertain whether the
+collected responses are accurate reflections of their beliefs. It
+is a common and tolerable threat to validity in many existing
+similar empirical studies, which assume that the majority of
+responses truly reflect what respondents truly believe.
+VI. RELATED WORK
+On the one hand, there has been work on SSC security.
+For instance, Ohm et al. [4] summarized 174 OSS packages
+used for real-world malicious SSC attacks. Blockchain has
+been applied to SSC security (e.g., [33]–[35]. Especially,
+Marjanovi´c et al. [36] used blockchain-based techniques to
+record the software composition details. On the other hand,
+there are limited scholarly papers on SBOMs. Martin et al. [21]
+introduced the concept of the SBOM and listed nine possible
+use scenarios. In 2021, Carmody et al. [37] presented a high-
+level overview of how SBOMs help build resilient medical
+SSCs. They illustrated the benefits of SBOMs for software pro-
+ducers, consumers and regulators, as well as relevant progress
+on SBOMs. Back in 2019, Barclay et al. [38] introduced their
+ideas on applying BOMs to data ecosystems for transparency
+and traceability, where they detailed a conceptual model to
+combine a BOM (static) and a bill of lots (dynamic) to jointly
+record the static data components and the dynamic data of
+a specific experiment. Based on their previous work, as a
+step towards operationalizing the conceptual model, Barclay
+et al. [39] recently introduced their work using a BOM as a
+verifiable credential for transparency into the AI SSCs, which
+is a step towards AIBOM.
+VII. CONCLUSION AND FUTURE WORK
+SBOMs are essential to SSC security considering the trans-
+parency enabled by SBOMs and the subsequently enhanced
+accountability, traceability and security. In this study, we
+interviewed 17 and surveyed 65 SBOM practitioners on their
+perception of SBOM. Despite the promising SSC transparency
+and security enabled by SBOMs, there are still open challenges
+to be addressed. To accelerate the adoption of SBOMs, higher-
+quality SBOM generation, clearer benefits and use cases in
+SBOM consumption, and lower barriers in SBOM sharing are
+prerequisites which need to be further studied, mitigated and
+addressed. In addition, SBOMs for AI software (i.e., AIBOM)
+are an inevitable trend given the popularity of AI and AI
+software, and AIBOMs need to consider the co-evolution of
+data/model/code/configuration.
+ACKNOWLEDGMENT
+The authors would like to thank all the interview and survey
+participants for their great help and support. This work could
+never have been accomplished without them.
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+

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+The Numerical Flow Iteration for the
+Vlasov–Poisson Equation
+Matthias Kirchhart∗
+R. Paul Wilhelm∗
+Abstract
+We present the numerical ow iteration (NuFI) for solving the Vlasov–Poisson equation. In a certain
+sense specied later herein, NuFI provides infnite resolution of the distribution function. NuFI exactly
+preserves positivity, all Lp-norms, charge, and entropy. Numerical experiments show no energy drift.
+NuFI is fast, requires several orders of magnitude less memory than conventional approaches, and
+can very eciently be parallelised on GPU clusters. Low delity simulations provide good qualitative
+results for extended periods of time and can be computed on low-cost workstations.
+1 Introduction
+1.1 The Vlasov–Poisson Equation
+Our problem of interest is a simplied model for the evolution of plasmas in their collisionless limit, as they
+occur in, for example, nuclear fusion devices. In dimensionless form it is given by the following system:
+∂tf + v · ∇xf − E · ∇vf = 0,
+(1)
+E := −∇xφ,
+(2)
+−∆xφ = ρ,
+(3)
+ρ(t, x) := ¯ρ −
+
+Rd f(t, x, v) dv▷
+(4)
+Here, f = f(t, x, v) is the electron distribution function, i. e., f(t, x, v) ≥ 0 describes the probability density
+of electrons having velocity v ∈ Rd and location x ∈ Rd at time t ∈ R. The Vlasov–Poisson equation for
+Plasmas is typically investigated for d ∈ {1, 2, 3}. In this work we will assume periodic boundary conditions
+in x, i. e., Ω := [0, L]d with L > 0, such that for all Cartesian basis vectors ei ∈ Rd and all k ∈ Z one has
+f(t, x + Lkei, v) = f(t, x, v).
+To obtain a well-posed problem, non-negative initial data of f needs to be supplied, usually at time
+t = 0:
+f(t = 0, x, v) = f0(x, v) for a given x-periodic f0 ∈ L1(Ω × Rd) ∩ L∞(Rd × Rd), f0 ≥ 0 a. e.
+(5)
+In many computational benchmarks f0 is a smooth function which is given as a relatively simple expression.
+Usually f0 decays exponentially as |v| → ∞, so in computational practice one pretends that one has
+f(t, x, v) = 0 whenever |v| ≥ vmax for some user-dened parameter vmax.
+Equation (4) denes the charge density ρ; the parameter ¯ρ stems from the assumption of a uniform ion-
+background and needs to be chosen such that overall neutrality is preserved, i. e., ∀t ≥ 0 :
+
+Ω ρ(t, x) dx = 0.
+Neglecting collisions and the magnetic eld, the Vlasov equation (1) then describes the evolution of f
+under the inuence of the self-consistent electric eld E = E(t, x), given in terms of the electric potential
+φ = φ(t, x), which in turn is given as the solution of the Poisson equation (3).
+∗Applied and Computational Mathematics, RWTH Aachen University, Schinkelstraße 2, 52062 Aachen, Germany.
+E-Mail: [email protected] and [email protected]
+1
+
+0
+π
+2
+π
+3π
+2
+2π
+5π
+2
+3π
+7π
+2
+4π
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+x
+fτ,h(30, x, 0)
+t = 30
+0
+π
+2
+π
+3π
+2
+2π
+5π
+2
+3π
+7π
+2
+4π
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+x
+fτ,h(100, x, 0)
+t = 100
+Figure 1: Cross-sections at v = 0 of approximations to distribution function f(t, x, v) for the two stream
+instability benchmark in d = 1, see Section 4.2, at times t = 30 and t = 100. Computed using
+the numerical ow iteration as described below, using Nx = 64 and Nv = 256. The emergence
+of laments which grow ever ner with time is clearly visible. Due to their nite resolution, all
+mesh-based approaches which directly discretise f will eventually fail to reproduce these laments
+accurately.
+1.2 Challenges in Numerical Algorithms for the Vlasov–Poisson Equation
+Supercially, numerically solving the Vlasov–Poisson equation might seem simple: assuming for the moment
+the electric eld E was known, the Vlasov equation (1) is a linear advection equation that can be solved
+using classical numerical schemes, e. g., nite dierences with upstream discretisation. From such an
+approximation, the charge density ρ could be computed directly; solving the Poisson equation (3) also is
+a classical problem for which there are many ecient algorithms available. The resulting approximation
+of the electric potential can then be fed back into the Vlasov solver to proceed by one time-step into the
+future.
+While possible in theory, such a simple approach faces many problems in practice. Grid- and particle-
+based methods that directly discretise f face the curse of dimensionality: the 2d-dimensional (x, v)-space
+results in extremely large memory requirements. For such schemes, simulating the three-dimensional case
+(d = 3) usually is – if at all – only feasible on very large high-performance computers. At the same time,
+such schemes typically have very low arithmetic intensity, i. e., a low FLOP/byte count. Modern computer
+architectures struggle to deliver good performance for such schemes.
+Even if these issues could be overcome, we want to particularly emphasise the diculty due to so-called
+laments, i. e., very ne details that develop in the solution over time. This is illustrated in Figure 1, which
+shows cross-sections of a numerical solution to a well-known benchmark problem at two dierent times t.
+The increasing lamentation in the solution manifests itself as oscillations with frequencies that rapidly
+increase over time t. As the laments become too small for a given resolution, conventional discretisations
+often begin to violate several of the conservation laws discussed in Section 2.5. In this sense, these schemes
+then give unphysical results. For example, avoiding overshoots and negative values of f near laments is
+very dicult for many high-order schemes.
+1.3 Overview
+The rest of this article is structured as follows. Section 2 introduces the numerical ow iteration and
+discusses mathematical and practical issues. In Section 3 we elaborate on relationships with other methods
+found in the literature. Numerical experiments are discussed in Section 4. We conclude with an outlook to
+possible future extensions in Section 5.
+2 The Numerical Flow Iteration
+To understand NuFI, we will rst need to introduce the exact ow associated with the Vlasov–Poisson
+equation. We then proceed by introducing the components of NuFI and also discuss several important
+conservation properties.
+2
+
+2.1 The Exact Flow and Solution to the Vlasov–Poisson Equation
+Assume we knew the velocity v ∈ Rd and position x ∈ Rd of some imaginary ‘particle’ at time s ∈ R. If we
+furthermore assume that the electric eld E was known for all times t, we can trace the state
+
+ˆx(t), ˆv(t)
+
+of
+that imaginary particle both forward (t > s) and backward (t < s) in time by solving the following initial
+value problem:
+d
+dt ˆx(t) = ˆv(t),
+ˆx(s) = x,
+d
+dt ˆv(t) = −E
+
+t, ˆx(t)
+
+,
+ˆv(s) = v▷
+(6)
+Equation (6) is the denition of the characteristic lines of the Vlasov equation (1). Assuming E is suciently
+smooth, as a consequence of the Picard–Lindel¨of theorem, it is well-known that this system has a unique
+solution for any (s, x, v) ∈ R × Rd × Rd. This leads us to the following denition:
+Denition 2.1 (Flow). The ow Φt
+s(x, v) of the Vlasov–Poisson equation is dened as the unique solution
+of the initial value problem (6):
+Φt
+s(x, v) :=
+
+ˆx(t), ˆv(t)
+
+▷
+We will sometimes use the term exact ow to distinguish Φt
+s from approximative, numerical ows.
+In other words: Φt
+s(x, v) tells us the position and velocity at some time t of a particle with given state
+(x, v) at time s. Note that system (6) is symplectic, and the following lemma is a well-known consequence
+that will be of great importance later on.
+Lemma 2.2 (Volume Preservation of the Exact Flow). The ow Φt
+s : Rd × Rd → Rd × Rd is a
+dieomorphism that preserves volume in phase-space:
+∀s, t ∈ R : ∀(x, v) ∈ Rd × Rd :
+det ∇(x,v)Φt
+s(x, v) ≡ 1▷
+With help of the ow Φt
+s, the exact solution to the Vlasov–Poisson system takes a very simple form.
+Lemma 2.3 (Solution Formula). Let the non-negative distribution function f be given at initial time
+t = 0, i. e., assume we were given f0(x, v) ≥ 0 such that f(0, x, v) = f0(x, v) for all (x, v) ∈ Rd × Rd. Then
+the (exact) solution to the Vlasov–Poisson equation is given by:
+∀(t, x, v) ∈ R × Rd × Rd :
+f(t, x, v) = f0
+
+Φ0
+t(x, v)
+
+▷
+2.2 The Numerical Flow and Solution to the Vlasov–Poisson Equation
+In NuFI, the exact numerical ow Φ is replaced by a numerical approximation Ψ. For the moment we
+will still assume the electric eld E was known for all times t. For simplicity of the exposition, we will
+restrict ourselves to the case Φ0
+t ≈ Ψ0
+nτ, n ∈ N0, that is, we only consider discrete times t = nτ ≥ 0 with a
+user-dened time-step τ > 0. However, extensions to arbitrary t are certainly possible.
+Due to the symplecticity of system (6), it is natural to approximate Φ0
+t using a symplectic one-step
+method for ordinary dierential equations. In this work, we will use the well-known St¨ormer–Verlet
+method.[1] Noting that we are going backwards in time, in mathematical notation we have:
+Ψ0
+nτ = Ψ0
+τ ◦ Ψτ
+2τ ◦ Ψ2τ
+3τ ◦ Ψ3τ
+4τ ◦ · · · ◦ Ψ(n−1)τ
+nτ
+,
+(7)
+where an individual step Ψ(k−1)τ
+kτ
+(x, v), k ∈ {1, ▷ ▷ ▷ , n}, is given as:
+vk− 1
+2 := v + τ
+2E(kτ, x),
+xk−1 := x − τvk− 1
+2 ,
+vk−1 := vk− 1
+2 + τ
+2E
+
+(k − 1)τ, xk−1
+
+,
+Ψ(k−1)τ
+kτ
+(x, v) := (xk−1, vk−1)▷
+(8)
+We believe that it is important to point out that Ψ0
+nτ can easily and eciently be implemented on
+computer hardware, at a cost of just one evaluation of E per time-step. In the following C++ snippet vec
+stands for a d-dimensional vector of oating point values.
+3
+
+void
+numerical_flow ( size_t n, double tau , vec &x, vec &v )
+{
+if ( n == 0 ) return;
+// Omit the
+following
+single
+line for
+Psi_tilda:
+v += (tau /2)*E(n*tau ,x);
+while ( --n )
+{
+x -= tau * v;
+// Inverse
+signs; we are
+v += tau * E(n*tau ,x); // going
+backwards in time!
+}
+x -= tau*v;
+v += (tau /2)*E(0,x);
+}
+This code also mentions the closely related function ˜Ψ0
+nτ, which is dened such that Ψ0
+nτ(x, v) = ˜Ψ0
+nτ
+
+x, v +
+τ
+2E(nτ, x)
+
+, which we will need later on.
+The St¨ormer–Verlet method is second order accurate in time: |Φ0
+nτ(x, v) − Ψ0
+nτ(x, v)| = O(τ 2). Moreover,
+this method is symmetric, i. e., running the method backwards in time is the same as running the forward
+method on the time-reversed Vlasov–Poisson equation. With the numerical ow in place, it is natural to
+dene approximations of the following shape:
+f(nτ, x, v) = f0
+
+Φ0
+nτ(x, v)
+
+≈ f0
+
+Ψ0
+nτ(x, v)
+
+=: fτ(nτ, x, v)▷
+(9)
+Several remarks are in order:
+• Both Ψ0
+nτ and ˜Ψ0
+nτ can be computed at arbitrary locations (x, v) ∈ Rd × Rd. The St¨ormer–Verlet
+method only discretises in time, there is no limit to the (x, v)-resolution of the numerical ow. It is in
+this sense when we say that in NuFI the distribution function f has innite resolution in phase-space.
+• It is not necessary to explicitly store f, only the initial data f0 and the electric eld E need to be
+accessible.
+• To evaluate Ψ0
+nτ, the electric eld E is only required at the discrete time steps iτ, i = 0, ▷ ▷ ▷ , n.
+• To evaluate ˜Ψ0
+nτ, the electric eld is only needed at previous times iτ, i = 0, ▷ ▷ ▷ , n − 1, i. e., the
+current E(nτ, ·) is not required.
+In practice, the exact electric eld E(t, x) will be replaced with a numerical approximation Eτ,h ≈ E;
+the meaning of h will be claried later. The resulting fully discrete numerical ow Ψ0
+nτ,h ≈ Ψ0
+nτ will then
+also be distinguished with an additional index h. However, regardless of errors in Eτ,h, just like the exact
+ow Φ0
+nτ, one of the remarkable key properties of symplectic integrators is that the numerical ows Ψ0
+nτ
+and respectively Ψ0
+nτ,h also preserve volume in phase-space. For completeness we repeat the simple proof.
+Lemma 2.4 (Volume Preservation of the Numerical Flow). Regardless of errors in the electric
+eld Eτ,h ≈ E, the numerical ows Ψ0
+nτ and Ψ0
+nτ,h of the St¨ormer–Verlet method are dieomorphisms
+which preserve volume in phase-space:
+∀n ∈ N0 : ∀(x, v) ∈ Rd × Rd :
+det ∇(x,v)Ψ0
+nτ(x, v) ≡ det ∇(x,v)Ψ0
+nτ,h(x, v) ≡ 1▷
+Proof. We only describe Ψ0
+nτ as the proof for Ψ0
+nτ,h is identical. Because of the chain rule and the properties
+of the determinant one has:
+det ∇(x,v)Ψ0
+nτ =
+n
+
+k=1
+det ∇(x,v)Ψ(k−1)τ
+kτ
+▷
+(10)
+Similarly, a single step Ψ(k−1)τ
+kτ
+is composed of the three sub-steps given in (8). The Jacobian matrix of,
+e. g., the rst sub-step is given by:
+∇(x,v)
+
+x
+v + τ
+2E(kτ, x)
+
+=
+
+I
+0
+τ
+2∇xE(kτ, x)
+I
+
+▷
+(11)
+4
+
+This is a triangular matrix with unit diagonal, so its determinant also is one. This is also the case for the
+other sub-steps, and thus det ∇(x,v)Ψ0
+nτ ≡ 1.
+2.3 Structure of the Numerical Flow Iteration
+So far we have assumed that the electric eld E was known exactly. In practice, however, this is of course
+not the case and it needs to be computed from the charge density ρ, which itself is dened in (4). Thus,
+only some approximation Eτ,h ≈ E will be available, where h denotes a discretisation parameter that will
+be described later.
+At initial time t = 0 the charge density ρ(t = 0, x) can be computed by integrating f0 along v. This can,
+for example, be done by using numerical quadrature giving an approximation ρτ,h(0, x) ≈ ρ(0, x). The
+approximate electric eld Eτ,h(0, x) then results from the solution of the Poisson equation (3), which can
+itself be discretised with some reasonable numerical method.
+From this point, we proceed iteratively in time. Thus, assume that we had already computed ρτ,h, φτ,h,
+and Eτ,h at times iτ, i = 0, ▷ ▷ ▷ , n − 1 for some n ∈ N. We seek to compute ρτ,h(nτ, x) using the numerical
+approximation f(nτ, x, v) ≈ f0
+
+Ψ0
+nτ,h(x, v)
+
+. However, Ψ0
+nτ,h cannot be evaluated, because Eτ,h(nτ, x) is
+still unknown. Luckily, this circular dependency can be resolved by using ˜Ψ0
+nτ,h and a simple change of
+variables:
+ρ(nτ, x) = ¯ρ −
+
+Rd f0
+
+Φ0
+nτ(x, v)
+
+dv
+Φ⇝Ψτ,h
+≈
+¯ρ −
+
+Rd f0
+
+Ψ0
+nτ,h(x, v)
+
+dv = ¯ρ −
+
+Rd f0
+˜Ψ0
+nτ,h(x, v + τ
+2Eτ,h(nτ, x))
+
+dv
+ˆv:=v+ τ
+2 Eτ,h(nτ,x)
+=
+¯ρ −
+
+Rd f0
+˜Ψ0
+nτ,h(x, ˆv)
+
+dˆv▷
+(12)
+In other words, for the computation of ρ, we can safely replace Ψ0
+nτ,h with ˜Ψ0
+nτ,h without any additional
+error. Additionally, unlike Ψ0
+nτ,h, its sibling ˜Ψ0
+nτ,h is computable! By replacing the last integral with a
+suitable quadrature rule, we can thus compute ρτ,h(nτ, x) ≈ ρ(nτ, x) and proceed. This is the numerical
+ow iteration, which can be summarised as follows:
+1. Compute ρτ,h(t = 0, x) by integrating f0 along v, using numerical quadrature.
+2. Compute and store φτ,h(t = 0, x) using ρτ,h(t = 0, x).
+3. For n = 1, 2, 3, ▷ ▷ ▷ :
+a) Compute an approximation ρτ,h(nτ, x) ≈ ρ(nτ, x) using (12) and numerical quadrature.
+b) Compute and store φτ,h(nτ, x) using ρτ,h(nτ, x).
+In the limit h → 0, the last integral in (12) is evaluated without any error and the Poisson equation
+−∆φh,0 = ρh,0 is also solved exactly.
+Denition 2.5 (Semi-discrete Approximation). The semi-discrete approximation fτ,0 is dened as
+fτ,0(nτ, x, v) := f0
+
+Ψ0
+nτ,0(x, v)
+
+,
+where at every time-step the last integral in (12) and the Poisson equation −∆φτ,0 = ρτ,0 are solved exactly.
+Note that except for the initial time t = 0, we will still usually have Ψ0
+nτ ̸= Ψ0
+nτ,0, E ̸= Eτ,0, φ ̸= φτ,0,
+and ρ ̸= ρτ,0 due to the errors introduced by the time discretisation. Consequently, we usually also have
+fτ,0 ̸= fτ with fτ from (9), which uses the exact eld E.
+2.4 Numerical Approximations of ρ and φ
+While the distribution function f shows increasing lamentation over time, numerical evidence has shown
+that this is often not the case for the electric potential φ and eld E, and to a somewhat lesser extent also
+for the charge density ρ. An illustration of this is given in Figure 2.
+5
+
+0
+π
+2
+π
+3π
+2
+2π
+5π
+2
+3π
+7π
+2
+4π
+−08
+−06
+−04
+−02
+0
+02
+04
+06
+08
+x
+ρ(t, x)
+E(t, x)
+ϕ(t, x)
+Figure 2: Even for very strongly lamented distributions f, the electric eld E and potential φ often remain
+comparatively smooth. This example shows approximations of f (left) as well as ρ, E, and φ
+(right) for the two stream instability benchmark (d = 1) at t = 100. See Section 4.2 for more
+details.
+For this reason, unlike f, the charge density ρ and especially the electric potential φ can eciently be
+approximated on relatively coarse grids. In this work we sample the values of ρ on a Cartesian grid of
+mesh-width hx =
+L
+Nx , for some Nx ∈ N. For each grid node xi, i = 0, ▷ ▷ ▷ , N d
+x − 1, we approximate ρ(nτ, xi)
+using (12). The integral is approximated using the mid-point rule on a Cartesian grid reaching from −vmax
+to vmax in each of the d components of v, having mesh width hv = 2vmax
+Nv :
+ρ(nτ, xi) ≈ ρτ,h(nτ, xi) := ¯ρ − hd
+xhd
+v
+Nd
+v −1
+
+j=0
+f0
+˜Ψ0
+nτ,h(xi, vj)
+
+▷
+(13)
+Here, the vj are the mid-points of the cells of the Cartesian grid in v-direction. This requires the
+evaluation of f0 ◦ ˜Ψ0
+nτ,h at N d
+xN d
+v locations (xi, vj) and is by far the computationally most expensive step
+of NuFI. Note however, that the evaluation of f0 ◦ ˜Ψ0
+nτ,h is an embarrassingly parallel operation. The
+summation can thus be eciently implemented on GPUs using so-called ‘atomic additions’. It can also be
+parallelised to entire clusters of GPUs using simple reductions on partial results of ρτ,h.
+While in principle any quadrature rule could be used, the mid-point rule has the advantage of achieving
+exponential convergence for compactly supported smooth functions.[2] As mentioned in the introduction,
+in many cases f0 is a smooth function that exponentially decays as |v| → ∞ and can thus eectively be
+considered as compactly supported. Hence, before the occurrence of lamentation while f still is smooth,
+we can expect very accurate results. At the same time, the mid-point rule has the advantage of being
+simple and stable: only positive terms appear in the sum.
+We have thus dened ρτ,h at the grid-points xi, between which we will interpolate. In this work we assume
+that f, and thus also ρ and φ are periodic in x. It thus makes sense to use trigonometric interpolation:
+ρτ,h(nτ, x) =
+
+α
+cαeiα 2πx
+L ,
+(14)
+where α is a multi-index and the coecients cα can eciently be computed from the point-values ρτ,h(nτ, xi)
+at the nodes xi using the fast Fourier transform (FFT). In this representation, solving the Poisson equation
+−∆φτ,h = ρτ,h corresponds to a trivial scaling of the coecients cα. The electric energy 1
+2∥∇xφτ,h∥2
+L2(Ω)
+can also be computed easily using Parseval’s identity.
+While the spectral representation of φτ,h is essentially optimal from a mathematical point of view, it
+comes with a practical diculty. When computing the numerical ow Ψ0
+nτ,h (or its relative ˜Ψ0
+nτ,h), the
+electric eld and thus −∇xφτ,h needs to be evaluated very often at arbitrary locations x. This evaluation
+is extremely expensive, as a summation over all modes α is necessary. For this reason, we follow a dierent
+approach.
+Instead, we compute the inverse Fourier transform and thereby eciently obtain the point values
+φτ,h(nτ, xi) at the grid nodes xi, i = 0, ▷ ▷ ▷ , N d
+x − 1. Afterwards we compute the Cartesian, periodic,
+tensor-product spline interpolant of fourth order (piece-wise, coordinate-wise cubical polynomials of global
+coordinate-wise smoothness C2). This representation is equally memory ecient, but a single evaluation
+6
+
+0.25
+0.2
+0.15
+0.1
+0.05
+10
+12of −∇xφ only takes a constant amount of time, independent of the resolution hx, resulting in signicant
+speed-ups.
+Denition 2.6 (Fully Discrete Approximation). The fully discrete approximation fτ,h is dened as:
+fτ,h(nτ, x, v) := f0
+
+Ψ0
+nτ,h(x, v)
+
+,
+where Ψ0
+nτ,h is the numerical ow that is computed using the numerical approximation Eτ,h ≈ E that results
+from taking the gradient of the spline interpolant of φτ,h.
+2.5 Conservation Properties
+What really makes NuFI stick out of the crowd, are its remarkable conservation properties. To avoid the
+technical details associated with the x-periodic setting, in this subsection we consider the whole-space case
+with ¯ρ = 0 instead, but note that these results also carry over to the periodic setting.
+Theorem 2.7 (Conserved Quantities). Let F(nτ, x, v) either denote the exact solution F(nτ, x, v) =
+f0
+
+Φ0
+nτ(x, v)
+
+, or a discrete approximation, F(nτ, x, v) = fτ,h(nτ, x, v) = f0
+
+Ψ0
+nτ,h(x, v)
+
+, τ > 0, h ≥ 0.
+Then, regardless of errors in the approximate electric eld Eτ,h ≈ E that is used in the computation of
+Ψ0
+nτ,h, F fulls for all nτ, n ∈ N0:
+• the maximum principle: 0 ≤ F(nτ, x, v) ≤ ∥f0∥L∞(Rd×Rd),
+• conservation of all Lp(Rd × Rd) norms, 1 ≤ p ≤ ∞:
+∥F(nτ, ·, ·)∥Lp(Rd×Rd) = ∥f0∥Lp(Rd×Rd),
+(15)
+• conservation of kinetic entropy:
+
+Rd×Rd
+F(nτ, x, v) ln
+
+F(nτ, x, v)
+
+dvdx =
+
+Rd×Rd
+f0 ln f0 dvdx,
+(16)
+• and, more generally, for any function g : R → R for which the following integrals are dened:
+
+Rd×Rd
+g
+
+F(nτ, x, v)
+
+dvdx =
+
+Rd×Rd
+g
+
+f0(x, v)
+
+dvdx▷
+(17)
+Proof. For brevity, we will write Ξ ∈ {Φ0
+nτ, Ψ0
+nτ,h}. The maximum principle directly follows from the fact
+that F = f0 ◦ Ξ and f0 ≥ 0. Conservation of the L∞-norm follows because Ξ is a dieomorphism and
+thus Ξ(Rd × Rd) = Rd × Rd. Conservation of entropy (g(x) = x ln x) and the other Lp-norms (g(x) = xp)
+follows from the last statement. For the last statement, we transform the integral and use the volume
+preservation of Ξ to obtain:
+
+Rd×Rd
+g
+
+F(nτ, x, v)
+
+dvdx =
+
+Rd×Rd
+g
+
+f0
+
+Ξ(x, v)
+
+dvdx =
+
+Rd×Rd
+g
+
+f0(x, v)
+
+| det ∇(x,v)Ξ−1(x, v)|
+
+
+
+≡1
+dvdx =
+
+Rd×Rd
+g
+
+f0(x, v)
+
+dvdx▷
+(18)
+The semi-discrete approximation fτ,0 additionally conserves momentum.
+Theorem 2.8 (Conservation of Momentum). Let F(nτ, x, v) either denote the exact solution F =
+f0
+
+Φ0
+nτ(x, v)
+
+, or the semi-discrete approximation F(nτ, x, v) = fτ,0(nτ, x, v) from Denition 2.5. Then F
+fulls the conservation of momentum:
+∀n ∈ N0 :
+
+Rd×Rd
+vF(nτ, x, v) dvdx =
+
+Rd×Rd
+vf0(x, v) dvdx▷
+(19)
+7
+
+Proof. The result for the exact solution is a classical matter; we thus only give the proof for fτ,0. For n = 0
+the result is trivial as fτ,0(0, x, v) ≡ f0(x, v). Now assuming the result held for some n ∈ N0, we will show
+that it also holds for k := n + 1. Abbreviating Ξ := ˜Ψ0
+kτ,0, we obtain:
+
+Rd×Rd
+vfτ,0(kτ, x, v) dvdx =
+
+Rd×Rd
+vf0
+
+Ξ(x, v + τ
+2Eτ,0(kτ, x))
+
+dvdx▷
+(20)
+We now perform the same change of variables as in (12), however we note that due to the product vfτ,0
+things become slightly more involved. Letting ˆv = v + τ
+2Eτ,0
+
+kτ, x
+
+, and noting that in this section we
+assume the whole-space case with ¯ρ = 0, we obtain:
+
+Rd×Rd
+ˆvf0
+
+Ξ(x, ˆv)
+
+dˆvdx + τ
+2
+
+Rd
+Eτ,0(kτ, x)ρτ,0(kτ, x) dx▷
+(21)
+Note that Eτ,0 = −∇φτ,0 and ρτ,0 = −∆φτ,0. Moreover, for any function φ that is smooth enough
+and decays suciently fast at innity we obtain using integration by parts, interchanging the order of
+dierentiation, and the fact that the Laplacian is self-adjoint:
+
+Rd ∇φ∆φ dx = −
+
+Rd φ∇∆φ dx = −
+
+Rd φ∆∇φ dx = −
+
+Rd ∆φ∇φ dx▷
+(22)
+Thus, this integral equals its own negative, and therefore the second integral in (21) is zero. As for the rst
+integral, we perform another change of variables and let ˆx = x − τ ˆv:
+
+Rd×Rd
+ˆvf0
+
+Ξ(x, ˆv)
+
+dˆvdx =
+
+Rd×Rd
+ˆvf0
+
+Ψ0
+nτ,0(ˆx, ˆv + τ
+2Eτ,0(nτ, ˆx))
+
+dˆvdˆx▷
+(23)
+Thus, by performing a nal change of variables on ˆv and repeating the same arguments as above, we obtain
+in total:
+
+Rd×Rd
+vfτ,0
+
+(n + 1)τ, x, v) dvdx =
+
+Rd×Rd
+vfτ,0
+
+nτ, x, v) dvdx▷
+(24)
+Hence by induction, the result follows.
+Finally, the exact solution also satises conservation of energy. Unfortunately, it is yet unclear to what
+extent this is the case for NuFI and we will point to our numerical experiments at the end of this article.
+Theorem 2.9 (Conservation of Energy). The exact solution f(t, x, v) = f0
+
+Φ0
+t(x, v)
+
+and electric eld
+E satisfy the conservation of energy:
+d
+dt
+
+
+1
+2
+
+Rd×Rd
+v2f(t, x, v) dvdx + 1
+2
+
+Rd
+E(t, x)2 dx
+
+
+ = 0▷
+(25)
+2.6 Complexity
+2.6.1 Memory Complexity
+Throughout this work we will assume that f0 is given as a compact mathematical expression and can thus
+essentially be stored at zero cost. In this case, NuFI essentially only requires us to store φτ,h for each
+time-step.
+It cannot be overstated that due to the missing v-dependence of φτ,h, storing the spline coecients of φ
+on a grid requires several orders of magnitude less memory than storing f. Especially when d = 3, memory
+consumption can be reduced millions of times. Let n ∈ N denote the number of time-steps that should be
+computed. We then need to store φτ,h(kτ, ·) for k = 0, 1, ▷ ▷ ▷ , n. This results in:
+Memory Requirement = (n + 1)N d
+x × size of one oating point value▷
+(26)
+8
+
+d
+Nx = Nv
+Memory of φτ,h per step
+Memory for storing f
+2
+128
+0.125 MiB
+2 048 MiB
+2
+256
+0.5 MiB
+32 768 MiB
+2
+512
+2 MiB
+524 288 MiB
+3
+128
+0.015 GiB
+32 768 GiB
+3
+256
+0.125 GiB
+2 097 152 GiB
+3
+512
+1 GiB
+134 217 728 GiB
+Table 1: Memory requirements for storing the electric potential φ on a d-dimensional grid compared
+to directly storing the distribution function f on a 2d-dimensional grid for various example
+discretisations.
+While in NuFI memory requirements do grow with the number of time-steps, the data required for an
+individual step is negligible compared to storing f on a 2d dimensional grid, requiring N d
+xN d
+v oating point
+values. Assuming 8 bytes per oating point value, the dierent memory requirements are illustrated in
+Table 1.
+This means that, even for simulations with d = 3 and Nx = 256, φτ,h can be completely stored for
+several hundred time-steps in the memory of modern GPUs.
+2.6.2 Computational Complexity
+By far, the computationally most expensive operation in NuFI is the evaluation of ρh(nτ, xi) using numerical
+quadrature. The cost of all other operations is negligible: quadrature is carried out on the 2d dimensional
+(x, v)-space while all other operations are carried out on the lower-dimensional x-space.
+For simplicity, we will assume that f0 can be evaluated eciently at O(1) cost. In each time-step
+k = 1, 2, ▷ ▷ ▷ , n, the numerical ux Ψ0
+kτ,h needs to be evaluated at N d
+xN d
+v quadrature nodes. Evaluating
+Ψ0
+kτ,h in turn requires k + 1 evaluations of the approximate electric eld Eτ,h; giving a total cost of
+approximately:
+N d
+xN d
+v
+n
+
+k=1
+(k + 1) = O
+n2
+2 N d
+xN d
+v
+
+(27)
+evaluations of Eτ,h. For this reason it is crucial to have ecient routines for the evaluation of Eτ,h available.
+In our approach using splines, the cost of a single evaluation of Eτ,h is independent of n, Nx and Nv.
+Thus, unlike conventional methods, the cost of NuFI grows quadratically with the number of time-steps.
+This might make the method look unattractive. However, NuFI has a much higher FLOP/byte ratio than
+approaches that directly store f, so it achieves much higher performance on modern computer systems.
+Additionally, also due to the low storage requirements, it easy to parallelise on clusters. Finally, no other
+method known to the authors conserves as many physical properties as NuFI does. It is maybe for this
+reason, as our numerical experiments will conrm, that NuFI only requires relatively coarse resolutions Nx,
+Nv to achieve qualitatively good results.
+3 Similarities and Dierences to Related Literature and Methods
+A key idea in NuFI is to approximate and follow the phase-ow Φt
+s and use the method of characteristics
+to exploit exact conservation of values of f along the phase-ow. In a sense this is a similar idea to
+Lagrangian schemes like Smooth Particle Hydrodynamics (SPH),[3],[4],[5] Particle-In-Cell (PIC),[5],[6],[7] and
+interpolation- and approximation-based particle methods.[8],[9] The key dierences between particle methods
+like PIC and NuFI are the direct discretisation in phase-space and that one traces the characteristics
+forward in time instead of backwards.
+In particle methods one tries to discretise the phase-space directly via sampling the initial distribution
+function f0 at a set of points in the phase-space called ‘particles’. The particles are then traced along
+their respective characteristics. For the non-linear case one then has to compute the electric eld from
+these particles and their respective carried values of f. The dierent Lagrangian schemes basically dier
+9
+
+Hardware
+CPU
+GPU
+Workstation Laptop
+Intel Xeon E-2276M @ 2.8 GHz,
+6 cores, 2 threads per core
+Nvidia Quadro T1000 Mobile,
+4 GB DDR5-VRAM
+Claix GPU Cluster
+2×Intel
+Xeon
+Platinum
+8160
+‘Skylake’ @ 2.1 GHz, 24 Cores
+each, per node
+2×Nvidia
+Volta
+V100-SXM2,
+16 GB HBM2-RAM, per node
+Table 2: Overview of the hardware congurations used for the numerical experiments.
+in this aspect, i. e., how the electric eld is computed from the samples. For the often-used PIC schemes
+the particles are mapped to a grid in x, thereby computing an approximation to ρ from which E can be
+obtained via a Poisson solver on the same grid.
+While in principle PIC has several desirable conservation properties, it struggles with excessive levels
+of noise caused by the ne lamentation and steep gradients in the solution f. This makes remeshing
+every few time-steps necessary.[10] Remeshing reduces the noise in the simulation but introduces a certain
+numerical diusion similar to semi-Lagrangian methods.[11] Due to the steep gradients, methods that
+approximate the solution f directly will suer of overshoots in the numerical solution.[9],[12]
+Related to Lagrangian methods and in particular PIC are the ‘semi-Lagrangian’ methods.[12],[13],[14],[15]
+Similar to purely Eulerian approaches[16],[17],[18] which discretise the distribution function directly and
+update the values on grid-nodes, semi-Lagrangian schemes also use a full grid-based discretisation of f in
+the phase-space. However, to avoid too restrictive time-step constraints and to reduce numerical diusion
+they trace the ‘movement of grid-points’ along the characteristics to use the values of the previous time-step
+approximation of f for interpolation or approximation in the current time-step. The possibility of choosing
+large time-steps also exists in NuFI.
+The use of direct discretisations in the whole phase-space via a grid is expensive as discussed in
+Section 2.6.1. Ways to overcome this are discussed in for instance the papers by Kormann on decomposition
+of the solution into a tensor train format and Einkemmer on comparing the discontinuous Galerkin with a
+spline-based semi-Lagrangian approach.[19],[20]
+4 Numerical Experiments
+In this section we describe the results of several numerical experiments. For this we used two dierent
+hardware congurations available to us: our local workstation laptop as well as the Claix GPU Cluster of
+RWTH Aachen University, see Table 2.
+4.1 Weak Landau Damping (d = 1)
+The rst test case is commonly called weak Landau Damping. The initial condition is
+f0(x, v) :=
+1
+√
+2π e− v2
+2 
+1 + α cos(kx)
+
+,
+(x, v) ∈ [0, L] × R
+(28)
+with k = 0▷5, α = 0▷01, L = 4π. The velocity space is cut at vmax = 10. We use dierent spatial resolutions
+Nx and Nv, but all simulations use the same time-step τ =
+1
+16.
+The initial state (28) is a small perturbation to the Maxwellian distribution
+fM(v) =
+1
+√
+2π e− v2
+2
+(29)
+which is an equilibrium of the Vlasov–Poisson equation (1). It is known that this particular perturbation
+gets periodically damped to zero in a weak sense as t → ∞. Thus, this benchmark consists of reproducing
+the correct damping rate for the electric eld. The so-called Landau damping rate is γE = 0▷153359, half
+of the corresponding damping rate for the electric energy γenergy = 0▷306718.[19]
+10
+
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+10−18
+10−16
+10−14
+10−12
+10−10
+10−8
+10−6
+10−4
+10−2
+t
+1
+2‖Eτ,h‖2
+L2
+single, Nx = 8, Nv = 32
+double, Nx = 8, Nv = 32
+single, Nx = 32, Nv = 128
+double, Nx = 32, Nv = 128
+single, Nx = 128, Nv = 512
+double, Nx = 128, Nv = 512
+Figure 3: Electric energy for the weak Landau damping benchmark (d = 1) displayed for simulations run in
+single and double precision arithmetic. Except for the nest resolution, results for single and
+double precision are indistinguishable.
+For d = 1 meaningful simulations can still be carried out on the workstation laptop. For such hardware
+computations in single precision are usually signicantly faster and provide sucient accuracy when d = 1.
+At the highest resolution, the computation of all 1 600 time-steps took 1.7 seconds in total when using
+single precision. As the laptop only uses a consumer grade GPU chip, the double precision computation
+was signicantly slower and took 28.3 seconds.
+The results are illustrated in Figure 3. For low and medium resolutions the results of single and double
+precision computations are indistinguishable. Only for high resolutions a dierent behaviour after t ≈ 60
+is observed: on the one hand, when computing in single precision no damping eect can be observed
+after t ≈ 62 as the electric energy arrived at machine precision level with respect to the single precision,
+suggesting that Landau damping can at most be resolved up to the chosen oating point precision. On the
+other hand, the damping eect is still observable until t ≈ 80 for double precision, however, with slightly
+slower rate. For single precision one does not observe recurrence phenomena: instead the electric energy
+oscillates with same amplitude after t ≈ 62. The simulation with double precision shows a slight recurrence
+at t ≈ 83.
+The correct damping rate and oscillation frequency are reproduced by all simulations. The highest
+resolution simulation captures the correct damping rate γenergy until t ≈ 62. Lower resolutions capture the
+correct damping rate only until their respective recurrence timings.
+4.2 Two Stream Instability (d = 1)
+Next we investigate the two stream instability benchmark. The initial condition is
+f0(x, v) :=
+1
+√
+2π e− v2
+2 v2
+1 + α cos(kx)
+
+,
+(x, v) ∈ [0, L] × R
+(30)
+with k = 0▷5, α = 0▷01, L = 4π. The velocity space is cut at vmax = 10, time integration uses a time step
+of τ =
+1
+16. We again consider simulations in low (Nx = 8, Nv = 32), medium (Nx = 32, Nv = 128), and
+high resolution (Nx = 128, Nv = 512).
+This benchmark simulates two colliding streams of electrons with opposing velocity which are both
+slightly perturbed from equilibrium. After an initial damping phase the streams start mixing, leading to
+an increasingly turbulent behaviour. In particular, the distribution function f develops an increasing level
+of lamentation and a ‘vortex’ over time.
+11
+
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+10−9
+10−6
+10−3
+100
+t
+1
+2∥Eτ,h∥2
+L2
+Nx = 8, Nv = 32
+Nx = 32, Nv = 128
+Nx = 128, Nv = 512
+Figure 4: Electric energy for the two stream instability benchmark (d = 1) displayed for simulations, run in
+double precision and using time-step τ =
+1
+16.
+Figure 4 shows the evolution of electric energy until T = 100. Until t ≈ 10 damping eects on the
+two particle streams are still dominating the overall dynamic. Between t ≈ 10 and t ≈ 25 a vortex in
+the phase-space starts forming. After t ≈ 25 the mixing process is dominant, where turbulence and
+lamentation in the solution make resolving f increasingly complicated. Until t ≈ 25 all three simulations
+are able to capture the underlying dynamics in the electric energy correctly. After t ≈ 25 the low resolution
+simulation is still able to capture the dynamics qualitatively, however, with a signicant error. After t ≈ 40
+the medium resolution simulation also starts to deviate from the high resolution one, albeit to a much
+lesser extent. The medium and high resolution simulation are able to resolve the slow oscillation in the
+electric energy after t ≈ 25. The high resolution run is able to reproduce the electric energy until t = 100
+without signicant numerical noise in the plot.
+Figure 6 shows the distribution function fτ,h for several times t. At t ≈ 5 the electric eld damping seizes
+to be dominant. At this time the dierence to the initial datum of f is barely perceivable. Around t ≈ 25
+the forming of a vortex in phase-space starts setting in which comes with reaching the global maximum in
+electric energy. After t ≈ 25 the vortex rotates periodically, creating an increasing number of laments
+for later t. The low resolution is also able capture the overall dynamics correctly in a qualitative sense,
+however, one observes increasing distortions for the low resolution simulation. For t = 100 the vortex seems
+to move along the x-axis for the low resolution simulation, which is an incorrect dynamic.
+The higher resolution simulation is able to correctly reproduce the dynamics until the end (t = 100).
+In addition, NuFI is able to reproduce the ne lamentation expected from the analytic solution. The
+pixel-artefacts which can be seen in Figure 6 are due to the nite resolution of the plot (2 048 sampling
+points in each direction) and would disappear in plots of suciently high resolution.
+Figure 5 shows the relative errors produced by simulation for several resolution settings in time and
+phase-space as well as dierent choice of oating point representation. The reference solution was computed
+using Nx = 512, Nv = 2 048 and τ =
+1
+32.
+When varying the phase-space resolution one observes that until t ≈ 20 all errors stay at approximately
+the same level. Then the mixing process with accompanied with increasing lamentation in the solution
+sets in, which leads to an increase of error for all resolutions. After t ≈ 26 errors seem to stabilise on a
+xed level again; ranging from 10−3 to 10−5 for the relative L2 error for the simulation in single precision
+and from 10−1 to 10−2 with respect to the L∞-norm. Similar error behaviour can be observed for double
+precision, suggesting that the chosen oating point precision makes no longer a dierence when entering
+the turbulent phase of the simulation in this case. Note however, that for early times errors in double
+precision can be up to eight orders of magnitude smaller.
+When varying the time-step-size and xing the resolution in phase-space we observe an initial jump in
+error between t = 0 and t ≈ 2. The error growth is much reduced for the smaller time steps. This implies
+that at this stage of the simulation the errors in the quadrature and the Poisson solver are close to machine
+precision 10−8, and that it is the time discretisation error which is dominating. After t ≈ 2 the errors levels
+stabilise and do not signicantly further increase until the onset of lamentation around t ≈ 26.
+12
+
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+24
+26
+28
+30
+10−10
+10−9
+10−8
+10−7
+10−6
+10−5
+10−4
+10−3
+t
+∥E − Eτ,h∥L2∥E∥L2
+hx = L32, hv = vmax64
+hx = L64, hv = vmax128
+hx = L128, hv = vmax256
+(a) L2-error, single precision
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+24
+26
+28
+30
+10−5
+10−4
+10−3
+10−2
+10−1
+t
+∥E − Eτ,h∥L∞∥E∥L∞
+hx = L32, hv = vmax64
+hx = L64, hv = vmax128
+hx = L128, hv = vmax256
+(b) L∞-error, single precision
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+24
+26
+28
+30
+10−16
+10−14
+10−12
+10−10
+10−8
+10−6
+10−4
+10−2
+t
+∥E − Eτ,h∥L2∥E∥L2
+hx = L32, hv = vmax64
+hx = L64, hv = vmax128
+hx = L128, hv = vmax256
+(c) L2-error, double precision
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+24
+26
+28
+30
+10−8
+10−7
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+t
+∥E − Eτ,h∥L∞∥E∥L∞
+hx = L32, hv = vmax64
+hx = L64, hv = vmax128
+hx = L128, hv = vmax256
+(d) L∞-error, double precision
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+24
+26
+28
+30
+10−8
+10−7
+10−6
+10−5
+10−4
+10−3
+10−2
+t
+∥E − Eτ,h∥L2∥E∥L2
+∆t = 12
+∆t = 14
+∆t = 18
+(e) L2-error, single precision
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+24
+26
+28
+30
+10−4
+10−3
+10−2
+10−1
+t
+∥E − Eτ,h∥L∞∥E∥L∞
+∆t = 12
+∆t = 14
+∆t = 18
+(f) L∞-error, single precision
+Figure 5: Relative errors in electric eld with respect to the L2− and L∞−norm for the two stream
+instability benchmark in d = 1. With τ =
+1
+16 or Nx = 64, Nv = 256 xed respectively. Reference
+solution computed using Nx = 512, Nv = 2 048 and τ =
+1
+32.
+13
+
+t = 5
+t = 25
+t = 50
+t = 100
+Figure 6: The distribution function fτ,h(t, x, v) for the two stream instability benchmark. Left Nx = 8,
+Nv = 32 and right Nx = 128, Nv = 512. Both computed using τ =
+1
+16 as time-step. The plot
+resolution is 2 048 points in both x- and v-direction.
+14
+
+fh
+5
+4
+0.25
+3
+2
+0.2
+1
+A
+0
+0.15
+-1
+0.1
+-2
+-3 
+0.05
+-4 
+-5.
+0
+2
+4
+6
+8
+10
+12
+Xfh
+5
+4
+0.25
+3
+2
+0.2
+1
+A
+0
+0.15
+-1
+0.1
+-2
+-3 
+0.05
+-4 
+-5.
+0
+2
+4
+6
+8
+10
+12
+X5
+4
+0.25
+3
+2
+0.2
+1
+0
+0.15
+-1
+0.1
+-2
+-3
+0.05
+-4
+-5
+0
+2
+4
+6
+8
+10
+125
+4
+0.25
+3
+2
+0.2
+1
+0
+0.15
+-1
+0.1
+-2
+-3
+0.05
+-4
+-5
+0
+2
+4
+6
+8
+10
+125
+4
+0.25
+3
+2
+0.2
+0
+0.15
+-1
+0.1
+-2
+-3
+0.05
+-4
+-5
+0
+2
+4
+6
+8
+10
+12
+X5
+4
+0.25
+3
+2
+0.2
+0
+0.15
+-1
+0.1
+-2
+-3
+0.05
+-4
+-5
+0
+2
+4
+8
+10
+125
+4
+0.25
+3
+2
+0.2
+0
+0.15
+-1
+0.1
+-2
+-3
+0.05
+-4
+-5 
+0
+2
+4
+6
+8
+10
+125
+4
+0.25
+3
+2
+0.2
+1
+0
+0.15
+-1
+0.1
+-2
+-3
+0.05
+-4
+-5
+0
+2
+4
+8
+10
+120
+10
+20
+30
+40
+50
+10−6
+10−4
+10−2
+100
+102
+t
+1
+2∥Eτ,h∥2
+L2
+Nx = 32, Nv = 128
+Nx = 64, Nv = 256
+Nx = 128, Nv = 512
+Figure 7: Electric energy for the strong Landau damping benchmark in d = 2, with time-step set to τ =
+1
+16.
+4.3 Strong Landau Damping (d = 2)
+For the two-dimensional case we consider the strong Landau damping or non-linear Landau damping
+benchmark. The initial condition is
+f0(x, y, u, v) := 1
+2π e− v2+u2
+2
+
+1 + α(cos(kx) + cos(ky))
+
+,
+(x, y, u, v) ∈ [0, L]2 × R2
+(31)
+with k = 0▷5, α = 0▷5, and L = 4π. The velocity space is cut at vmax = 10, we chose τ =
+1
+16 as time-step.
+All computations are carried out using double precision arithmetic.
+The initial datum is a strong perturbation of the Maxwellian distribution function. We thus expect a
+strong growth in the electric eld strength. In this sense, this case is more comparable to the two stream
+instability than the weak Landau damping benchmark.
+Figure 7 shows the evolution of electric energy over time. After a initial periodic damping period until
+t ≈ 10 with electric energy ranging between 102 and 10−1 the electric eld gets excited again. The electric
+energy increases periodically until t ≈ 42 after which it gets damped again.
+While the high resolution simulation is able to resolve the dynamics in the electric energy correctly until
+t = 50, the low resolution simulation starts struggling after the excitement sets in, at t ≈ 20 divergence from
+the high resolution solution becomes observable. In particular after t ≈ 30 the low resolution simulation is
+also no longer able to capture the correct peaks and lows of the electric energy. However, the oscillating
+nature and overall dynamics are still reproduced qualitatively correctly.
+In Figure 8 we plot relative errors with dierent resolutions in phase space. The reference solution was
+computed using Nx = 256, Nv = 1 024 and τ =
+1
+32.
+Initially the relative L2-errors are again close to machine precision for the high resolution simulation,
+similar to the d = 1 case. For low resolution the simulation errors are around 10−8 with respect to the
+L2-norm and 10−4 in the L∞-norm. However, in contrast to the d = 1 two stream benchmark the errors
+increase signicantly faster. The errors increase continuously until t ≈ 20 and level out at a range between
+10−1 for low resolution in L2- as well as L∞-norms and 10−4 in L2- as well as 10−2 in L∞-norm for the
+high resolution simulation.
+Figures 9 and 10 illustrate the electric eld of high and low resolution simulations at various times t.
+There is no observable qualitative or quantitative dierence between the two simulations in these plots,
+suggesting that even the low resolution is able able to correctly capture the dynamics in the electric eld
+until at least t = 50.
+4.4 Two Stream Instability (d = 3)
+For the three-dimensional case we again consider the two stream instability benchmark. The initial condition
+is
+f0(x, y, z, u, v, w) :=
+1
+2(2π)
+3
+2
+
+e− (v−v0)2
+2
++ e− (v+v0)2
+2
+
+e− u2+w2
+2
+
+1 + α(cos(kx) + cos(ky) + cos(kz)
+
+,
+(32)
+15
+
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+24
+26
+28
+30
+10−16
+10−13
+10−10
+10−7
+10−4
+10−1
+t
+∥E − Eτ,h∥L2∥E∥L2
+Nx = 32, Nv = 128
+Nx = 64, Nv = 256
+Nx = 128, Nv = 512
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+22
+24
+26
+28
+30
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+t
+∥E − Eτ,h∥L∞∥E∥L∞
+Nx = 32, Nv = 128
+Nx = 64, Nv = 256
+Nx = 128, Nv = 512
+Figure 8: L2- and L∞ errors with respect to a reference solution for the strong Landau damping benchmark
+in d = 2. Computed with time-step τ =
+1
+16. The reference solution was computed using Nx = 256,
+Nv = 1 024.
+with (x, y, z, u, v, w) ∈ [0, L]3 × R3 where k = 0▷2, α = 10−3, L = 4π, and v0 = 2▷4. These parameters are
+the same as the ones chosen by Kormann.[19] The velocity space is cut at vmax = 10 and we use τ =
+1
+16
+as time-step. We carry out simulations at dierent resolutions; the nest uses Nx = 32, Nv = 64, and
+therefore more than eight billion quadrature nodes in phase-space. Using eight nodes of the Claix cluster
+with two GPUs each, the computation of all n = 800 time-steps took 3:58 hours. Again, only double
+precision arithmetic is considered.
+Figure 11 shows the evolution of electric energy over time. In contrast to the d = 1 benchmark, the
+amplitude of the perturbation of the equilibrium is smaller, which leads to a delayed excitement of the
+electric eld as well as delayed vortex-formation. The peak electric energy is reached at t ≈ 35. The
+lower resolution simulation is not able to capture some of the peak-timings and after t ≈ 30 exhibits a
+faster oscillation than the higher resolution simulation. However, even with low resolution of only 8 points
+in spatial and 16 points in velocity-direction the NuFI is still able to capture the overall dynamics in a
+qualitatively correct manner up to t ≈ 30.
+Figure 12 shows a representative cross-section of the electric eld at z = 0. The electric eld aligns along
+the x-axis with positive direction for approximately y > 15 and negative direction for y < 15.
+4.5 On the Conservation of Energy
+From Section 2.5 we know that all Lp-norms of fτ,h as well as kinetic entropy are conserved exactly by
+NuFI. While the conservation is exact, the actual numerical values of the integrals in (15) and (16) can
+only be approximated by means of quadrature formulæ.
+The same holds true for the total energy: its value can also only be approximated by means of quadrature.
+Additionally, however, we cannot expect an exact conservation of energy due to various discretisation errors.
+This even holds for the semi-discrete scheme where h → 0 with a xed time-step τ. In particular, in many
+mechanical systems, symplectic time integrators usually do not conserve energy exactly; however the error
+in total energy remains bounded and does not grow over time. In other words: while these schemes usually
+do not conserve energy exactly, they are free of a systematic energy drift. Thus the question arises whether
+this is also the case in NuFI.
+To at least partially address this question, we consider the two stream instability benchmark in d = 1,
+τ =
+1
+32. At time t = 0 the initial data f0 and corresponding charge density ρ are very smooth; thus
+the conserved quantities and the total energy can accurately be computed using numerical quadrature.
+The values obtained at t = 0 can thus serve as reference values. We then proceed as follows: in every
+time-step the integrals are evaluated using the same quadrature rule as in (13). The results are illustrated
+in Figure 13. For the entropy and Lp-norms the deviations from the value at t = 0 are quadrature errors
+only. The value for total energy contains both quadrature and discretisation errors. However, we observe
+that the relative errors of the approximate total energy are of the same magnitude as the quadrature errors
+for the other conserved quantities. We thus conclude that the error in total energy is smaller than the
+quadrature error; a systematic energy drift – if present – could only be detected by using more quadrature
+points. As this is the case for all tested resolutions, we conclude that an energy drift is unlikely.
+16
+
+t = 0
+t = 25
+t = 50
+Figure 9: The electric eld Eτ,h for the strong Landau damping benchmark in d = 2 at dierent times t.
+Left we display the magnitude |E(t, x)| of the electric eld, the right hand side illustrates the
+direction. The simulation was run with τ =
+1
+16, Nx = 32, Nv = 128.
+17
+
+10
+8
+J.
+0.
+0.2
+10E
+12
+10
+8
+0
++
+0
+N
+4
+6
+8
+10
+120.08
+12
+0.07
+10
+0.06
+0.05
+0.04
+0.03
+ 0.02
+0.01
+10
+212
+10
+8
+.-
+↑
+0
+t
+0
+4
+6
+8
+10
+1212
+10
+0.2
+0.15
+0.1
+0.05
+10En
+12
+10
+8
+6
+2
+0
+0
+N
+4
+6
+10
+12 t = 0
+t = 25
+t = 50
+Figure 10: The same as Figure 9, but this time with high resolution Nx = 128, Nv = 512. Both simulations
+qualitatively produce the same electric eld.
+18
+
+10
+8
+J.
+0.
+0.2
+10E
+12
+10
+8
+0
++
+0
+N
+4
+6
+8
+10
+120.08
+12
+0.07
+10
+0.06
+0.05
+0.04
+0.03
+ 0.02
+0.01
+10
+212
+10
+8
+.-
+↑
+0
+t
+0
+4
+6
+8
+10
+1212
+10
+0.2
+0.15
+0.1
+0.05
+10En
+12
+10
+8
+6
+2
+0
+0
+N
+4
+6
+10
+12 0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+10−3
+10−2
+10−1
+100
+101
+102
+103
+104
+105
+t
+1
+2∥Eτ,h∥2
+L2
+Nx = 8, Nv = 16
+Nx = 16, Nv = 32
+Nx = 32, Nv = 64
+Figure 11: Electric energy for the two stream instability benchmark in d = 3. Time-step τ =
+1
+16.
+4.6 Computational Time and Scaling
+As mentioned before, NuFI can be very eciently implemented on GPU clusters. In this section, we thus
+look at the computational time and scaling behaviour of NuFI. The code for these computations is available
+from the authors per request. We consider the cases d = 1 and d = 2 only, as they allow larger numbers of
+samples: for d = 3 the number of quadrature nodes grows too quickly and we would end up with graphs
+containing only a few computational results.
+First we consider the computational time on a single node of the Claix cluster, see Table 2. For the
+case d = 1 only one GPU and single precision was used, both GPUs and double precision was used for
+d = 2. Starting with Nx = 8, Nv = 32, we measured the total computational time for n = 480 time-steps
+of τ =
+1
+16, i. e., simulations were stopped at tend = 30. For each subsequent simulation the values of Nx
+and Nv were doubled.
+The computational times are displayed in Figure 14. For d = 1 the computation time stays below 10−1 s
+until roughly one million quadrature nodes. Before this point, there are still spare execution units in the
+GPU thus computational time remains approximately constant. Afterwards we observe linear scaling until
+roughly one billion points. At this point there is again a upwards kink in the graph. We believe that this is
+due to caching eects: until this points the coecients of φτ,h for a single time-step t into the 128 KiB
+large cache of the GPU. For the d = 2 we observe linear scaling starting from the second measurement
+point.
+Next we consider a strong scaling experiment in d = 2, with Nx = 64, Nv = 256, and n = 480 time-steps,
+in double precision. Starting with a single node, we measure the computational time with an increasing
+number of computational nodes of the Claix cluster. The results are illustrated in Figure 15. We observe
+almost optimal strong scaling, i. e., when doubling the number of computational nodes the computation
+time is cut in half. After the measurement point with eight computational nodes there is slight kink in the
+plot which we believe is caused by the base calculations which take the same amount of time independent of
+the number of nodes like the Poisson solver, which uses a serial, non-parallelised implementation. We plan
+to carry out larger simulations on bigger machines, once corresponding computational resources become
+available to us.
+5 Conclusion
+The numerical ow iteration is a new method for numerically solving the Vlasov–Poisson equation with
+excellent conservation properties, low memory requirements, and high arithmetic intensity. It is ideally
+19
+
+t = 0
+t = 25
+t = 50
+Figure 12: The electric eld Eτ,h for the two stream benchmark in d = 3, at dierent times t for z = 0.
+Left: magnitude |Eτ,h|2, right: directions in the xy-plane. The simulation was run with τ =
+1
+16,
+Nx = 32, Nv = 64, i. e., using 8 589 934 592 quadrature points.
+20
+
+10°
+30
+25
+20
+10
+10
+15
+20
+25
+30En
+30
+25
+-
+20
+7
+15
+10
+2
+n
+0
+t.
+0
+n
+10 
+15
+20
+25
+30 Eh
+OE
+0.12
+ 25 
+0.1
+20
+0.08
+0.06
+10
+0.04
+0.0Z
+00
+5
+10 
+15
+20
+25En
+30
+25
+20
+15
+10
+0
+0
+n
+10
+15
+20
+25
+30Eh
+30
+25
+20
+10
+00
+5
+10
+15 ,
+20
+25 
+30En
+30
+25
+20
+15
+10
+0
+0
+n
+10
+15
+20
+25
+300
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+−15%
+−10%
+−05%
+0%
+05%
+10%
+15%
+t
+Quadrature Error of ∥fτ,h∥L1
+Nx = 64, Nv = 128
+Nx = 128, Nv = 256
+Nx = 256, Nv = 512
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+−15%
+−10%
+−05%
+0%
+05%
+10%
+15%
+t
+Quadrature Error of ∥fτ,h∥L2
+Nx = 64, Nv = 128
+Nx = 128, Nv = 256
+Nx = 256, Nv = 512
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+−15%
+−10%
+−05%
+0%
+05%
+10%
+15%
+t
+Quadrature Error of Entropy
+Nx = 64, Nv = 128
+Nx = 128, Nv = 256
+Nx = 256, Nv = 512
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+−15%
+−10%
+−05%
+0%
+05%
+10%
+15%
+t
+Approximate Error of Total Energy
+Nx = 64, Nv = 128
+Nx = 128, Nv = 256
+Nx = 256, Nv = 512
+Figure 13: Quadrature errors when computing the L1-norm, L2-norm, and kinetic entropy of fτ,h for the
+two stream instability benchmark in d = 1 using τ =
+1
+32. These quantities are conserved exactly,
+their numerical value, however, can only be approximated using quadrature. The values are
+compared to the approximations at time t = 0. Total energy cannot be expected to be conserved
+exactly, and cannot be evaluated exactly, either. However, its approximate error is of the same
+order of magnitude as the quadrature error of the exactly preserved quantities and no drift is
+visible.
+102
+103
+104
+105
+106
+107
+108
+109
+1010
+10−1
+101
+103
+105
+NxNv
+tcomp (in s)
+(a) d = 1
+104
+105
+106
+107
+108
+109
+1010
+10−1
+101
+103
+105
+N 2
+xN 2
+v
+tcomp (in s)
+(b) d = 2
+Figure 14: Computational times for n = 480 time-steps and Nx = 2k, Nv = 2k+2, k = 3, 4, 5, ▷ ▷ ▷ . Left:
+d = 1, single precision, one GPU of the Claix cluster. Right: d = 2, double precision, one node
+with two GPUs. Complexity grows linearly with the number of quadrature points N d
+xN d
+v . For
+d = 1 caching eects become visible at 109 quadrature points.
+21
+
+20
+21
+22
+23
+24
+24
+25
+26
+27
+28
+29
+nodes
+tcomp (in s)
+Figure 15: Strong scaling with the number of nodes for d = 2, Nx = 64, Nv = 256, n = 480 time-steps,
+and double precision, where both GPUs on each node of the Claix cluster are used. We observe
+almost optimal strong scaling; for 16 nodes parallel eciency is only slightly reduced.
+suited for modern ultra-parallel high-performance computing hardware and workstations alike. We believe
+that these properties will nally enable suciently resolved simulations of the Vlasov–Poisson equation for
+the most important case d = 3, already in ‘Tier 2’ data centres.
+Several extensions to the scheme are possible. The approach can directly be applied to multi-species
+systems and can be extended to the full Maxwell equations.
+In this work, for simplicity, we used entirely uniform discretisations. However, one can easily envision
+adaptive schemes: φτ,h can be stored using an adaptive space–time discretisation, adaptive numerical quad-
+rature can be used, as well as adaptive time-stepping schemes. However, as usual, adaptive schemes make
+eective parallelisation signicantly harder and require sophisticated load-balancing and synchronisation
+mechanisms.
+Inhomogeneous right hand sides g(t, x, v) of the Vlasov equation (1) can also easily be taken into account.
+In this case the solution formula from Lemma 2.3 becomes:
+f(t, x, v) = f0
+
+Φ0
+t(x, v)
+
++
+ t
+0
+g
+
+s, Φs
+t(x, v)
+
+ds▷
+(33)
+The last integral can then be eciently approximated using the numerical ow Ψτ,h and the trapezoidal
+rule. This way collision operators could also be incorporated into the simulation.
+Acknowledgments
+Matthias Kirchhart started this work as a member of RWTH Aachen University, where he also received
+funding from the German research foundation (DFG), project number 432219818, ‘Vortex Methods for
+Incompressible Flows’.
+The idea, analysis, and content were created there.
+He now works for Intel
+Deutschland GmbH, which kindly permitted the completion of this article.
+Paul Wilhelm is recipient of a scholarship from the German national high performance computing
+organisation (NHR).
+We would like to acknowledge the work of Jan Eifert, who helped in the implementation of the code.
+Without these fundings and support, this work would not have been possible.
+References
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+23
+

QNA0T4oBgHgl3EQfDP_p/content/tmp_files/2301.02002v1.pdf.txt

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+1 
+ 
+Ultrafast chiroptical switching in UV-excited molecules 
+ 
+Vincent Wanie1*, Etienne Bloch2, Erik P. Månsson1, Lorenzo Colaizzi1,3, Sergey Ryabchuk3,4, 
+Krishna Saraswathula1,3, Andrea Trabattoni1,5, Valérie Blanchet2, Nadia Ben Amor6,  
+Marie-Catherine Heitz6, Yann Mairesse2, Bernard Pons2*, Francesca Calegari 1, 3, 4* 
+ 
+1Center for Free-Electron Laser Science, Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany 
+2Université de Bordeaux - CNRS - CEA, CELIA, UMR5107, F-33405 Talence, France  
+3Physics Department, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany 
+4The Hamburg Centre for Ultrafast Imaging, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany 
+5Institute of Quantum Optics, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany 
+6CNRS, UPS, LCPQ (Laboratoire de Chimie et Physique Quantiques), FeRMI, 118 Route Narbonne, F-31062 Toulouse, France 
+*Email: [email protected]; [email protected]; [email protected]   
+ 
+Molecular chirality is a key design property for many technologies including bioresponsive 
+imaging 1, circularly polarized light detection 2 and emission 3, molecular motors 4,5 and 
+chiroptical switches 6,7. Imaging and manipulating the primary steps of transient chirality is 
+therefore central for controlling numerous physical, chemical and biological properties that 
+arise from chiral molecules in response to external stimuli. So far, the manifestation of 
+electron-driven chiral dynamics in neutral molecules has not been demonstrated at their 
+natural timescale. Here, we use time-resolved photoelectron circular dichroism (TR-PECD) 
+to image the dynamics of coherent electronic motion in neutral chiral molecules and 
+disclose its impact on the molecular chiral response.  We report on a rapid sign inversion of 
+the chiroptical signal occurring in ~7 fs following the UV-excitation of methyl-lactate 
+molecules. The populated electronic coherences can be used for chiroptical switching 
+where the amplitude and direction of the net photoelectron current generated by PECD is 
+controlled. The interpretation of the results is supported by theoretical modelling of both 
+the molecular photoexcitation to Rydberg states and the subsequent photoionization via 
+PECD. Our findings establish a general method to investigate electron dynamics in a variety 
+of chiral systems with high sensitivity and pave the way to a new scheme for enantio-
+sensitive charge-directed reactivity in neutral chiral molecules. 
+ 
+
+2 
+ 
+Chirality is a peculiar property that characterizes the majority of biochemical systems: a chiral 
+molecule exists in two geometrical configurations that are non-superimposable mirror images 
+of each other - defined as (R) and (S) enantiomers - exhibiting different physical and chemical 
+properties when interacting with another chiral entity. This chiral recognition is central to 
+many fields of applied sciences, including enantioselective catalysis, drug engineering and 
+biophysics 7,8. Capturing the primary steps of chiral recognition and the mechanisms dictating 
+the outcomes of a chiral interaction would thus have a significant impact in various fields 
+working with chiral property of matter. At the ultrafast electronic timescale for instance, the 
+opportunity to steer electrons responsible for chemical activity notably promises a way to 
+control the outcome of enantio-sensitive reactions 9 via charge-directed reactivity 10–12. 
+However, the ability to track and control electron-driven chiral interactions as well as other 
+electron dynamics in biochemically-relevant neutral molecules in general remains to be 
+demonstrated.  
+ 
+In this context, the temporal resolution provided by attosecond technologies developed in 
+the past twenty years gives access to the fastest electron dynamics of matter on their natural 
+timescale. Seminal pump-probe experiments using attosecond light pulses have revealed 
+valence electron dynamics in atoms 
+13, autoionization dynamics in molecules 
+14, 
+photoionization delays in solids 15,16 as well as electron-driven charge migration in 
+biomolecules 12,17. However, the intrinsically high photon energy of attosecond light sources 
+inevitably leads to target ionization, which results in probing ultrafast dynamics of cationic 
+states. Investigating the light-induced electron dynamics of biochemically-relevant neutral 
+molecules with high temporal resolution requires new experimental approaches. Two 
+important technological challenges must be addressed. First, the pump pulse must have well-
+
+3 
+ 
+defined characteristics: (i) a photon energy below the ionization threshold, (ii) a broadband 
+energy spectrum to trigger electron motion among multiple electronic states and (iii) a time 
+duration that provides a prompt excitation before any nuclear motion can take place, 
+together with sufficient temporal resolution. Because of the low ionization potential of most 
+molecular systems, laser pulses with such characteristics are confined to a narrow spectral 
+region covering the ultraviolet (UV) and the vacuum-UV ranges, which also avoids triggering 
+complex high-order, strong-field multiphoton driven processes that typically do not occur in 
+nature 18,19. 
+Second, the spectroscopic observable must be carefully chosen. In other words, how to probe 
+electron dynamics with an increased sensitivity compared to typical ultrafast spectroscopic 
+methods such as all-optical transient absorption spectroscopy or time-resolved 
+photoelectron and photoion spectroscopies 20,21? Since circularly polarized light possesses a 
+chiral character, a promising approach is to use the chiroptical response of bio-relevant 
+systems to investigate their ultrafast dynamics. 
+ 
+It has been known since 1976 that the photoionization of randomly oriented chiral molecules 
+by circularly polarized radiation yields a photoelectron angular distribution (PAD) which 
+presents a forward-backward asymmetry along the light propagation axis 22,23. This 
+asymmetry, referred to as PhotoElectron Circular Dichroism (PECD), is orders of magnitude 
+larger than the conventional absorption circular dichroism 24, and it is therefore well suited 
+to the study of chiral features in the gas phase.  PECD was shown to be strongly sensitive to a 
+wealth of structural molecular properties, including the nature of ionized orbitals and 
+continuum states 25, isomerism and chemical substitution 26,27 and the vibrational structure 
+of the cation 28. While structural features are generally gained from static measurements, the 
+
+4 
+ 
+availability of femtosecond and attosecond pulses provides a valuable tool to characterize 
+and manipulate the dynamical properties of chiral neutral molecules at the electron 
+timescale. So far, the strength of PECD has only been demonstrated in the time domain for 
+the investigation of nuclear dynamics, internal conversion and photoionization delays in chiral 
+molecules 29–32.  
+ 
+Here, we employ a novel laser-based technology delivering few-femtosecond UV pulses 33 to 
+perform TR-PECD measurements with unprecedented temporal resolution and extend the 
+method to the investigation of electron-driven chiral interactions in neutral methyl-lactate 
+(ML) – a derivative of lactate, which has regained substantial interest due to its recently 
+uncovered metabolic functions 34. A linearly polarized UV pulse promptly launches a coherent 
+electronic wavepacket in the bio-relevant molecule while a time-delayed circularly polarized 
+near-infrared (NIR) probe triggers ionization from the transient wavepacket (Fig.1a). The 
+chiroptical molecular response is characterized by the preferential direction of emission of 
+the ejected electron through PECD. Numerical simulations modelling the experiment show 
+that the NIR pulse tracks electronic beatings initiated by the coherent superposition of 
+Rydberg states created by the broadband UV pump pulse. The resulting oscillations of the 
+charge density can be used for chiroptical switching in which the net photoelectron current 
+from the initially isotropic sample is reversed on a sub-10 fs timescale when adjusting the 
+pump-probe time delay. 
+ 
+Fig. 1 shows an overview of the experimental approach used to investigate in real-time the 
+chiral dynamics of ML. Following sudden photoexcitation of (S)-ML enantiomers just below 
+the ionization threshold by a few-fs UV pulse, a circularly polarized near-infrared (NIR) probe 
+
+5 
+ 
+pulse leads to photoionization (a). For each pump-probe delay 𝑡, the photoelectron angular 
+distributions (PADs) are collected with a velocity map imaging spectrometer (VMIs) for both 
+left (ℎ = +1) and right (ℎ = −1) circular polarizations of the NIR probe pulse. These raw 
+distributions are then fitted using a pBasex inversion algorithm 29 to yield the PAD 
+𝑆(")(𝜖, 𝜃, 𝑡), where 𝜖 is the kinetic energy of the photoelectron and 𝜃 its emission angle with 
+respect to the light propagation axis (see section 1.2 of the supplement). The differential PECD 
+image is then defined as the normalized difference 𝑃𝐸𝐶𝐷(𝜖, 𝜃, 𝑡) = 2
+$("#)(%,',())$(%#)(%,',()
+$("#)(%,',()*$(%#)(%,',() 
+and its evolution is observed as a function of the pump-probe delay 𝑡. Figure 1b displays 
+𝑃𝐸𝐶𝐷(𝜖, 𝜃, 𝑡) for 𝑡 = 5, 11, 17 and 26 fs, which captures an inversion of the photoelectron 
+forward-backward asymmetry in about 7 fs. 
+ 
+The inversion procedure consists in fitting the VMIs images according to 𝑆(")(𝜖, 𝜃, 𝑡) =
+∑
+𝑏+
+(")(𝜖, 𝑡)𝑃+(cos𝜃)
+,-
++./
+ where 𝑃+(cos𝜃) are Legendre polynomials and 𝑁 = 3 is the total 
+number of photons absorbed to reach the continuum from the ground state: the pump-
+induced excitation involves 2 photons while ionization consists in the absorption of one NIR 
+probe photon. 𝑏/
+(")(𝜖, 𝑡) corresponds to the total (angle-integrated) photoionization cross 
+section. In the case of an achiral sample, the PAD is symmetric with respect to the light 
+propagation axis so that the 𝑆(")(𝜖, 𝜃, 𝑡) expansion is restricted to even 𝑛’s. For chiral 
+samples, the asymmetric contribution to the photoelectron yield emerges from the additional 
+𝑏+
+(") amplitude coefficients with odd 𝑛.  Besides 𝑃𝐸𝐶𝐷(𝜖, 𝜃, 𝑡), it is convenient to introduce 
+an angularly-integrated quantity to characterize the whole chiroptical response at fixed 
+kinetic energy. Defining it as the difference of electrons emitted in the forward and backward 
+hemispheres for ℎ = +1, normalized to the average number of electrons collected in one 
+
+6 
+ 
+hemisphere, we obtain the so-called multiphotonic (MP)-PECD 35, 𝑀𝑃 − 𝑃𝐸𝐶𝐷(𝜖, 𝑡) =
+2𝛽0
+(*0)(𝜖, 𝑡) −
+0
+, 𝛽1
+(*0)(𝜖, 𝑡) +
+0
+2 𝛽3
+(*0)(𝜖, 𝑡) where 𝛽+
+(*0)(𝜖, 𝑡) =
+4&
+("#)(%,()
+4'
+("#)(%,().  
+In our analysis the 𝑏3 coefficient has been omitted due to its negligible contribution. Figure 2 
+shows the multiphotonic PECD as a function of the delay 𝑡 for three 𝜖 ranges defined by the 
+time- and energy-resolved map of the 𝛽0 (𝜖, 𝑡) coefficient (see Fig. S2d of the supplement). 
+The results are shown for (S)-ML but the measurements were also performed in (R)-ML – the 
+comparison of enantiomeric responses is shown in Fig. S4 of the supplement. For the lowest 
+𝜖 range between 25-100 meV (a), it is striking that the photoelectron emission asymmetry 
+reverses in the laboratory frame in ~7 fs. A clear modulation of the asymmetry remains over 
+few tens of fs, which is also observed at higher 𝜖  between 100-300 meV (b) and 300-720 meV 
+(c). Importantly, such a clear modulation hardly shows up in the time-resolved 𝛽/ (𝜖, 𝑡) 
+coefficient associated to the photoelectron yield (see Section 1.2 of the supplement). This 
+latter observable provides a limited signal to noise ratio compared to the odd coefficients 
+𝛽,+*0(𝜖, 𝑡)	evaluated from differential measurements, highlighting the capabilities of TR-
+PECD over conventional photoelectron spectroscopy. In the following, we aim at identifying 
+the origin of the modulation, taking into account that the timescale possibly involves 
+electronic and/or nuclear degrees of freedom. 
+ 
+We modeled the experiment including both the UV photoexcitation and the NIR 
+photoionization steps. A detailed description of the theoretical method is provided in Section 
+2 of the supplement. Briefly, we treat the photoexcitation and photoionization as sequential 
+perturbative processes. The electronic spectrum of ML and the two-photon excitation 
+amplitudes are obtained via large-scale time-dependent density functional theory 36. 
+
+7 
+ 
+Ionization from the excited states is described using the continuum multiple scattering X𝛼 
+approach 37,38. Importantly, we work within the frozen-nuclei approximation.  
+We illustrate the results of our calculations in Figure 3. The pump pulse leads to the 
+population of excited states stemming mainly from excitation of the highest occupied 
+molecular orbital (HOMO) of the ML ground state (see section 2.1 of the supplement). We 
+thus present in panel (a) the two-photon excitation cross sections for the states associated to 
+almost pure HOMO excitation and leading, through subsequent photoionization by the probe 
+pulse, to the emission of photoelectrons with kinetic energies 𝜖 = 250 and 500 meV, 
+respectively. These 𝜖 values are representative of the second and third energy ranges 
+discriminated in the experimental data, respectively – the case of low energy photoelectron 
+dynamics (𝜖 = 50 meV) is discussed in Section 2.2.3 of the supplement. Including excitations 
+from the HOMO states of panel (a) in the dynamical calculations yield the time-resolved MP-
+PECD displayed in panels (b) and (d). The calculations are started at 𝑡 = 10 fs to ensure no 
+temporal overlap between the pump and probe pulses. The computed asymmetry presents 
+clear modulations as a function of the pump-probe delay. The power spectra of the MP-PECD 
+signals, obtained by Fourier analysis, are compared to their experimental counterparts in 
+panels (c) and (e). An excellent agreement is found at 𝜖 = 250 meV where the oscillatory 
+pattern of the MP-PECD is traced back to the pump-induced coherent superposition of 3d and 
+4p Rydberg states highlighted in Figure 3(a). This coherent superposition leads to quantum 
+beatings with ~15 fs period, associated to ~300 meV frequency, which survive long after the 
+pump pulse vanishes. We note that the most stable geometries of methyl-lactate do not 
+possess any vibrational mode in the vicinity of 2200 cm-1 (~15 fs) 39. Similarly, the coherent 
+superposition of 4p and 4d,f Rydberg states explains the oscillatory feature of the MP-PECD 
+at 𝜖 = 500 meV. A small mismatch of ~60 meV is observed between the experimental and 
+
+8 
+ 
+theoretical power spectra in Figure 3(e). This is typical of the error made in quantum 
+chemistry computations of excited state energies. The convergence of our results with 
+respect to the number of excited states included in the dynamical calculations is illustrated in 
+Section 2.2.3 of the supplement.  
+ 
+In our fixed-nuclei description, the electron coherences leading to oscillatory MP-PECD do not 
+vanish and even lead to an overestimation of the MP-PECD amplitude at all 𝑡. In the 
+experiments, these coherences are found to decrease as time elapses, through decreasing 
+MP-PECD amplitudes (Fig. 2), but are preserved on a relatively long timescale, up to ~40 fs. 
+The time it takes for decoherence to occur in photoionized 40–47 and photoexcited 18,48 
+molecules is currently the subject of extensive investigations. Electronic wavepackets are 
+subject to three main decoherence sources: (i) the decrease of the overlap between nuclear 
+wavepackets on different electronic states, (ii) the dephasing of the different wavepacket 
+components, and (iii) the change of electronic state populations induced by non-adiabatic 
+couplings 42. Describing the coupled electron and nuclear dynamics in an energy range where 
+tens of electronic states lie (see Figure S11 of the supplement) is beyond state-of-the-art 
+theoretical approaches. Therefore, we alternatively performed classical molecular dynamics 
+calculations on the ground state of cationic ML to which all the HOMO Rydberg states involved 
+in the pump-probe dynamics correlate upon ionization. Two main classes of trajectories 
+showed up, converging towards two different isomeric forms of the ML cation. Within each 
+class of trajectories, the Rydberg state energies of neutral ML were found to remain 
+approximately parallel to each other and to the ML cation along the reaction path (see Figure 
+S11 of the supplement). This favors the overlap of the nuclear wavepackets associated to 
+different electronic Rydberg states over an extended time duration and thus minimizes the 
+
+9 
+ 
+role of decoherence mechanisms (i) and (ii). This also strengthens the frozen-nuclei 
+assumption with regard to the description of electronic quantum beatings which are dictated 
+by energy differences and should therefore remain basically the same from 𝑡 = 0 fs onwards. 
+The most probable source of decoherence in the present investigation is non-adiabatic 
+dynamics, not only between the states populated by the pump but also with the lower-lying 
+states reached by internal conversion soon after the prompt excitation. This information is 
+encoded in the ~40 fs lifetime of the transient 𝑏/ signal shown in Fig. S3 of the supplement. 
+ 
+Our findings on fast evolving MP-PECD and related chiroptical switching, driven by long-
+lasting electronic coherences, can be highlighted by reducing the number of transient excited 
+states to two. In the case of 𝜖 = 250 meV, the main MP-PECD oscillation frequency is 291 meV 
+and corresponds to the coherent superposition of the 3d and 4p Rydberg states, respectively 
+located at 𝐸15 = 8.834 eV and 𝐸26 = 9.120 eV on the energy scale (Fig. 3a). For a single 
+molecular orientation 𝑹H, the excited electron wavepacket thus reads, at time 𝑡 after the pump 
+pulse vanishes, ΦJ𝑹H, 𝒓, 𝑡L = ∑
+𝐴7(𝑹H)Ψ7(𝒓)exp(−𝑖𝐸7𝑡/ℏ)
+7.15,26
+ where 𝐴8J𝑹HL are the two-
+photon transition amplitudes associated to the excited states Ψ7(𝒓). The associated electron 
+density can be partitioned as 
+                            𝜌J𝑹H, 𝒓, 𝑡L = 𝜌8+9:"J𝑹H, 𝒓L + 𝜌9;:<<J𝑹H, 𝒓LcosVJ𝐸26 − 𝐸15L𝑡/ℏW                      (1) 
+where 𝜌8+9:"J𝑹H, 𝒓L = 𝐴15
+, J𝑹HLΨ15
+, (𝒓) + 𝐴26
+, J𝑹HLΨ26
+, (𝒓), with 𝑹H such that 𝐴8J𝑹HL ∈ ℝ, and 
+𝜌9;:<<J𝑹H, 𝒓L = 2𝐴15J𝑹HL𝐴26J𝑹HLΨ15(𝒓)Ψ26(𝒓). Figure 4(a) shows the coherent part  
+𝜌J𝑹H, 𝒓, 𝑡L − 𝜌8+9:"J𝑹H, 𝒓L of the electron density, oscillating back-and-forth along the 
+molecular structure with a period 𝑇 = 2𝜋ℏ/(𝐸26 − 𝐸15) of 14.4 fs. Ionization of the 3d and 
+
+10 
+ 
+4p state superposition leads, after averaging over the orientations 𝑹H, to the total 
+photoelectron yield which can be decomposed similarly to (1): 
+                                𝑏/
+(±0)(𝜖, 𝑡) = 𝑏/(&)*+
+(±0) (𝜖)	+	𝑏/),*--
+(±0) (𝜖)cosVJ𝐸26 − 𝐸15L𝑡/ℏW.                           (2) 
+The computed yield is presented in Figure 4(b) for 𝜖 = 250	meV, showing how the coherent 
+state superposition leading to 	𝑏/),*--
+(±0) (𝜖)cosVJ𝐸26 − 𝐸15L𝑡/ℏW	modulates the incoherent sum 
+𝑏/(&)*+
+(±0) (𝜖) of individual cross sections. The unnormalized MP-PECD can in turn be written as: 
+    𝑀𝑃-𝑃𝐸𝐶𝐷(𝜖, 𝑡) = 𝑀𝑃-𝑃𝐸𝐶𝐷8+9:"(𝜖)	+	𝑀𝑃-𝑃𝐸𝐶𝐷9;:<<(𝜖)cos ]
+>?./)?01@(
+ℏ
+− Δ𝜙`             (3) 
+where the additional phase Δ𝜙 arises from the interference of the state-selective continuum 
+partial wave amplitudes building the asymmetry of the photoelectron yield (see the 
+supplement) – as usual, this interference is washed out at the level of the total photoelectron 
+yield 37,49.  The temporal evolution of the unnormalized two-state MP-PECD is shown in Figure 
+4(c) from which we extract the time delay	Δ𝑡 = 1.8 fs associated to Δ𝜙 = 0.79 rad. The MP-
+PECD reverses sign within one period of the oscillation since the asymmetries of single 3d- 
+and 4p-mediated pathways, contributing to the incoherent MP-PECD (dashed red line in Fig. 
+4c) around which the coherent part oscillates, are opposite for 𝜖 = 250	meV. Such a 
+chiroptical switching depends not only on the transient bound resonances but also on the 
+dichroism encoded by ionization. In this respect, we note that a photoexcitation electron 
+circular dichroism configuration 50 in which molecules are photoexcited by a circularly 
+polarized pump pulse and subsequently ionized with a linearly polarized probe would reduce 
+the degrees of freedom to only the transient bound resonances. 
+ 
+In conclusion, we have shown how the prompt creation of an electronic wavepacket by a few-
+fs UV pulse dictates the chiral properties of methyl-lactate in the first instants following 
+
+11 
+ 
+photoexcitation, allowing for ultrafast chiroptical switching. The high sensitivity of TR-PECD 
+that allows us to track electronic coherences is intrinsic to its differential, background-free 
+nature. A complete modeling of the experiment identifies how the chiral character of Rydberg 
+states contributes to chiroptical switching and additional trajectory calculations describe how 
+the electron dynamics driving the effect can preserve the coherence over several tens of 
+femtoseconds. The experimental results demonstrate that our spectroscopic method can 
+provide insights on the nature of chiral interactions at the few-fs timescale, and notably on 
+the role of the primary electron dynamics in the light-induced chiral response of complex 
+molecular systems such as chiral biomolecules and organometallic complexes. Offering a 
+route to investigate the fundamental origin of chiral recognition that is ubiquitous in 
+biological phenomena 51, the possibility to control the photoelectron emission direction in the 
+laboratory frame also offers the potential to engineer petahertz switching devices based on 
+chiral interactions. Most importantly, these results provide new perspectives towards the 
+possibility to achieve enantio-sensitive charge-directed reactivity in neutral molecules to 
+steer the outcome of photophysical and photochemical reactions 9. 
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+Golubev, N. v., Begušić, T. & Vaníček, J. On-the-Fly Ab Initio Semiclassical Evaluation 
+of Electronic Coherences in Polyatomic Molecules Reveals a Simple Mechanism of 
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+Physics 14, 484–489 (2018). 
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+gas phase. Angewandte Chemie International Edition 47, 6970–6992 (2008). 
+  
+
+15 
+ 
+ 
+ 
+Fig. 1: Light-induced chiral dynamics of methyl-lactate. (a) A few-femtosecond linearly polarized UV pulse creates an 
+electronic wavepacket of Rydberg states via 2-photon absorption. The red and blue structure illustrates a sketch of a time-
+evolving Rydberg wavepacket in the neutral molecule that is probed by a time-delayed circularly polarized NIR pulse. The 
+probing step leads to the ejection of an electron along the light propagation axis defined by the vector k. The large bandwidth 
+of the probe pulse allows two or more Rydberg states to reach the same electron continuum state and thus probe their 
+interference. (b) The resulting photoelectron angular distribution is recorded by a velocity map imaging spectrometer. For 
+each time delay, an image is recorded for both left (LCP) and right circular polarization (RCP) of the probe pulse. The 
+differential image PECD(ϵ,θ,t) defined in the main text is shown for time delays of 5, 11,17 and 26 fs for photoelectrons with 
+kinetic energies from 25 to 300 meV along the radial coordinate.  
+
+5 fs
+11 fs
+17 fs
+26 fs delay16 
+ 
+ 
+ 
+Fig. 2: Energy-resolved analysis. Temporal evolution of the MP-PECD in (S) - methyl-lactate for three different kinetic energy 
+ranges of photoelectrons: 25-100 meV (a), 100-300 meV (b) and 300-720 meV (c). The standard error of the mean over 5 
+measurements is shown by the shaded areas and the solid blue lines show the fit of the oscillations from t = 0 fs (see the 
+corresponding Fourier analysis in Fig. 3d,e). The change of sign in (a) identifies a reversal of the photoelectron emission 
+direction in the laboratory frame, generating a net photoelectron current 𝑗!
+"! along the light propagation direction k.   
+
+25-10
+20
+60
+(b)
+100-300 meV
+20
+20
+40
+60
+(c)
+300-720 meV
+2017 
+ 
+ 
+ 
+Fig. 3: Modelling of the experiment. (a) Two-photon absorption (TPA) cross sections for the excited states stemming from 
+almost pure HOMO excitation. The cross sections have been convoluted with the UV-pump intensity squared. The blue and 
+green curves correspond to the spectral probe intensity, down-shifted in energy to elicit the transient Rydberg states leading 
+to photoelectrons with energies 𝜖 = 250 and 𝜖 = 500 meV through ionization by one photon centred at frequency 
+	𝜔 = 1.75 eV. (b) Calculated MP-PECD for photoelectrons with 𝜖 = 250 meV (green) compared to the experiment (blue). 
+The calculations start at 𝑡 = 10 fs corresponding to the end of the pump-probe overlap region (yellow area). (c) 
+Corresponding power spectra from a Fourier analysis. The frequency axis is displayed for beatings of excited states with an 
+energy spacing between 150 meV (27.6 fs) and 500 meV (8.3 fs). The main peak from the computed curve is at 291 meV 
+(14.2 fs). The power spectrum of the experimental data shows a peak frequency at 280 meV (14.8 fs) and was evaluated up 
+to 𝑡 = 35  fs where the oscillations vanish. (d) Calculated MP-PECD for photoelectrons with 𝜖 = 500 meV (green) compared 
+to the experiment (blue). (e) Corresponding power spectra with a central component at 269 meV (15.4 fs) for the computed 
+curve. The power spectrum of the experimental data is shown with a central frequency at ~329 meV (12.6 fs). 
+
+20
+TPA cross section (a.u.)
+(a)
+[1-NIR] 250 mev [I1-NIR] 500 meV
+15
+4df
+10
+3d
+5
+4p
+0
+8.8
+9.0
+9.2
+9.4
+9.6
+Energy (eV)
+15
+30
+(a.u.)
+(b)
++- Experiment
+(c)
+Exp.
+Th.
+Theory (250 meV)
+MP-PECD (%)
+10
+20
+MP-PECD (%)
+0.8
+0.6
+5
+10
+0.4
+0
+0.2
+10
+、
+200 300 400 500
+Frequency (meV)
+10
+40
+(a.u.)
+(d)
+Experiment
+e)
+Exp
+Th.
+Theory (500 meV)
+MP-PECD (%)
+MP-PECD (%)
+ spectrum
+0.8
+5
+20
+0.6
+0
+0.4
+-5
+FT
+-10
+40
+T
+0
+0
+10
+20
+30
+40
+50
+60
+200 300 400 500
+Delay (fs)
+Frequency (meV)18 
+ 
+ 
+Fig. 4: Electron-driven dynamics in the case of quantum beating monitored by 3d and 4p states. (a) Temporal evolution of 
+the coherent part of the electron density over one period of the quantum beating between the 3d and 4p states (see equation 
+(1) of the text). (b) Photoelectron yield as a function of the pump-probe delay for 𝜖 = 250 meV. The quantum beating leads 
+to an oscillatory behavior of the yield which is in phase with the variation of the electron density shown in (a), as expected 
+from equations (1) and (2). (c) MP-PECD as a function of the pump-probe delay for 𝜖 = 250 meV. The dichroism is delayed 
+by 𝛥𝜙 = 0.79 rad (1.8 fs) with respect to the variation of the electron density because of the interferences between the 
+continuum partial wave amplitudes (see equation (3)).  
+METHODS 
+ 
+Experimental setup 
+The experiments were carried out with a 1 kHz titanium:sapphire laser (FemtoPower, Spectra- 
+Physics), delivering 25-fs, 12-mJ pulses at 800 nm. 5.6 mJ were used for spectral broadening 
+in a 2.3-m long hollow-core fiber (few-cycle inc.) filled with a pressure gradient of helium gas. 
+The fiber setup seeds an all-vacuum Mach-Zehnder-like interferometer with 5-fs near-
+infrared (NIR) pulses. One arm is used for the generation of the UV-pump pulse via third- 
+harmonic generation in neon gas (Fig. S1a). A pair of silicon superpolished substrates (Gooch 
+& Housego) is used at Brewster angle to attenuate the residual part of the NIR driving field by 
+3 orders of magnitude while reflecting ~16% of the UV radiation (50 nJ). In the second arm of 
+the interferometer, the remaining part of the NIR beam is focused to the experimental region 
+by a toroidal mirror (f = -900 mm) followed by a motorized zero-order quarter-waveplate 
+(B. Halle) in order to control the helicity of the circularly polarized probe pulses (16 μJ), with 
+
+10.0 fs
+13.6 fs
+17.2 fs
+20.8 fs
+24.4 fs
+7.2 fs
+MP-PECDcross
+MP-PECDincoh19 
+ 
+an estimated intensity of 5×1012 Wcm−2 (Fig. S1b). Liquid (S)-methyl-lactate (97% 
+enantiomeric excess, Sigma-Aldrich) was evaporated and transported to a velocity map 
+imaging spectrometer to measure the photoelectron angular distribution as a function of the 
+pump-probe time delay. 
+Computation of TR-PECD 
+ 
+At time 𝑡 after the pump pulse vanishes, the electron wavepacket formed in a ML molecule 
+whose orientation in the laboratory frame is characterized by 𝑹H reads 
+ΦJ𝑹H, 𝒓, 𝑡L = ∑ 𝒜8J𝑹HLΨ8(𝒓)exp(−i𝐸8𝑡/ℏ)
+8
+, 
+where Ψ8(𝒓) are excited states with energies 𝐸8 and two-photon absorption amplitudes from 
+the ground state 𝒜8J𝑹HL. These states, energies and transition amplitudes, have been 
+obtained by large-scale TDDFT 36 calculations, detailed in Section 2.1 of the supplement. In 
+the spectral region spanned by the pump pulse, most of the excited states have a Rydberg 
+character and stem from the excitation of the ML HOMO (see Fig. S5 of the supplement).  
+The absorption of one NIR photon of the probe pulse leads to the ejection of a photoelectron 
+with wavevector 𝒌′H  in the molecular frame. The associated ionization dipole is 
+𝒅𝒌C
+",D:EJ𝑹H, 𝑡L = f 𝒜8J𝑹HLg𝐼0)-FG(𝜔8) < Ψ𝒌C
+())|𝒆𝒉
+m. 𝒓|Ψ8 > exp(−i𝐸8𝑡/ℏ)
+8
+ 
+where 𝐼0)-FG(𝜔8) is the spectral intensity of the probe pulse at frequency 
+ 𝜔8 = 𝑘C,/2 + 𝐼6 − 𝐸8, with 𝐼6 the ML ionization potential, 𝒆𝒉
+m is the circular polarization of 
+the probe pulse (ℎ = ±1) and Ψ𝒌C
+()) is the ingoing scattering state associated to the electron 
+ejected in the continuum. Neither the scattering state nor the excited states explicitly depend 
+on 𝑡 since the calculations are made assuming that the nuclei remain frozen at their 
+equilibrium locations at all 𝑡 (see Section 2.3 of the supplement). Ψ𝒌2
+())(𝒓)	is obtained in the 
+
+20 
+ 
+framework of the X𝛼 approximation for the exchange electron interaction 37,38, detailed in 
+Section 2.2.2 of the supplement.  
+Rotating the ionization dipole into the laboratory frame allows us to define the orientation-
+averaged differential ionization cross section as 
+𝑑𝜎s(")
+𝑑Ω𝒌
+(𝑘, 𝜃, 𝜑, 𝑡) ∝ w 𝑑𝑹H|𝒅𝒌
+",EI4J𝑹H, 𝑡L|, 
+where 𝑘 = 𝑘′ and (𝜃, 𝜑) are the spherical angles characterizing the direction 𝒌H of electron 
+ejection in the laboratory frame -- 𝜃 is defined with respect to the pulse propagation direction 
+𝒛y. While the cross section can be put in the closed form  
+5JK(+)
+5L𝒌 (𝑘, 𝜃, 𝜑, 𝑡) = ∑
+∑
+𝑏<,,8
+(")(𝑘, 𝑡)𝑌<
+,8(𝜃, 𝜑)
+,
+8.),
+M
+<./
+, 
+where 𝑌<
+,8(𝜃, 𝜑) are spherical harmonics, we show in section 2.2.2 of the supplement that 
+the MP-PECD is 
+𝑀𝑃 − 𝑃𝐸𝐶𝐷(𝜖, 𝑡) =
+0
+4','
+("#)(%,() {2√3𝑏0,/
+(*0)(𝜖, 𝑡) −
+√O
+, 𝑏1,/
+(*0)(𝜖, 𝑡) +
+√00
+2 𝑏3,/
+(*0)(𝜖, 𝑡)}. 
+where 𝜖 =
+ℏ5P5
+,D , with 𝑚 the electron mass, is the photoelectron kinetic energy. The 𝑏<,,8
+(") 
+coefficients basically depend on partial-wave ionization amplitudes weighted by the primary 
+excitation factors, as shown in Section 2.2.2 of the supplement.  
+ 
+ 
+ 
+
+21 
+ 
+ACKNOWLEDGEMENTS 
+We thank F. Remacle and O. Smirnova for their constructive feedback and K. Pikull for the 
+excellent technical support. 
+AUTHOR CONTRIBUTIONS 
+V.W., E.B., V.B., Y.M., B.P. and F.C. conceived the experiment. V.W., E.B., E.P.M., L.C., S.R., K.S. 
+and A.T. performed the experiments. V.W., E.B. and Y.M., carried out the data analysis. 
+M.C.H., N.B.A. and B.P. calculated the molecular and electronic properties of methyl-lactate. 
+B.P. performed the PECD calculations. M.C.H. performed the classical trajectory simulations. 
+V.W., Y.M., B.P. and F.C. drafted the manuscript. All authors contributed to the discussion of 
+the results and the editing of the manuscript. 
+COMPETING INTERESTS 
+The authors declare no competing interests. 
+FUNDING 
+We acknowledge financial support from the European Research Council under the ERC-2014-
+StG STARLIGHT (grant no. 637756), European Union’s Horizon 2020 research and innovation 
+program No. 682978 – EXCITERS, the German Research Foundation (DFG)—SFB-925—project 
+170620586 and the Cluster of Excellence Advanced Imaging of Matter (AIM). 
+DATA AVAILABILITY 
+The data that support the findings of this study are available from the corresponding author 
+upon reasonable request. 
+CODE AVAILABILITY 
+The code used for the simulations contained in this study is available from the corresponding 
+authors upon reasonable request. 
+ 
+

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+Friend-training: Learning from Models of Different but Related Tasks
+Mian Zhang†∗, Lifeng Jin⋄, Linfeng Song⋄, Haitao Mi⋄, Xiabing Zhou† and Dong Yu⋄
+†Soochow University, Suzhou, China
+[email protected], [email protected]
+⋄Tencent AI Lab, Bellevue, WA, USA
+{lifengjin,lfsong,haitaomi,dyu}@tencent.com
+Abstract
+Current self-training methods such as standard
+self-training, co-training, tri-training, and oth-
+ers often focus on improving model perfor-
+mance on a single task, utilizing differences in
+input features, model architectures, and train-
+ing processes. However, many tasks in natural
+language processing are about different but re-
+lated aspects of language, and models trained
+for one task can be great teachers for other re-
+lated tasks. In this work, we propose friend-
+training, a cross-task self-training framework,
+where models trained to do different tasks are
+used in an iterative training, pseudo-labeling,
+and retraining process to help each other for
+better selection of pseudo-labels.
+With two
+dialogue understanding tasks, conversational
+semantic role labeling and dialogue rewriting,
+chosen for a case study, we show that the
+models trained with the friend-training frame-
+work achieve the best performance compared
+to strong baselines.
+1
+Introduction
+Many different machine learning algorithms, such
+as self-supervised learning (Mikolov et al., 2013;
+Devlin et al., 2019; Liu et al., 2021), semi-
+supervised learning (Yang et al., 2021) and weakly
+supervised learning (Zhou, 2018), aim at using un-
+labeled data to boost performance. They have been
+of even greater interest recently given the amount
+of unlabeled data available. Self-training (Scudder,
+1965) is one semi-supervised learning mechanism
+aiming to improve model performance through
+pseudo-labeling and has been successfully ap-
+plied to computer vision (Lee et al., 2013; Chen
+et al., 2021), natural language processing (Dong
+and Schäfer, 2011; Bhat et al., 2021) and other
+fields (Wang et al., 2019; Kahn et al., 2020).
+The main challenge of self-training is how to
+select high-quality pseudo-labels.
+Current self-
+training algorithms mainly focus on a single task
+∗Work done when interning at Tencent AI Lab.
+predicate: 喜欢 (like)
+arg-0: 
+arg-1: 宫崎骏(Hayao Miyazaki)
+知道久石让吗？
+(Do you know Joe Hisaishi?)
+我很喜欢他。
+(I like him so much.)
+知道啊，宫崎骏的很多电影配
+乐都是久石让的，比如《幽灵
+公主》。
+(Yes, I do. Many of Hayao 
+Miyazaki’s movie soundtracks 
+are composed by Hisaishi, such 
+as Princess Mononoke).
+cross-task supervision
+DR
+CSRL
+context
+current utterance
+rewritten utterance
+predicate-arguments
+我很喜欢久石让。
+(I like Joe Hisaishi so much.)
+Figure 1: An example of cross-task supervision be-
+tween a CSRL parser and a DR system in a di-
+alogue.
+久 石 让( Joe Hisaishi ) from the rewrit-
+ten utterance provides cross-task supervision to 宫
+崎骏( Hayao Miyazaki ), the predicted arg-1 of 喜
+欢(like) from the CSRL parser, while 我( I ) to the pre-
+dicted arg-0.
+when assessing the quality of pseudo-labels and suf-
+fer from gradual drifts of noisy instances (Zhang
+et al., 2021). This work is motivated by the observa-
+tion that learning targets of tasks represent different
+properties of the inputs, and some properties are
+shared across the tasks which can be used as super-
+vision from one task to another. Such properties
+include certain span boundaries in dependency and
+constituency parsing, and some emotion polarities
+in sentiment analysis and emotion detection. Two
+dialogue understanding tasks, conversational se-
+mantic role labeling (CSRL) and dialogue rewriting
+(DR), are also such a pair, with shared properties
+such as coreference and zero-pronoun resolution.
+As shown in Figure 1, the rewritten utterance can
+be used to generate cross-task supervision to the
+arguments of predicate 喜欢(like). We leverage the
+cross-task supervision from friend tasks – different
+but related tasks – as a great criterion for assessing
+the quality of pseudo-labels.
+In this work, we propose friend-training, the
+first cross-task self-training framework. Compared
+to single-task self-training, friend-training exploits
+supervision from friend tasks for better selec-
+arXiv:2301.13683v1  [cs.CL]  31 Jan 2023
+
+tion of pseudo-labels. To this end, two novel mod-
+ules are proposed: (1) a translation matcher, which
+maps the pseudo-labels of different tasks for one
+instance into the same space and computes a match-
+ing score representing the cross-task matching de-
+gree of pseudo-labels from different tasks; (2)
+an augmented (instance) selector, which leverages
+both the confidence of pseudo-labels from task-
+specific models and the matching score to select
+instances with pseudo-labels of high quality as new
+training data. We choose CSRL and DR as friend
+tasks to conduct a case study for friend-training,
+and specify the translation matcher and augmented
+selector for friend-training between these tasks. Ex-
+perimental results of domain generalization and
+few-shot learning show friend-training surpasses
+both classical and state-of-the-art semi-supervised
+learning algorithms by a large margin. To summa-
+rize, contributions from this work include:
+• We propose friend-training, the first cross-task
+self-training framework which exploits super-
+vision from friend tasks for better selection of
+pseudo-labels in the iterative training process.
+• We provide specific modeling of friend-training
+between CSRL and DR, with a novel translation
+matcher and a novel augmented selector.
+• Extensive experiments with CSRL and DR
+demonstrate the effectiveness of friend-training,
+outperforming several strong baselines.
+2
+Related Work
+Self-training Self-training (Scudder, 1965; An-
+gluin and Laird, 1988; Abney, 2002; Lee et al.,
+2013) is a classical semi-supervised learning frame-
+work (Chapelle et al., 2009) which has been widely
+explored in recent years. The general idea of self-
+training is to adopt a trained model to pseudo-label
+easily acquired unlabeled data and use them to
+augment the training data to retrain the model it-
+eratively. This paradigm shows promising effec-
+tiveness in a variety of tasks: including text classi-
+fication (Mukherjee and Awadallah, 2020; Wang
+et al., 2020a), image classification (Xie et al., 2020;
+Zoph et al., 2020), machine translation (He et al.,
+2020) and model distillation (Mukherjee and Has-
+san Awadallah, 2020).
+Co-training (Blum and
+Mitchell, 1998) and tri-training (Zhou and Li, 2005)
+are similar iterative training frameworks to self-
+training but with a different number of models or
+considering different views of the training data,
+both of which see wide adoption in NLP (Mihalcea,
+2004; McClosky et al., 2006; Wan, 2009; Li et al.,
+2014; Caragea et al., 2015; Lee and Chieu, 2021;
+Wagner and Foster, 2021). These frameworks aim
+at improving performance with multiple models
+trained on one task, without directly leveraging the
+benefit of supervision from related tasks.
+Multi-task Learning Multi-task learning (Caru-
+ana, 1997; Yang et al., 2021) seeks to improve the
+learning performance of one task with the help of
+other related tasks, among which two lines of work
+are related to ours: (1) semi-supervised multi-task
+learning (Liu et al., 2007; Li et al., 2009) combines
+semi-supervised learning and multi-task learning.
+Liu et al. (2007) exploited unlabeled data by ran-
+dom walk and used a task clustering method for
+multi-task learning. Li et al. (2009) integrated ac-
+tive learning (MacKay, 1992) with the model in
+Liu et al. (2007) to retrieve data that are most infor-
+mative for labeling. Although these works tried to
+utilize unlabeled data to enhance multi-task learn-
+ing, our work differs from them in incorporating su-
+pervised signals among tasks to select high-quality
+pseudo-labels for updating models, which is an iter-
+ative training process without additional human an-
+notation. (2) Task grouping (Kumar and III, 2012;
+Standley et al., 2020; Fifty et al., 2021) aims to
+find groups of related tasks and employs multi-task
+learning to each group of tasks, with one model for
+each group. Our work focuses on training single-
+task models, but task grouping techniques can be
+used to look for possible friend tasks.
+Conversational Semantic Role Labeling CSRL
+is a task for predicting the semantic roles of
+predicates in a conversational context.
+Wu
+et al. (2021) leveraged relational graph neural net-
+works (Schlichtkrull et al., 2018) to model both the
+speaker and predicate dependency, achieving some
+promising results. However, the current dataset (Xu
+et al., 2021) for CSRL is limited to mono-domain.
+High-quality labeled data for new domains are
+needed to empower more applicable CSRL models.
+Dialogue Rewriting DR is commonly framed as a
+sequence-to-sequence problem which suffers large
+search space issue (Elgohary et al., 2019; Huang
+et al., 2021). To address it, Hao et al. (2021) cast
+DR to sequence labeling, transforming rewriting
+an utterance as deleting tokens from an utterance
+or inserting spans from the dialogue history into an
+utterance. Jin et al. (2022) improved the continuous
+span issue in (Hao et al., 2021) by first generating
+
+multiple spans for each token and slotted rules and
+then replacing a fixed number rules with spans.
+3
+Friend-training
+Friend-training is an iterative training framework to
+jointly refine models of several friend tasks. Differ-
+ent from self-training, friend-training injects cross-
+task supervision into the selection of pseudo-labels.
+We first briefly describe self-training before pre-
+senting friend-training.
+3.1
+Self-training
+Classic self-training aims at iteratively refining a
+model of a single task by using both labeled data
+and a large amount of unlabeled corpus. At each
+iteration, the model first assigns the unlabeled data
+with pseudo-labels. Subsequently, a set of the unla-
+beled instances with pseudo-labels is selected for
+training, presumably with information for better
+model generalization. Then cross-entropy of model
+predictions and labels on both gold and pseudo-
+labeled data is minimized to update the model:
+L =
+N
+�
+i=1
+yi log yi
+pi
++ λ
+N′
+�
+i=1
+y′
+i log y′
+i
+p′
+i
+,
+(1)
+where the left term is the loss for the labeled data
+and the right for the unlabeled data while λ is a
+coefficient to balancing them; N(N′) is the number
+of instances, y (y′) is the label and p (p′) is the
+output probability of the model.
+Self-training is usually limited to only one task,
+but there are thousands of NLP tasks already pro-
+posed and many of them are related.
+Models
+trained for one task can be great teachers for other
+related tasks. We explore this cross-task supervi-
+sion in self-training by incorporating two novel
+modules introduced in subsection 3.2.
+3.2
+Friend-training
+For friend-training with two tasks,1 we have two
+classifiers fa and fb trained on two different tasks
+with labeled training sets La and Lb, with expected
+accuracies ηa and ηb, respectively. The two datasets
+are created independently and the prediction tar-
+gets of the two tasks are partially related through
+a pair of translation functions Fa : ˆYa → Σ
+and Fb : ˆYb → Σ, where Σ is the set of possi-
+ble sub-predictions that all possible predictions
+1We focus on the two-friend version of friend-training in
+this work, however, friend-training can easily be extended to
+more than two friends.
+of the two tasks ˆYa and ˆYb can be reduced to.
+| ˆYa| ≥ |Σ|, | ˆYb| ≥ |Σ|.
+We assume that the
+translation functions are general functions with
+the expected probability of generating a translation
+ϵF =
+1
+|Σ|. The translation functions are determin-
+istic and always map the gold labels of the friend
+tasks for the same input to the same translation.
+Both classifiers make predictions on the unla-
+beled set U at iteration k. Some instances Uk
+F
+with pseudo-labels are chosen as new training data
+based on the results of the translation functions,
+φa(x) = Fa(fa(x)) and φb(x) = Fb(fb(x)), and
+some selection criteria, such as total agreement. If
+total agreement is used as the selection criterion,
+the probability of erroneous predictions for fa in
+these instances is
+Prx[fa(x) ̸= f∗
+a(x)|φa(x) = φb(x)]
+=1 − ηaPrx[φa(x) = φb(x)|fa(x) = f∗
+a(x)]
+Prx[φa(x) = φb(x)]
+,
+(2)
+with f∗ being the optimal classifier.
+Because both classifiers are very different due to
+training data, annotation guidelines, models, pre-
+diction targets, etc., being all different, the two
+classifiers are very likely to be independent of each
+other. Under this condition Equation 2 becomes
+1 − ηa(ηb + ϵF(1 − ηb))
+Prx[φa(x) = φb(x)]
+=1 −
+Z
+Z + ηbϵF(1 − ηa) + E ,
+(3)
+where Z = ηa(ηb + ϵF(1 − ηb)) and E = ϵ2
+F(1 −
+ηa)(1 − ηb). We give the detailed derivation of
+Equation 2 and 3 in Appendix A.1. This indi-
+cates that the quality of the picked instances is
+negatively correlated with the number of false pos-
+itive instances brought by the noisy translation
+ηbϵF(1 − ηa), and the number of matching nega-
+tive instances E. When ϵF is minimized by choos-
+ing translation functions with a sufficiently large
+co-domain Σ, the probability of error instances
+chosen when two classifiers agree approaches 0.
+This also indicates that even when 1 − ηa is large,
+i.e. fa performs badly, if the co-domain is large,
+the error rate of the chosen instances can still be
+kept very low.2 As the dependence between the
+2Intuitively, this means independent classifiers trained to
+do different tasks are unlikely to predict the same but wrong
+sub-prediction for a given input, if the sub-prediction includes
+a large number of decisions.
+
+two classifiers grows in training, the probability
+of error instances also increases. When they are
+completely dependent on each other, Equation 2
+becomes 1 − ηa, i.e. classic self-training.
+Based on this formulation, two additional mod-
+ules are needed: (1) a translation matcher that
+maps predictions of two models trained on different
+tasks into the same space and computes a matching
+score; (2) an augmented (instance) selector which
+selects instances with pseudo-labels for the clas-
+sifiers considering both the matching score of the
+translated predictions and the model confidences.
+Translation Matcher Given the prediction of mod-
+els of two friend tasks fa(x) and fb(x), the transla-
+tion matcher M leverages translation functions Fa
+and Fb to get the translated pseudo-labels and com-
+putes a matching score m for the pair of pseudo-
+labels, which represents the similarity of the pair
+in the translation space:
+ma,b = M (Fa(fa(x)), Fb(fb(x))) ,
+(4)
+with total agreement being 1. This matching score
+serves as a criterion for the selection of high quality
+pseudo-labels with cross-task supervision.
+Augmented Selector Apart from pseudo-label sim-
+ilarity, other information about pseudo-label quality
+can be found from model confidence, which self-
+training algorithms specifically utilize, to augment
+matching scores. The augmented selector consid-
+ers both the confidence of the pseudo-labels from
+task models, denoted as {ca, cb}, and the matching
+scores:
+qτ = Sτ(ma,b, cτ),
+(5)
+where qτ ∈ {0, 1} represents the selection result
+of the pseudo-label for task τ ∈ a, b. Therefore,
+instances with low matching scores but high con-
+fidence may also be selected as the training data.
+The complete algorithm is shown in Algorithm 1.
+4
+Friend Training between CSRL and
+DR
+To verify the effectiveness of friend-training, we
+select two dialogue understanding tasks as friend
+tasks to conduct friend-training experiments for a
+case study: conversational semantic role labeling
+(CSRL) and dialogue rewriting (DR). While both
+require skills such as coreference and zero-pronoun
+resolution, the two tasks focus on different proper-
+ties of the dialogue utterance: (1) CSRL focuses
+on extracting arguments of the predicates in the
+Algorithm 1: Two-task friend-training
+Input
+:Labeled data sets for two friend
+tasks, La, Lb; an unlabeled data set
+U; task models fa, fb.
+Output :Refined fa, fb.
+Pre-train fτ with Lτ (τ ∈ a, b);
+while not until the maximum iteration do
+Lu
+a = ∅; Lu
+b = ∅;
+for z in U do
+Generate fa(z), fb(z) and ca, cb;
+ma,b ← Equation 4;
+qa, qb ← Equation 5;
+if qτ = 1 (τ ∈ a, b) then
+Lu
+τ = Lu
+τ + {z, vτ};
+end
+Update fτ with Lτ, Lu
+τ by Equation 1
+(τ ∈ a, b);
+end
+Return fa, fb;
+utterance from the whole dialogue history; (2) DR
+aims to rewrite the last turn of a dialogue to make it
+context-free and fluent by recovering all the ellipsis
+and coreference in the utterance. Figure 2 provides
+an overview of friend-training between the above
+two tasks. Next, we first introduce the task mod-
+els and then specify the translation matcher and
+augmented selector for applying friend-training.
+4.1
+Task Models
+Task Definition A dialogue consists of N tempo-
+rally ordered utterances {u1, ..., uN}. (1) Given
+utterance ut and K predicates {pred1, ..., predK}
+of ut, a CSRL parser predicts spans from the di-
+alogue as arguments for all predicates. (2) A dia-
+logue rewriter rewrites ut to make it context-free
+according to its context {u1, ..., ut−1}.
+Dialogue Encoder We concatenate dialogue con-
+text {u1, ..., ut−1} and the current utterance ut as
+a sequence of tokens {x1, ..., xM} and encode it
+with BERT (Devlin et al., 2019) to get the contex-
+tualized embeddings:
+E = e1, ..., eM = BERT(x1, ..., xM) ∈ RH×M.
+Encoders for CSRL and DR share no parameters,
+but for simplicity, we use the same notation E for
+their outputs.
+Conversational Semantic Role Labeling With
+the contextualized embeddings, we further gener-
+ate predicate-aware utterance representations G =
+
+Unlabeled 
+data
+CSRL Parser
+Dialogue 
+Rewriter
+Translation
+Matcher
+SSRL Parser
+(𝑝𝑟𝑒𝑑1, 𝒜1)
+Translation Matcher
+𝑢1
+𝑢2
+𝑢3
+𝑢3
+′
+𝑢3
+′
+(𝑝𝑟𝑒𝑑2, 𝒜2)
+Augmented 
+Selector
+(𝑝𝑟𝑒𝑑1, ℬ1)
+(𝑝𝑟𝑒𝑑2, ℬ2)
+(𝑝𝑟𝑒𝑑1, 𝒜1)
+(𝑝𝑟𝑒𝑑2, 𝒜2)
+GM()
+Friend-training
+confidence score
+matching score
+geometric mean
+arguments matching
+GM()
+𝑚1
+𝑚2
+𝑚′
+Step1
+Step2
+Step3
+Figure 2: The overview of the friend-training process between CSRL and DR for one dialogue instance which
+has three utterances and the last utterance contains two predicates. Step1: the unlabeled dialogue is labeled by
+the CSRL parser and dialogue rewriter, resulting in predictions of arguments for the predicates (CSRL) and the
+rewritten utterance (DR), respectively. Step2: Pseudo-labels of both tasks are fed into the translation matcher
+to get their matching scores: the translation matcher first conducts sentence-level semantic role labeling (SSRL)
+on the rewritten utterance u′
+3 and then compares the results with those of the CSRL parser for matching scores.
+Step3: The threshold-based augmented selector makes the final decision of whether to add each pseudo-label to
+the training data considering both their confidence and matching scores. Best viewed in color.
+{g1, ..., gM} ∈ RH×M as Wu et al. (2021) by ap-
+plying self-attention (Vaswani et al., 2017) to E
+with predicate-aware masking, where a token is
+only allowed to attend to tokens in the same utter-
+ance and tokens from the utterance containing the
+predicate:
+Maski,j =
+�
+1
+if u[i] = u[j] or u[j] = u[pred],
+0
+otherwise,
+where u[m] denotes the utterance containing token
+xm and u[pred] denotes the one with the predicate.
+The predicate-aware representations are then pro-
+jected by a feed-forward network to get the distri-
+bution of labels for each token:
+Pc = softmaxcolumn-wise(WcG + bc) ∈ RC×M,
+where Wc and bc are learnable parameters and C
+is the number of labels. The labels follow BIO se-
+quence labeling scheme: B-X and I-X respectively
+denote the token is the first token and the inner
+token of argument X, where O means the token
+does not belong to any argument. The output of
+the CSRL parser for K predicates are denoted as
+{A1, ..., AK}, where set Ak containing the argu-
+ments for predk.
+Dialogue Rewriting Following Hao et al. (2021),
+we cast DR as sequence labeling. Specifically, a
+binary classifier on the top of E first determines
+whether to keep each token for in utterance ut in
+the rewritten utterance:
+Pd = softmaxcolumn-wise(WdE + bd) ∈ R2×M,
+where Wd and bd are learnable parameters. Next,
+a span of the context tokens is predicted to be in-
+serted in front of each token. In practice, two self-
+attention layer (Vaswani et al., 2017) are adopted
+to calculate the probability of context tokens being
+the start index or end index of the span:
+Pst = softmaxcolumn-wise(Attnst(E)) ∈ RM×M,
+Ped = softmaxcolumn-wise(Attned(E)) ∈ RM×M,
+where Pst
+i,j (Ped
+i,j) denotes the probability of xi be-
+ing the start (end) index of the span for xj. Then
+by applying argmax to P, we could obtain the start
+and end indexes of the span for each token:
+sst = argmaxcolumn-wise(Pst) ∈ RM,
+sed = argmaxcolumn-wise(Ped) ∈ RM,
+The probability of the span to be inserted in front
+of xm is Pst
+sst
+m,m × Ped
+sed
+m ,m when sst
+m ⩽ sed
+m. When
+sst
+m > sed
+m, it means no insertion. The output of the
+dialogue rewriter for ut is denoted as u′
+t.
+4.2
+Translation Matcher
+To translate the outputs (pseudo-labels) from
+the CSRL parser {A1, ..., AK} and the dialogue
+rewriter u′
+t into a same space, we leverage a nor-
+mal sentence-level semantic role parser with fixed
+
+Step2Step3parameters to greedily extract arguments from the
+rewritten utterance u′
+t for the K predicates, denoted
+as {B1, ..., BK} (Appendix A.5 shows an example).
+So the common target space Σ is the label space of
+CSRL, which is large enough to make the error rate
+of chosen instances keep very low (see the analysis
+in subsection 3.2). The matching score mk ∈ [0, 1]
+for predk is calculated based on the edit distance
+between Ak and Bk:
+mk = 1 −
+dist(⊕Ak, ⊕Bk)
+max(len(⊕Ak), len(⊕Bk)),
+where dist() calculates the edit distance between
+two strings, len() returns the length of a string
+and ⊕Ak denotes the concatenation of arguments
+in set Ak in a predefined order of arguments3
+(empty strings means arguments do not exist). Fur-
+thermore, we obtain the overall matching score
+m′ ∈ [0, 1] for the rewritten utterance u′
+t as fol-
+lows:
+m′ = GM(m1, ..., mK),
+where GM() represents the geometric mean.
+4.3
+Augmented Selector
+The augmented selector selects high-quality
+pseudo-labels according to both their matching
+scores and confidence. For CSRL, we calculate
+the confidence score for each predicate based on
+the output of the softmax layer. Specifically, we
+obtain the confidence of an argument for predk by
+multiplying the probability of its tokens, denoted
+as {ak1, ..., ak|Ak|}. We then use the geometric
+mean of all the confidence of arguments belonging
+to predk as the confidence for predk. The overall
+score sk ∈ [0, 1] for predk is calculated as follows:
+sk = αGM(ak1, ..., ak|Ak|) + (1 − α)mk,
+where hyper-parameter α gives a balance between
+the matching score and the confidence. For DR, we
+multiply the probabilities of spans to be inserted
+and of decisions on whether to keep tokens or not
+as the model confidence of u′
+t, denoted as bt. The
+overall score rt ∈ [0, 1] of u′
+t is as follows:
+rt = βbt + (1 − β)m′,
+where a larger value of hyper-parameter β places
+more importance on the model confidence. α and
+β are set to be 0.2 for both tasks in the experiments.
+3Argument concatenating order: ARG0, ARG1, ARG2, ARG3,
+ARG4, ARGM-TMP, ARGM-LOC, ARGM-PRP
+Pick thresholds are set for sk and rt to control the
+number and quality of selected pseudo-labels. We
+analyze the effects of different values of thresholds
+in subsection 5.4.
+5
+Experiments
+5.1
+Setup
+Datasets We use five dialogue datasets in our ex-
+periments with domains spanning movies, celebri-
+ties, book reviews, products, and social networks.
+For CSRL, we use DuConv (Xu et al., 2021) and
+WeiboCSRL and for DR, REWRITE (Su et al.,
+2019) and RESTORATION (Pan et al., 2019). The
+datasets of the same task differ in domains and
+sizes. WeiboCSRL is a newly annotated CSRL
+dataset for out-of-domain testing purposes. More-
+over, we use LCCC-base (Wang et al., 2020b) as the
+unlabeled corpus, which is a large-scale Chinese
+conversation dataset with 79M rigorously cleaned
+dialogues from various social media. More details
+on the annotation of WeiboCSRL and the proper-
+ties of the datasets could be found in Appendix A.2.
+Experiment Scenarios Our main experiments in-
+volve two scenarios. (1) Domain generalization:
+we use DuConv as the training data in the source
+domain and WeiboCSRL for out-of-domain eval-
+uation, while for DR, REWRITE is used for
+training and RESTORATION for evaluation. (2)
+Few-shot learning: we randomly select 100 cases
+from DuConv and REWRITE as the training data
+for CSRL and DR, respectively, and conduct in-
+domain evaluation, which means models of both
+the tasks are co-trained with only a few samples of
+each task. The unlabeled data for both scenarios
+are 20k dialogues extracted from LCCC-base. Im-
+plementation details are provided in Appendix A.3.
+Evaluation We follow Wu et al. (2021) to report
+precision (Pre.), recall (Rec.), and F1 of the ar-
+guments for CSRL and Hao et al. (2021) to report
+word error rate (WER) (Morris et al., 2004), Rouge-
+L (R-L) (Lin, 2004) and the percent of sentence-
+level exact match (EM) for DR.
+5.2
+Baselines
+We
+compare
+friend-training
+with
+six
+semi-
+supervised training paradigms: two standard tech-
+niques such as standard self-training (SST) (Scud-
+der, 1965) and standard co-training (SCoT) (Blum
+and Mitchell, 1998), as well as four recent methods
+such as mean teacher (MT) (Tarvainen and Valpola,
+2017), cross pseudo supervision (CPS) (Chen
+
+WeiboCSRL
+RESTORATION
+Method
+Pre.
+Rec.
+F1
+R-L
+EM
+WER(⇓)
+Base
+57.97
+54.47
+56.16
+82.78
+25.25
+28.69
+Multitask-Base
+53.66
+54.32
+53.99
+81.68
+22.49
+32.44
+SST (Scudder, 1965)
+60.85
+56.54
+58.62
+85.22
+32.97
+22.22
+MT (Tarvainen and Valpola, 2017)
+58.42
+55.71
+57.03
+83.76
+28.82
+26.49
+CPS (Chen et al., 2021)
+60.34
+52.87
+56.36
+85.60
+32.68
+22.78
+SCoT (Blum and Mitchell, 1998)
+57.33
+54.13
+55.69
+84.51
+29.25
+24.87
+STBR (Bhat et al., 2021)
+60.77
+58.04
+59.38
+85.79
+33.78
+23.30
+STea (Yu et al., 2021)
+60.10
+55.13
+57.50
+85.75
+34.23
+22.17
+FDT (Ours)
+65.29(↑4.44)
+58.63(↑2.09)
+61.78(↑3.16)
+86.82(↑1.60)
+38.22(↑5.25)
+20.31(↑1.91)
+(a) Domain generalization for models trained with DuConv and REWRITE.
+DuConv
+REWRITE
+Method
+Pre.
+Rec.
+F1
+R-L
+EM
+WER(⇓)
+Base
+29.50
+21.90
+25.14
+73.44
+3.60
+39.98
+Multitask-Base
+22.43
+20.63
+21.49
+78.97
+11.70
+40.46
+SST (Scudder, 1965)
+34.16
+27.49
+30.46
+80.93
+27.80
+31.02
+MT (Tarvainen and Valpola, 2017)
+36.32
+30.69
+33.27
+81.66
+33.00
+31.66
+CPS (Chen et al., 2021)
+37.14
+29.47
+32.86
+79.56
+23.30
+32.60
+SCoT (Blum and Mitchell, 1998)
+38.37
+26.15
+31.10
+78.58
+22.31
+33.79
+STBR (Bhat et al., 2021)
+32.37
+25.21
+28.34
+82.37
+29.80
+30.31
+STea (Yu et al., 2021)
+39.34
+28.78
+33.25
+83.04
+31.57
+30.36
+FDT (Ours)
+40.41(↑6.25)
+30.82(↑3.33)
+34.97(↑4.51)
+82.83(↑1.90)
+34.20(↑6.40)
+27.87(↑3.15)
+FDT-S (Ours)
+40.12
+33.41
+36.46
+83.11
+37.10
+26.88
+Fully-trained Base
+69.83
+68.53
+69.17
+89.47
+52.30
+20.54
+(b) Few-shot learning for models trained with DuConv and REWRITE.
+Table 1: Test results for domain generalization and few-shot learning. Base denotes the task models trained with
+data from a single task. Multitask-Base denotes the base model of CSRL and DR sharing the same dialogue
+encoder. Results are averaged across three runs. ⇓ means lower is better. For few-shot learning, performance of
+the base models trained with the full training set from the single task is provided for reference.
+et al., 2021), self-training with batch reweight-
+ing (STBR) (Bhat et al., 2021) and self-teaching
+(STea) (Yu et al., 2021). See Appendix A.4 for
+more details.
+5.3
+Main Results
+Table 1 shows the comparison between friend-
+training (FDT) and the baselines mentioned in sub-
+section 5.2. FDT achieves the best overall perfor-
+mance over the baselines by significant margins
+in both domain generalization and few-shot learn-
+ing scenarios, which demonstrates the effective-
+ness of FDT in different experimental situations
+to utilize large unlabeled corpora. Moreover, we
+show the absolute improvements of FDT over SST
+in parentheses (↑). As we could see, in few-shot
+learning, FDT obtain 4.51 and 3.15 higher abso-
+lute points over SST on F1 of DuConv and WER
+of REWRITE, respectively, than those of domain
+generalization, which are 3.16 and 1.91 points, re-
+vealing that FDT could realize its potential easier
+in few-shot learning. Besides, for few-shot learn-
+ing, we further consider the situation where a full-
+trained base model from the friend task is available,
+denoted as FDT-S. As we could see, when the tar-
+get task is CSRL, FDT-S makes a gain of 1.49
+points on F1 over FDT and when the target task
+is DR, FDT-S outperforms FDT on WER by 0.99
+points and EM by 2.90 points, indicating that more
+reliable supervision from friend task could further
+enhance the few-shot learning of the target task.
+5.4
+Analysis
+In this section, we conduct experiments to analyze
+how selected parameters and settings interact with
+model performance in FDT.
+Pick Thresholds We vary the pick thresholds of
+CSRL and DR in domain generalization scenario
+and track the model performance: we fix the pick
+threshold of the friend task to the best (see Ap-
+pendix A.3) when varying that of the evaluating
+task. As illustrated in Figure 3a, when the thresh-
+olds increase gradually, the models become better
+with higher F1 for CSRL and lower WER for DR.
+We attribute this to wrong pseudo-labels being fil-
+tered out by the augmented selector of FDT. Then
+
+0.1
+0.3
+0.5
+0.7
+0.9
+(a) pick threshold
+55
+56
+57
+58
+59
+60
+F1
+CSRL
+DR
+19
+20
+21
+22
+23
+24
+25
+WER
+(a) The effect of pick thresholds.
+10/10
+30/30
+50/50
+70/70
+90/90
+%train(CSRL/DR)
+40
+44
+48
+52
+56
+60
+F1
+BASE
+STBR
+STea
+FDT
+(b) F1 on CSRL test set.
+10/10
+30/30
+50/50
+70/70
+90/90
+%train(CSRL/DR)
+20
+24
+28
+32
+36
+40
+44
+WER
+BASE
+STBR
+STea
+FDT
+(c) WER on DR test set.
+Figure 3: Sub-figures (b) and (c) show the model performance of the comparing methods with different strengths
+of base models; the dashed horizontal line represents the performance of FDT with a fully trained base model.
+10/10
+30/30
+50/50
+70/70
+90/90
+10/100
+30/100
+50/100
+70/100
+90/100
+%train(CSRL/DR)
+52
+54
+56
+58
+60
+62
+64
+66
+68
+F1
+53.98
+58.7
+59.84
+60.2
+60.71
+54.87
+57.89
+58.51
+58.95
+59.31
+FDT
+FDT-SF
+(a) The performacne on CSRL test set.
+10/10
+30/30
+50/50
+70/70
+90/90
+100/10
+100/30
+100/50
+100/70
+100/90
+%train(CSRL/DR)
+20
+21
+22
+23
+24
+25
+26
+WER
+25.26
+22.18
+20.89
+20.61
+20.37
+22.93
+21.15
+20.8
+20.76
+20.71
+FDT
+FDT-SF
+(b) The performance on DR test set.
+Figure 4: The role of co-updating in friend-training.
+the model performances hit the peaks and drop as
+the thresholds keep increasing in the interval of
+high values, which is owed to high thresholds pro-
+ducing insufficient pseudo-labels for iterative train-
+ing. Automatically choosing proper pick thresholds
+is worth to be explored in the future.
+The Strength of Base Model To understand and
+compare how performance of models before friend-
+training or self-training influences their final per-
+formance, we compare STBR, STea and FDT with
+the base models trained on different percentages of
+labeled data in the source domain when evaluating
+on out-domain testing data. Specifically, we follow
+domain generalization settings and use a variable
+percentage of labeled data to conduct experiments.
+For CSRL and DR, respectively, we set the
+amount of labeled data as {10%/10%, 30%/30%,
+50%/50%, 70%/70%, 90%/90%}. The results are
+shown in Figure 3b and Figure 3c. We can see
+that all the methods adopting self-training to make
+use of unlabeled data surpass the base model by a
+significant margin, whether when given a weak or
+strong base model, demonstrating the effectiveness
+of self-training paradigm. Moreover, FDT achieves
+the best results across the evaluating percentages
+of labeled data: when the base model has a good
+amount of training data, such as those trained on
+30% labeled data and above, the performance of
+FDT is significantly better than STBR and STea,
+proving that FDT leverages the features learned
+from labeled data more effectively with cross-task
+supervision.
+The Role of Co-updating We also explore the
+case where one of the models of the friend tasks is
+fully trained and does not have to be updated. We
+consider FDT-SF, FDT with a fixed fully trained
+base model from the friend task in domain gener-
+alization4. As illustrated in Figure 4, FDT-SF sur-
+passes FDT when given a weak base model for the
+evaluating task because of the strong supervision
+from the friend task. However, FDT outperforms
+FDT-SF when the evaluating task is given a fairly-
+trained model, which demonstrates the benefits of
+co-updating the models in friend-training.
+6
+Conclusion
+We propose friend-training, the first cross-task self-
+training framework, which leverages supervision
+from friend tasks for better selection of pseudo-
+labels. Moreover, we provide specific modeling
+of friend-training between conversational seman-
+tic role labeling and dialogue rewriting. Experi-
+ments on domain generalization and few-shot learn-
+ing scenarios demonstrate the promise of friend-
+4Specifically, when the evaluating task is CSRL, the
+amount of labeled data for the two tasks are set as {10%/100%,
+30%/100%, 50%/100%, 70%/100%, 90%/100%}, and
+when the evaluating task is DR, {100%/10%, 100%/30%,
+100%/50%, 100%/70%, 100%/90%}.
+
+training, which outperforms prior classical or state-
+of-the-art semi-supervised methods by substantial
+margins.
+7
+Limitation
+We showed how the friend-training strategy can be
+applied to two dialogue understanding tasks in the
+case study here, but many other task pairs or task
+sets can be examined to fully explore the generality
+of the approach. Identifying friend tasks depends
+on expert knowledge in this work, but approaches
+for task grouping and task similarity may be used to
+automatically discover friend tasks. Besides, with
+the proliferation of cross-modal techniques, tasks
+of different modalities are expected to act as friend
+tasks as well. Also, designing translation functions
+and matchers for friend tasks in the friend-training
+framework requires an understanding of the rela-
+tionship between the friend tasks, but prompting
+and model interpretability methods could poten-
+tially be applied for easing this process.
+8
+Acknowledgement
+We thank the anonymous reviewers for their help-
+ful comments and the support of National Nature
+Science Foundation of China (No.62176174).
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+A
+Appendix
+A.1
+Error rates
+We have two classifiers fa and fb trained on two
+different tasks with labeled training sets La and
+Lb, with expected accuracies ηa and ηb, respec-
+tively. The prediction targets of the two tasks are
+partially related through a pair of translation func-
+tions Fa : ˆYa → Σ and Fb : ˆYb → Σ, where Σ
+is the set of possible sub-predictions that all pos-
+sible predictions of the two tasks ˆYa and ˆYb can
+be reduced to. | ˆYa| ≥ |Σ|, | ˆYb| ≥ |Σ|. The sub-
+predictions can be a part of the whole prediction
+targets for both tasks, or some lossy transforma-
+tion of the prediction targets. For example, a sub-
+prediction for a POS-tagging task can be the POS
+tag of the first word (a part of the prediction) or the
+number of the NN tag in the whole prediction se-
+quence (a transformation of the prediction). We as-
+sume that the translation functions are general func-
+tions with the expected probability of generating
+a translation ϵF =
+1
+|Σ|; they are deterministic and
+always map the gold labels of the friend tasks for
+the same input to the same translation. Both clas-
+sifiers make predictions on the unlabeled set U at
+iteration k. Some instances Uk
+F with pseudo-labels
+are chosen as new training data based on the results
+of the translation functions, φa(x) = Fa(fa(x))
+and φb(x) = Fb(fb(x)), and some selection crite-
+ria, such as total agreement. If total agreement is
+used as the selection criterion, the probability of
+erroneous predictions for fa in these instances is
+Prx[fa(x) ̸= f∗
+a(x)|φa(x) = φb(x)]
+=1 − Prx[fa(x) = f∗
+a(x)|φa(x) = φb(x)]
+=1 − Prx[fa(x) = f∗
+a(x)]·
+Prx[φa(x) = φb(x)|fa(x) = f∗
+a(x)]
+Prx[φa(x) = φb(x)]
+=1 − ηa·
+Prx[φa(x) = φb(x)|fa(x) = f∗
+a(x)]
+Prx[φa(x) = φb(x)]
+,
+(6)
+with f∗ being the optimal classifier. If we consider
+the two classifiers very likely to be independent
+from each other, then the probability of the transla-
+tion of the predictions from the two classifiers be-
+ing the same given the prediction from classifier fa
+is correct, which is Prx[φa(x) = φb(x)|fa(x) =
+f∗
+a(x)], is the sum of the probability of the clas-
+sifier fb making the correct prediction ηb and the
+probability of an erroneous translation of the wrong
+prediction ϵF(1 − ηb). The probability of the trans-
+lations matching Prx[φa(x) = φb(x)] has four sit-
+uations: both predictions of the two classifiers are
+correct ηaηb; fa(x) is correct but fb(x) is wrong
+and being translated erroneously ηaϵF(1 − ηb);
+fb(x) is correct but fa(x) is wrong and being trans-
+lated erroneously ηbϵF(1 − ηa); both fa(x) and
+fb(x) are wrong but matching in the translation
+space ϵ2
+F(1 − ηa)(1 − ηb). Under these conditions
+Equation 6 becomes
+1 − ηa(ηb + ϵF(1 − ηb))
+Prx[φa(x) = φb(x)]
+=1 −
+Z
+Z + ηbϵF(1 − ηa) + E ,
+(7)
+where Z = ηa(ηb + ϵF(1 − ηb)) and E = ϵ2
+F(1 −
+ηa)(1 − ηb) which shows that the term ηbϵF(1 −
+ηa)+E needs to be small to make the probability of
+matching translations with predictions being wrong
+small. This indicates that the quality of the picked
+instances based on the total agreement criterion
+is negatively correlated with the number of false
+positive instances brought by the noisy translation
+ηbϵF(1−ηa), and the number of matching negative
+instances E. ϵF can be minimized by choosing
+translation functions with a sufficiently large co-
+domain Σ, which means that when the translation
+space is large enough, it is unlikely that the two
+classifiers totally agree in the translation space but
+do not agree in their own prediction target spaces.
+So the probability of them agreeing and making
+correct predictions is much larger than agreeing but
+making incorrect predictions while the probability
+of error instances chosen when two classifiers agree
+approaches 0, indicating that even when 1 − ηa is
+large, i.e. fa performs badly, if the co-domain is
+large, the error rate of the chosen instances can still
+be kept very low.
+A.2
+Datasets
+Annotation Procedure of WeiboCSRL The dia-
+logues we use for CSRL annotation are extracted
+from LCCC-base (Wang et al., 2020b), which
+consists of at least 4 turns and 80 total charac-
+ters to assure enough context for CSRL and DR.
+These dialogues and those used as unlabeled data
+for experiments in section 5 are from different
+parts of LCCC-base. For each dialogue, we an-
+notate the predicates in the last utterance with
+the guidance of frame files of Chinese Proposi-
+
+tion Bank5. For each predicate, the arguments
+we annotate are numbered arguments ARG0-ARG4
+and adjuncts ARGM-LOC, ARGM-MNR, ARGM-TMP and
+ARGM-NEG, whose definitions are shown in (Xue,
+2006). ARGM-MNR is not included for evaluation in
+section 5 because annotation of ARGM-MNR is lack-
+ing in DuConv, the training data for CSRL. In the
+end, we obtain 3891 annotated predicates.
+Dataset Details Table 2 shows the statistic of the
+datasets used in the experiments. DuConv (Xu
+et al., 2021) focuses on movies and celebrities and
+we adopt the same train/dev/test splitting as Xu
+et al. (2021). REWRITE (Su et al., 2019) contains
+20K dialogues with a wide range of topics crawled
+from Chinese social media platforms; the last ut-
+terance of each dialogue is rewritten to recover all
+co-referred and omitted information. RESTORA-
+TION (Pan et al., 2019) contains dialogues from
+Douban6, most of which are book, movie or prod-
+uct reviews. Compared with REWRITE, it contains
+more annotated dialogues, but around 40% of the
+last utterances require no rewriting.
+Domain
+#Instance(train/dev/test)
+DuConv
+movies and celebrities
+23361 / 2852 / 2977
+WeiboCSRL
+social media
+- / 1945 / 1946
+REWRITE
+social media
+16925 / 1000 / 1000
+RESTORATION
+book, movie and
+product reviews
+- / 5000 / 5000
+Table 2: Dataset statistics.
+A.3
+Implementation Details
+Dataset configuration of the tasks for the experi-
+mental scenarios are shown in Table 3.
+Preprocessing Details The maximum length of
+the input dialogue is set to 125. We transform the
+word-based labeling of DuConv to character-based
+labeling and we use the scripts7 provided by Hao
+et al. (2021) to generate token-level annotations for
+sequence-labeling-based DR. For unlabeled data,
+we discard dialogues with less than 4 turns to guar-
+antee sufficient context for CSRL and DR.
+Model Details We use pretrained BERT8 (Devlin
+et al., 2019) as the dialogue encoder for CSRL and
+DR. Both the values of hyper-parameter α and β
+are set to 0.2 and the pick thresholds are set to 0.6.
+We choose a state-of-the-art sentence-level seman-
+5https://verbs.colorado.edu/chinese/cpb/
+6https://www.douban.com
+7https://github.com/freesunshine0316/
+RaST-plus
+8https://huggingface.co/bert-base-chinese
+tic role labeling (SSRL) parser9 for the translation
+matcher which follows the same structure as (He
+and Choi, 2021).
+Training Details We adopt AdamW (Loshchilov
+and Hutter, 2019) to optimize models with a learn-
+ing rate of 4e-5 and batch size of 16. We use λ = 1
+to balance the loss of labeled and unlabeled data.
+Task
+Train
+Dev&Test
+DG
+CSRL
+DuConv (train)
+WeiboCSRL (dev,test)
+DR
+REWRITE (train)
+RESTORATION (dev,test)
+FSL
+CSRL
+DuConv (100 cases)
+DuConv (dev,test)
+DR
+REWRITE (100 cases)
+REWRITE (dev,test)
+Table 3: Dataset configuration of domain generaliza-
+tion (DG) and few-shot learning (FSL).
+A.4
+Baselines
+Standard self-training (Scudder, 1965) generates
+pseudo-labels to unlabeled data with a base model
+and uses them to train a new base model, which is
+repeated until convergence. Standard co-training
+(Blum and Mitchell, 1998) is similar to Standard
+self-training, but with two different base models
+dealing with the same task, generating pseudo-
+labels and adding the trusted ones for iterative train-
+ing. Mean teacher (Tarvainen and Valpola, 2017)
+maintains a teacher model on the fly, whose weights
+are the exponential moving average of the weights
+of a student model across iterations. Cross pseudo
+supervision (Chen et al., 2021), a state-of-the-art
+variant of self-training, maintains two networks
+with different initialization; the pseudo-label of
+one network is used to supervise the other network.
+Self-training with batch reweighting (Bhat et al.,
+2021) is a state-of-the-art self-training method that
+reweights the pseudo-labels in a batch when train-
+ing according to the confidence from the teacher
+model. Self-teaching (Yu et al., 2021), a state-of-
+the-art semi-supervised method that sequentially
+trains a junior teacher, a senior teacher and an ex-
+pert student to leverage the unlabeled data.
+For the hyper-parameters of the baselines, we
+keep the common hyper-parameters, such as learn-
+ing rate, batch size, optimizer, and so on, the same
+as our proposed method. And we adopt the values
+of method-specific hyper-parameters used in the
+original papers, such as the merging weight of soft
+and hard labels of self-teaching and the smoothing
+parameter for updating of mean teacher.
+9https://github.com/hankcs/HanLP
+
+Context:
+ch: [A]我有一个非常喜欢的女明星。[B]她叫什么名字？[A]布蕾克·莱弗利。[B]她很有名吗？
+en: [A] I have a favorite actress. [B] What’s her name? [A] Blake Lively. [B] Is she famous?
+Current utterance
+ch: [A]她是一个非常受关注的女明星。
+en: [A] She is a actress attracting much attention.
+Rewritten utterance
+ch: [A] 布蕾克·莱弗利是一个非常受关注的女明星。
+en: [A] Blake Lively is a actress attracting much attention.
+Predicates
+是(is)
+受(attract)
+CSRL
+ch: ARG1: 一个非常受关注的女明星
+en: ARG1: a actress attracting much attention
+ARG0: 布蕾克·莱弗利, ARG1: 关注
+ARG0: Blake Lively, ARG1: attention
+SSRL
+ch: ARG0: 布蕾克·莱弗利, ARG1: 一个非常受关注的女明星
+en: ARG0: Blake Lively, ARG1: a actress attracting much attention
+ARG0: 布蕾克·莱弗利, ARG1: 关注
+ARG0: Blake Lively, ARG1: attention
+Predicate matching score
+0.61
+1.0
+Predicate confidence
+0.95
+0.54
+Predicate overall score
+0.67
+0.90
+Utterance matching score
+0.81
+Utterance confidence
+0.92
+Utterance overall score
+0.83
+Table 4: Case study: [A] and [B] are the signatures of speakers. ch and en are the language abbreviations.
+A.5
+Case Study
+We show a representative case of selecting pseudo-
+labels in Table 4. There are two predicates in cur-
+rent utterance: 是(is) and 受(attract). For 是(is),
+the CSRL parser yields only ARG1 while SSRL
+parser gives the same ARG1 but more of ARG0
+based on the rewritten utterance. With the differ-
+ence in arguments, the overall score is not high and
+this predicate could be regarded as low-quality if a
+high pick threshold is set. For 受(attract), the CSRL
+and SSRL parsers give the same arguments, which
+are the right answer. However, if we only consider
+the model confidence of the predicate, which is
+0.54, this high-quality predicate are more likely
+to been discarded than consider the overall score,
+which is 0.90. And the rewritten utterance gets a
+high overall score, which is what we expected.
+A.6
+Discussion on Generalization of the
+Framework
+It is not uncommon at all for different language
+tasks sharing some information. With one case
+study presented in detail in the main body of the
+paper, we also provide a short example of a dif-
+ferent friend task pair – constituency parsing and
+dependency parsing – and explain how they can
+help each other and show the general nature of the
+friend-training framework.
+Early work (Magerman, 1995; Collins, 2003) has
+shown relationship between dependency and con-
+stituency parsing through head-finding rules, and
+Jin and Schuler (2019) show directly how common
+structures between dependency and constituency
+trees can be derived for parsing evaluation. In a
+dependency graph, a set of nodes with a single
+incoming edge is usually indicative of a phrase
+structure, such as a noun phrase, a verb phrase or
+a prepositional phrase. Such phrasal structures are
+well-marked in constituency treebanks, and could
+be used as the shared friend information for friend-
+training. Here is a sketch of how friend-training
+can be applied to this pair:
+1. Train a constituency parser and a dependency
+parser, presumably trained with a small num-
+ber of training instances, as the models for the
+friend-training framework.
+2. Run both parsers on a common set of unla-
+beled data for parsing results.
+3. Find phrases such as noun, verb or preposi-
+tional phrases in the predicted constituency
+trees.
+4. Compare with the dependency trees, and
+check if spans of such phrases have only a
+single incoming edge. If so, the constituency
+and dependency parsing results can be consid-
+ered agreeing, and added to the silver training
+set. If not, the silver annotation is discarded.
+5. Train the parsers again with the gold and silver
+training instances.
+As long as some shared information can be iden-
+tified between two seemingly different tasks, the
+
+noisy agreement between that partial target can
+provide valuable supervision between two tasks.
+The translation and matching between constituency-
+dependency targets are simpler compared to the
+CSRL-rewriting pair presented in the paper, partly
+because no model is required for the translation
+process. However the CSRL-rewriting pair is more
+significant because heuristics may be difficult or
+not obvious to design where ‘bridging’ tasks such
+as single-sentence SRL may be readily available.
+

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+1
+Real-Time Digital Twins:
+Vision and Research Directions for 6G and Beyond
+Ahmed Alkhateeb, Shuaifeng Jiang, and Gouranga Charan
+Abstract—This article presents a vision where real-time digital
+twins of the physical wireless environments are continuously
+updated using multi-modal sensing data from the distributed
+infrastructure and user devices, and are used to make communi-
+cation and sensing decisions. This vision is mainly enabled by the
+advances in precise 3D maps, multi-modal sensing, ray-tracing
+computations, and machine/deep learning. This article details this
+vision, explains the different approaches for constructing and
+utilizing these real-time digital twins, discusses the applications
+and open problems, and presents a research platform that can
+be used to investigate various digital twin research directions.
+I. INTRODUCTION
+Heading towards 6G, communication systems are increas-
+ingly featuring key trends such as the employment of large
+numbers of antennas and the use of higher frequency bands [1].
+These technology trends bring higher data-rate and multiplex-
+ing gains to the networks, but also impose critical challenges
+on the ability of these systems to support highly-mobile,
+energy-constrained, reliable, and low-latency applications. For
+example, the deployment of large antenna arrays is associated
+with high channel acquisition and beam sweeping overhead
+[2], [3], which makes it hard for these massive MIMO systems
+to support mobile applications, and the use of higher frequency
+bands at mmWave and sub-THz makes the wireless links very
+sensitive to line-of-sight (LOS) blockages, which challenges
+the reliability and latency of the networks [4].
+In this article, we present a novel vision in which a real-time
+digital twin is utilized to make operational physical, access,
+network, and application layer decisions to the real physical
+world in communication and sensing systems. The key features
+of the envisioned digital twin-based wireless systems can be
+summarized as follows:
+• Enabled by 3D maps and multi-modal sensing: The
+envisioned digital twin will leverage precise 3D maps and
+fuse multi-modal sensory data from distributed devices
+and infrastructure nodes to construct an accurate real-time
+digital replica of the physical world.
+• Capable of making real-time decisions: Leveraging
+advances in real-time 3D ray-tracing, efficient computing,
+and machine/deep learning, the envisioned digital twin
+will be capable of making real-time decisions for the
+wireless communication systems.
+• Continuously refined for better approximation: We
+envision the digital twin as a model that will be continu-
+ously refined to improve its approximation of the physical
+The authors are with the School of Electrical, Computer and Energy
+Engineering, Arizona State University, (Email: alkhateeb, s.jiang, gcha-
+[email protected]).
+world (including the electromagnetic and optical aspects)
+and to enhance its decision accuracy.
+• A global digital twin shared between devices: In its
+ultimate version, we envision this digital twin to be
+global and shared between devices such that all devices
+can jointly enhance it and benefit from it using their
+coordinated sensing and communication decisions.
+• Used for all communication layer decisions: With the
+real-time emulation of the physical world, the envisioned
+digital twin can be utilized to make physical-layer de-
+cisions such as channel prediction, as well as access,
+network, and application layer decisions.
+It is important to highlight here that the general concept
+of real-time digital twins has been studied before in the
+context of smart manufacturing, intelligent transportation, and
+healthcare [5]. For wireless communications, prior work has
+mainly focused on network operation topics such as edge
+computing, network optimization, and service management,
+in which digital twins are leveraged to simulate the real
+world at the network level [6], [7]. In contrast, we focus
+on utilizing the real-time digital twin to simulate the real
+world with a particular emphasis on the physical modeling
+of the environment and wireless signal propagation. To that
+end, the envisioned real-time digital twins produce real-time
+instantaneous and statistical information about the wireless
+channels, which could be leveraged to make decisions for
+the physical, access, network, or application layers of the
+communication systems.
+The goal of this article is to expose the potential of real-
+time digital twins for wireless communication and sensing
+systems in 6G and beyond. In the next sections, we discuss
+the key enabling technologies, the different approaches for
+constructing and utilizing these digital twins, and the various
+applications and future research directions. We also present
+a research platform that could be used to investigate many
+interesting digital twin research directions.
+II. TODAY’S TECHNOLOGY ADVANCES LEAD TO
+REAL-TIME DIGITAL TWINS
+The vision of building real-time digital twins is motivated
+by the recent advances in 3D maps, multi-modal sensing, ray
+tracing, and machine/deep learning. Next, we briefly discuss
+these four key enablers for real-time digital twins.
+Precise 3D Maps: 3D maps contain information about the
+positions, shapes, orientations, and materials of the commu-
+nication devices and other objects in the environment. The
+current application trends of AR/XR, autonomous driving,
+arXiv:2301.11283v1  [eess.SP]  26 Jan 2023
+
+2
+Radar
+LiDAR
+Camera
+IMU
+Position
+...
+High-Fidelity Sensing
+Real-Time Ray Tracing
+ML/DL Advances
+Precise 3D Map
+Real World
+Digital Replica
+Precise Real-Time
+3D Map
+Channel Info.
+Geometric Info.
+(Position/Mobility)
+Desicions
+Configurations
+Operations
+Twin
+Construction
+& Update
+Real-Time
+Ray Tracing
+Task Solving
+Vehicle
+Radar
+Vehicle 
+LiDAR
+Surveillance
+Camera
+User
+Position
+Digital Twin Guided
+Beamforming
+User
+IMU
+Fig. 1. This figure presents the general idea of the digital twin with the four key enablers: precise 3D map, high-fidelity sensing, real-time ray tracing, and
+ML/DL. The digital twin is constructed based on a real-time 3D map. By performing real-time ray tracing, the digital twin infers channel information. The
+channel and geometry information can facilitate various applications.
+and metaverse technologies created an increasing demand
+for precise 3D map data. In response to that, the 3D map
+data collection, processing, and management capabilities have
+been significantly advanced. Vehicle, airborne, and satellite
+3D imaginary sensors are now used to collect and build
+very accurate 3D maps. Further, the growing computational
+and database resources are making it possible to process and
+manage large-scale 3D maps, even at the scale of the full world
+like Nvidia OmniVerse [8]. Thanks to all these developments,
+precise 3D maps are becoming more affordable and accessible.
+High-Fidelity Sensing: Recent trends in sensing-aided
+communication,
+integrated
+sensing
+and
+communication
+(ISAC), and internet-of-things (IoT) tend to deploy multi-
+modal sensors, such as cameras, radars, LiDARs, positioning,
+at the infrastructure, user equipment, and IoT devices. These
+distributed sensors can complement each other since they have
+different observing angles and different types of information,
+such as position, shape, and mobility measures about the
+various stationary and moving objects. This sensing capability
+can be further enhanced by leveraging the recent advances in
+multi-modal data fusion [9]. As a result, it is becoming more
+feasible to acquire high-fidelity sensing information about the
+surrounding environment in nearly real-time, which is a key
+enabler for the envisioned digital twin.
+Real-Time Ray Tracing: Ray tracing simulators attempt
+to trace the wireless signal propagation paths between trans-
+mit and receive antennas, and generate the parameters, such
+as the angles of arrival/departure and complex path gains,
+of these propagation paths. A main limitation of the ray
+tracing simulators is that they typically require considerable
+computational overhead, and hence, incur high latency. The
+significant advances in parallel computing hardware and ray-
+tracing computational approaches over the last two decades,
+however, are enabling real-time ray-tracing for both wireless
+and optical signals. This means that, given precise real-time
+3D maps, the wireless channels between the (possibly mobile)
+transmitters and receiver can potentially soon be computed in
+the digital twins in real-time.
+Advances in Machine/Deep Learning: Machine and deep
+learning have demonstrated powerful capabilities in extracting
+features, approximating complex functions, and solving non-
+trivial optimization problems in wireless communication and
+sensing/perception. In the digital twins, these advances in
+machine/deep learning can be utilized to (i) enhance the
+quality and reduce the cost of building precise 3D maps, (ii)
+improve the efficiency of the multi-modal sensing in terms
+of sensory data processing, transferring, sharing, and fusion
+[10], and (iii) advance the ray tracing accuracy and reduce its
+latency and computational complexity.
+All these advances in precise 3D maps, high-fidelity sens-
+ing, real-time ray tracing, and machine/deep learning are
+making it more feasible to realize real-time digital twins.
+In particular, the precise 3D map and high-fidelity sensing
+complement each other; while the 3D maps mainly contain
+information about the static objects in the environment, the
+high-fidelity sensing can augment these maps with information
+about the dynamic objects in real-time. Conducting ray tracing
+on these real-time 3D maps leads to real-time wireless digital
+twins. Further, the advances in machine/deep learning can
+be utilized to enhance the various aspects of these real-time
+digital twins. The real-time digital twins open opportunities for
+novel capabilities and applications in wireless communication
+
+3
+Level 1: Local Information and Individual Decisions
+Level 2: Shared Information and Individual Decisions
+Level 3: Shared Information and Joint Decisions
+Local/Partial Digital Twin 
+of Vehicle 1
+Local/Partial Digital Twin 
+of Vehicle 2
+Vehicles 1 and 2 
+Equipped with Multi-modal Sensors
+Device 1 Local Decisions 
+(Ray-Tracing, Beam Pred...)
+Device 2 Local Decisions 
+(Ray-Tracing, Beam Pred...)
+Device 1 Local Decisions 
+(Ray-Tracing, Beam Pred...)
+Device 2 Local Decisions 
+(Ray-Tracing, Beam Pred...)
+Joint Decisions 
+(Ray-Tracing, Beam Pred...)
+Sensing Information is Projected to A Shared Digital Twin
+Sensing Information is Projected to A Shared Digital Twin
+V1
+V1
+V1
+V2
+V2
+V2
+Vehicle 1 Sensing Information is 
+Projected to A Local Digital Twin
+Vehicle 2 Sensing Information is 
+Projected to A Local Digital Twin
+Fig. 2. This figure presents our vision of how the digital twin system will operate in the real world. In particular, it shows three different operating modes
+(i) local information and individual decision, (ii) shared information and individual decision, and (iii) shared information and joint decision.
+and sensing systems. These digital twins can, for instance,
+be leveraged to compute channel or channel covariance in-
+formation, predict LOS link blockages, proactively predict
+handover and traffic steering, and even predict application-
+specific caching requirements.
+III. TRUE DIGITAL TWINS THAT KEEP LEARNING
+There are different approaches of how such a real-time
+digital twin could be leveraged. In this section, we first discuss
+two main approaches where digital twins are either only used
+for training machine learning models or for making real-time
+decisions. Then, we present our vision for true digital twins
+that keep learning and improving over time.
+Digital Twins for Training ML Models: With the pre-
+cise 3D maps and accurate ray-tracing simulators, we can
+build high-fidelity and site-specific synthetic datasets. These
+datasets could be utilized to train machine learning models
+for the various wireless communication and sensing tasks. In
+particular, these synthetic datasets could be generated in large-
+scale and with high variance, which is hard and expensive to
+collect in the real world. The machine learning models that are
+trained on these site-specific synthetic datasets could then be
+deployed to make inference/decisions for the physical world.
+Further, these models could be refined using limited real-world
+datasets for better and more robust performance. This approach
+relaxes the latency requirements as the digital twins are not
+used for real-time decisions. The drawback, however, is that
+these ML models are not benefiting from the global real-time
+sensing and awareness that the real-time digital twins have,
+which may limit their performance.
+Digital Twins for Real-Time Decisions: Another approach
+is to use these real-time digital twins to directly make real-time
+or near real-time decisions for the physical-world communi-
+cation and sensing systems. For example, an FDD massive
+MIMO basestation can use the real-time digital twin to predict
+the downlink channel or, at least, the dominant subspace of
+this channel, which saves large channel training and feedback
+overhead. Mobile users may also use this digital twin, for
+instance, to predict if their LOS links are going to be blocked
+by a moving scatterer or whether they need to switch to other
+beams. This approach can, therefore, benefit from the real-time
+nature of the digital twin and the richer awareness about the
+surrounding environment in making real-time decisions. The
+drawback, however, is that the decisions that are solely made
+based on the digital twin will be very sensitive to the modeling
+accuracy, which challenges the robustness of these decisions.
+True Digital Twins: The previous two approaches have a
+clear trade-off between the ability to benefit from the real-
+time awareness of the digital twins to make efficient decisions
+(e.g., in terms of wireless resources and mobility support)
+and the ability to ensure the robustness of these decisions.
+This motivates what we call true digital twins that can be
+thought of as machine learning models themselves that keep
+learning and improving their approximation of the physical
+world, and hence their decisions, over time. In particular,
+these digital twins could leverage learning agents and use
+prior decisions and feedback to enhance the modeling accuracy
+of the 3D maps, the processing and integration of the multi-
+modal sensing data, and the approximation fidelity of the ray
+tracing. These true digital twins can, therefore, be used to
+make decisions that are both real-time and accurate for the
+various wireless communication and sensing tasks.
+
+4
+IV. THREE DIGITAL TWIN LEVELS
+Constructing and utilizing digital twins requires interaction
+with the devices that are contributing to the digital twins
+with sensing data and leveraging the digital twins in making
+decisions. This, however, raises questions about the required
+level of coordination between these devices for both the
+sensing and communication tasks. For that, we envision that
+digital twins will evolve through three main digital twin
+levels of coordination as the computation, synchronization,
+and communication capabilities develop over time. Next, we
+briefly present these operating modes (levels) and highlight
+how they could function in the real world.
+Local information and individual decision: We envision
+the first level of the digital twin to incorporate local sensing
+and individual decisions. In particular, each device equipped
+with one or more sensors will be expected to collect its
+local sensing information. The acquired local data can then
+be projected onto a 3D map to generate a real-time digital
+twin. This, in general, will help generate a partial view of the
+entire map, resulting in a localized digital twin. This local real-
+time digital twin can then be utilized to facilitate the sensing
+and wireless communication decision-making process. An
+advantage to such a local approach is that the limited sensing
+information captured by each device can be processed locally
+(on the device or edge), reducing the downstream processing
+and decision-making latency. However, the partial nature of the
+generated digital twin limits the scope of the decision-making
+capabilities of the devices. For example, enabling advanced
+applications, such as future blockage prediction and handoff,
+may require access to a more global real-time digital twin.
+Shared information and individual decision: The pro-
+posed first level of the digital twin is limited by the local
+view of the wireless environment, which results in a partial
+digital twin. Generating an accurate real-time digital twin
+and reaping its benefits requires a global overview of the
+environment. Fusing the sensing information collected across
+different devices in the wireless environment at any given
+time can be a promising way to achieve this vision of a
+comprehensive digital twin. As such, this idea forms the basis
+of our vision of the second level of the digital twin. Similar to
+the first level, each device is expected to collect sensing data
+of its surrounding environment. However, here, we propose
+to fuse the information collected across devices to generate a
+detailed and thorough digital twin. The fusion operation can
+be performed at the edge or cloud. Even though information
+sharing across devices (or at the edge/cloud) is enabled,
+each device is envisioned to undertake its own sensing and
+communication decisions. This is due to the high computation,
+synchronization, and communication requirement for joint and
+globally-optimized decisions for all the devices.
+Shared information and joint/cooperative decision: With
+the increased computation, synchronization, and communica-
+tion capabilities of future devices, we envision that digital
+twins will evolve into a more global form where devices can
+also coordinate and jointly optimize their sensing and commu-
+nication decisions. Therefore, in this third level of the digital
+twins, devices are expected to share and fuse their sensing
+information, mostly at the edge, similar to the second level,
+to form a global and comprehensive real-time digital twin.
+In addition, the sensing and communication decisions will be
+jointly optimized either in a central or distributed way. This
+is expected to enhance both the sensing and communication
+performance of future wireless systems. These three digital
+twin levels are illustrated in Fig. 2, clarifying the differences
+in the information-sharing and decision-making approaches.
+V. APPLICATIONS
+The proposed digital twin leverages precise 3D maps and
+high-fidelity sensing to capture real-time information about the
+geometry and materials of the static and dynamic objects in
+the environment and to construct a real-time digital replica. By
+performing real-time ray tracing on this digital replica, we can
+infer various information about the communication channels,
+such as the propagation path parameters (path loss, delays,
+angles, etc.), channel and covariance information, link quality,
+and blockage status. Moreover, by utilizing the temporal and
+spatial consistency of the channels, the digital twin can predict
+future channel information in dynamic environments. The real-
+time and future channel predictions can be leveraged to make
+real-time and proactive decisions that can potentially improve
+the communication system operations in the physical, access,
+network, and application layers.
+Physical Layer: The physical layer operation has, by defini-
+tion, a clear dependancy on the wireless channels. Many phys-
+ical layer tasks, e.g., MIMO precoding and link adaptation,
+directly rely on partial or full knowledge about the communi-
+cation channels. However, the channel acquisition is typically
+associated with high overhead, especially in large-dimensional
+systems, which degrades the overall system efficiency. Real-
+time digital twins open novel opportunities to revolutionize the
+channel acquisition process: When the real-time 3D maps and
+ray tracing computations are sufficiently accurate, the digital
+twin could be directly used to accurately infer the channels,
+reducing or even eliminating the channel acquisition overhead.
+Further, in scenarios where the approximation is not sufficient
+to accurately estimate the full channels, the digital twin may
+still be used to predict the dominant channel subspaces and
+reduce the channel acquisition overhead [11], [12]. These
+digital twins can also be used to estimate the signal-to-noise
+ratio (SNR) of the communication links and improve the
+modulation and coding scheme selection. Another interesting
+physical layer application of the digital twin is to generate a
+massive amount of data for a given site in the real world. This
+site-specific data can then be utilized to optimize the traffic
+beam sets and the channel feedback compression codebooks.
+Access Layer: Current and future communication systems,
+especially at higher frequencies, employ large antenna arrays
+and highly-directional beams to achieve sufficient SNR and
+realize high data rates. Aligning these beams, however, typi-
+cally requires large beam training overhead that scales with
+the number of antennas. The real-time and future channel
+information predicted by the digital twin can facilitate the
+initial access (initial beam alignment) and beam management
+for these systems and reduce the beam training overhead.
+
+5
+Cloud
+Edge
+Ray-Tracing
+3D Map
+AI/ML
+D1: Sensing Data
+D2: Sensing Data
+Dn: Sensing Data
+Physical World
+Digital World
+E.g.: MIMO Precoding
+Channel Prediction
+Link Adaptation
+Initial Access
+Beam Management
+E.g.: Proactive Blockage Prediction
+E.g.: Service Migration
+Proactive Network Configuration
+Occlusion 
+Will Lead to Link Disruption
+E.g: Proactive Caching
+Flexible Application-Specific Decisions
+Physical Layer
+Access Layer
+Network Layer
+Application Layer
+Massive MIMO
+Basestation
+User
+Propagation Paths 
+Precoding Matrix
+Fig. 3. This figure presents some example communication applications that digital twins can enable across the different layers, such as the physical layer,
+MAC layer, network layer, and application layer. For example, in the MAC layer, the digital twin can proactively predict future blockages and initiate hand-off
+by tracking the mobility pattern of the different objects in the environment over time.
+Further, due to the increased penetration loss, high frequency
+communication links, e.g., in mmWave, experience sudden
+disturbance due to blockages. By tracking the motion of the
+user and other objects in the environment, the digital twins can
+proactively predict the occurance and duration of incoming
+blockages before they happen. This enables seamless han-
+dover control and improve the network reliability and latency
+performance. Digital twins can alsp enhance the access layer
+resource allocation, user scheduling, MU-MIMO user pairing,
+and interference management, among many other applications.
+Network Layer: In prior work, digital twins have been
+used to optimize the network layer operations and applications
+such as device and traffic monitoring, resource allocation,
+edge computing, and cyber security. We refer to this type
+of digital twins as the network-level digital twins since they
+typically focus on monitoring network status and modeling
+the network-level entities, services, and dynamics. Differently,
+the envisioned physical-level digital twins can provide fine-
+grained information about the communication links, which
+can also be used to improve the accuracy of the radio access
+network (RAN) modeling. This accurate RAN modeling can
+enhance the efficiency and reliability of various network-level
+operations. Further, the physical-level digital twins can provide
+real-time and future information about the user position and
+mobility characteristics, which could be leveraged to improve
+several edge computing operations. For instance, when a user
+is predicted to move out of the service area of the in-use edge
+server, proactive service migration can be triggered to improve
+the service quality. The physical-level digital twins can also
+work cooperatively and in an integrated manner with the
+network-level digital twins to enhance the end-user experience.
+Application Layer: New emerging applications, such as au-
+tonomous driving and AR/VR, pose more stringent reliability
+and latency requirements to advanced communication systems.
+The 6G is envisioned to support 9-nine reliability with 0.1ms
+latency for mission and safety-critical applications. Moreover,
+different applications often have diverse requirements for data
+throughput, reliability, and latency, thus different strategies
+when handling link instability. The digital twin can simulate
+the physical signal propagation and channels, which can offer
+fine-grained information on the communication link quality,
+both in real-time and proactively. This link quality information
+provides more flexibility for making efficient application-layer
+decisions in a way that respects this application diversity. For
+example, if a video streaming application knows ahead of time
+about the communication link disruption and blockage status,
+this application can pre-load a certain portion of the video,
+considering the duration of the cut-off, and achieve a seam-
+less user experience with efficient usage of communication
+resources. While the digital twin provides low-level and fine-
+grained information about the communication links, how to
+efficiently utilize this information is still to be investigated.
+VI. DIGITAL TWIN RESEARCH PLATFORM
+Here, we present a digital twin research platform based on
+the DeepSense 6G [13] and the DeepVerse 6G [14] datasets
+that can be used to investigate various digital twin research
+directions. Next, we briefly present these digital twin datasets
+and show how they can be used to investigate an example
+application of digital twin-assisted beam prediction.
+
+GPSPosition,Orientation
+Camera
+User
+LiDAR
+Radar
+Future BlockageFuture Blockage
+Prediction
+Prediction Algorithm
+InputSequenceEdgeServerEdges
+Cloud Server
+Service MigrationServerUE Movement6
+Digital Twin Datasets: The digital twins rely on co-existing
+real-world data and high-fidelity synthetic data generated using
+accurate 3D maps and ray tracing. For that, the DeepSense 6G
+and the DeepVerse 6G datasets are well-suited to facilitate the
+research, development, and application of digital twins. The
+DeepSense 6G is a large-scale multi-modal sensing and com-
+munication dataset collected in various real-world scenarios;
+the DeepVerse 6G is a synthetic dataset that can simulate high-
+fidelity multi-modal sensing and communication data from ray
+tracing. Combining real-world scenarios and their synthetic
+replicas from the two datasets, we present a digital twin
+research platform to investigate its efficacy, limitations, use
+cases, and applications in real-world systems.
+Example Application: In [15], the digital twin is used to
+first infer the channels with real-time 3D maps (containing
+user positions) and ray tracing. Then, these channels can be
+used to infer optimal beams. However, the high computational
+complexity of ray tracing may result in high latency, thus
+limiting real-time applications. For that, ML models can be
+employed to approximate the ray tracing simulation with
+accelerated computation. Furthermore, ML can be leveraged
+to learn the mapping function from 3D maps to the optimal
+beam in an end-to-end manner. This can potentially improve
+the computational efficiency since the ML has the flexibility
+to not model the unnecessary ray tracing for a specific com-
+munication problem. When the ML is trained solely using the
+digital replica, it is limited by the impairments in the 3D map
+and ray tracing. A small amount of real-world data can be
+utilized to fine-tune the ML models (i.e. transfer learning) and
+to even transcend the digital replica.
+Experimental Results: Scenario 1 of DeepSense is adopted
+as the real-world scenario, and the digital replica of this
+scenario is constructed and simulated. An ML model is first
+trained on the position and wireless beam data from the digital
+replica. Then it is fine-tuned on the real-world data. The top-2
+beam prediction accuracy is obtained by testing the model on
+unseen real-world data. After training on 200 digital replica
+data points, the ML model can achieve a high top-2 accuracy
+of 91.4%. Note that this digital twin approach does not need
+any real-world training data. Moreover, with transfer learning,
+a small number of real-world data points (less than 20) can
+quickly improve the ML model to go beyond the digital replica
+and achieve near-optimal performance. Next, the digital replica
+impairment is explicitly modeled by adopting a uniform beam
+steering codebook that is different from the beam codebook of
+the communication hardware. The uniform codebook leads to
+lower performance when fine-tuned on a very limited amount
+of real-world data. However, when more than 20 real-world
+data points are used for fine-tuning, the performance of the two
+codebooks becomes very similar. The mismatches between the
+real world and the digital replica can be efficiently calibrated
+by a small amount of real-world data.
+VII. FUTURE RESEARCH DIRECTIONS
+Digital twins are envisioned to be an integral component
+of 6G and beyond wireless communication systems and have
+recently piqued the interest of both the industry and academia
+0
+20
+40
+60
+80
+100
+Number of Real Data Points Used for Training
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Top-2 Accuracy Tested on Real-World Data
+Trained on real-world data 
+Transfer learining (measured codebook)
+Transfer learining (unifrom codebook)
+0
+5
+10
+15
+20
+0.86
+0.88
+0.9
+0.92
+0.94
+0.96
+Transfer learning
+Pre-trained on 200
+synthetic data points
+Fig. 4. This figure shows the top-2 accuracy performance by first training the
+NN on the synthetic data generated by the digital twin and then fine-tuning
+it on real-world data.
+alike. However, realizing this vision and exploring the true
+potential of digital twins requires overcoming the fundamental
+challenges and thoroughly investigating some of the important
+aspects. Next, we present some open research directions that
+need to be investigated toward enabling digital twin-aided
+next-generation wireless communication.
+Impact of Digital Twin on Communication Tasks: De-
+ploying digital twin-aided solutions in the real world will re-
+quire revisiting all the problem statements, such as LOS/NLOS
+beam prediction, blockage prediction, hand-off, etc., to evalu-
+ate the efficacy of these solutions accurately. In particular, it
+is necessary to investigate how much gain in performance can
+be achieved by utilizing the real-time digital twins compared
+to conventional and sensing-aided approaches. To that end,
+the recently published DeepVerse 6G synthetic dataset (with
+multi-modal sensing and communication data generated from
+3D models and ray-tracing) was created to mimic the real-
+world scenarios of the DeepSense 6G dataset, effectively
+making them a digital twin of each other. These two datasets
+combined can enable the development and evaluation of digital
+twin-aided applications in real-world communication systems.
+Communication-Sensing Trade-Off: In order to gener-
+ate an accurate real-time digital twin with minimal latency,
+the sensing data collected across different devices, in some
+cases, need to be transferred quickly to a central unit for
+further processing (digital twin levels 2 and 3). The data
+transfer rate is dependent on the available bandwidth of the
+communication system itself. As the amount of sensing data
+increases (for example, with the increase in sensing modalities
+or the number of devices), so does the requirement for com-
+munication bandwidth. While access to diverse and detailed
+sensing information can help generate a more accurate digital
+twin, which improves the performance of the communication
+system, transferring that sensing data will consume communi-
+cation resources, potentially offsetting any gains. A promising
+solution to overcome this challenge is to first process the
+
+7
+sensory data locally and extract relevant features, and then
+transfer the low-dimensional extracted features across devices.
+Investigating this trade-off and the possible solutions is an
+interesting open problem.
+Sensing Fusion and Coordination: As presented in Sec-
+tion IV, in the third level of the digital twin, devices can
+communicate and coordinate to make joint sensing and com-
+munication decisions. For instance, multiple devices in the
+same location will capture similar sensing data. Most of this
+data may be redundant, and transferring this data over the
+limited communication bandwidth for further processing will
+only increase the computational cost overhead without signif-
+icantly improving the generated digital twin. One solution to
+improve resource utilization can be to sense and transfer the
+optimum data required to generate a complete and accurate
+digital twin. Realizing such a solution will require the devices
+to communicate with each other and adopt efficient protocols
+for cooperative sensing. These challenges highlight the need
+for further investigation in this direction.
+Central and Distributed Decision: Generating an accurate
+and complete real-time digital twin necessitates the sharing
+and transferring of sensing data across devices. Given access
+to this digital twin of the real world, the next question is,
+how will this replica of the real world be utilized to facilitate
+the sensing and communication decision-making process? For
+instance, a user can make individual decisions in a distributed
+manner or adopt a centralized way that fosters a collaborative
+decision-making environment. Both these approaches have
+their advantages and limitations. For example, the individual
+distributed decisions are mostly made on the edge devices,
+which will minimize any latency involved with data transfer
+back and forth from the cloud devices. However, they generally
+require computation-intensive machine learning-based models,
+which will further increase the overhead in these resource-
+constrained edge devices. Developing a robust and efficient
+solution that can be deployed in the real world requires a
+detailed investigation of these different possibilities.
+VIII. CONCLUSION
+This article presented and discussed a vision for future
+wireless communication and sensing systems that would lever-
+age precise 3D maps, distributed multi-modal sensing, effi-
+cient ray-tracing computations, and advanced machine/deep
+learning to construct, update, and utilize real-time digital
+twins. As computations, communication, and synchronization
+capabilities evolve over time, we expect devices to gradu-
+ally coordinate in building and updating global digital twins
+using their sensing information and potentially make joint
+sensing and communication decisions for the physical, access,
+network, and application layers. The article also presented
+a research platform, based on the real-world DeepSense 6G
+dataset and its digital replica, DeepVerse 6G, to investigate
+various digital twin research directions, and showed how to
+use this platform for one example application.
+REFERENCES
+[1] T. S. Rappaport, Y. Xing, O. Kanhere, S. Ju, A. Madanayake, S. Mandal,
+A. Alkhateeb, and G. C. Trichopoulos, “Wireless communications and
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+beyond,” IEEE Access, vol. 7, pp. 78 729–78 757, 2019.
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+[Online]. Available: https://arxiv.org/abs/2301.07682
+

U9FIT4oBgHgl3EQfgSvJ/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf,len=387
+page_content='1  Real-Time Digital Twins:  Vision and Research Directions for 6G and Beyond  Ahmed Alkhateeb, Shuaifeng Jiang, and Gouranga Charan  Abstract—This article presents a vision where real-time digital  twins of the physical wireless environments are continuously  updated using multi-modal sensing data from the distributed  infrastructure and user devices, and are used to make communi-  cation and sensing decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This vision is mainly enabled by the  advances in precise 3D maps, multi-modal sensing, ray-tracing  computations, and machine/deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This article details this  vision, explains the different approaches for constructing and  utilizing these real-time digital twins, discusses the applications  and open problems, and presents a research platform that can  be used to investigate various digital twin research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' INTRODUCTION  Heading towards 6G, communication systems are increas-  ingly featuring key trends such as the employment of large  numbers of antennas and the use of higher frequency bands [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  These technology trends bring higher data-rate and multiplex-  ing gains to the networks, but also impose critical challenges  on the ability of these systems to support highly-mobile,  energy-constrained, reliable, and low-latency applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' For  example, the deployment of large antenna arrays is associated  with high channel acquisition and beam sweeping overhead  [2], [3], which makes it hard for these massive MIMO systems  to support mobile applications, and the use of higher frequency  bands at mmWave and sub-THz makes the wireless links very  sensitive to line-of-sight (LOS) blockages, which challenges  the reliability and latency of the networks [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  In this article, we present a novel vision in which a real-time  digital twin is utilized to make operational physical, access,  network, and application layer decisions to the real physical  world in communication and sensing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The key features  of the envisioned digital twin-based wireless systems can be  summarized as follows:  Enabled by 3D maps and multi-modal sensing: The  envisioned digital twin will leverage precise 3D maps and  fuse multi-modal sensory data from distributed devices  and infrastructure nodes to construct an accurate real-time  digital replica of the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Capable of making real-time decisions: Leveraging  advances in real-time 3D ray-tracing, efficient computing,  and machine/deep learning, the envisioned digital twin  will be capable of making real-time decisions for the  wireless communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Continuously refined for better approximation: We  envision the digital twin as a model that will be continu-  ously refined to improve its approximation of the physical  The authors are with the School of Electrical, Computer and Energy  Engineering, Arizona State University, (Email: alkhateeb, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='jiang, gcha-  ran@asu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  world (including the electromagnetic and optical aspects)  and to enhance its decision accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  A global digital twin shared between devices: In its  ultimate version, we envision this digital twin to be  global and shared between devices such that all devices  can jointly enhance it and benefit from it using their  coordinated sensing and communication decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Used for all communication layer decisions: With the  real-time emulation of the physical world, the envisioned  digital twin can be utilized to make physical-layer de-  cisions such as channel prediction, as well as access,  network, and application layer decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  It is important to highlight here that the general concept  of real-time digital twins has been studied before in the  context of smart manufacturing, intelligent transportation, and  healthcare [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' For wireless communications, prior work has  mainly focused on network operation topics such as edge  computing, network optimization, and service management,  in which digital twins are leveraged to simulate the real  world at the network level [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' In contrast, we focus  on utilizing the real-time digital twin to simulate the real  world with a particular emphasis on the physical modeling  of the environment and wireless signal propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' To that  end, the envisioned real-time digital twins produce real-time  instantaneous and statistical information about the wireless  channels, which could be leveraged to make decisions for  the physical, access, network, or application layers of the  communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  The goal of this article is to expose the potential of real-  time digital twins for wireless communication and sensing  systems in 6G and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' In the next sections, we discuss  the key enabling technologies, the different approaches for  constructing and utilizing these digital twins, and the various  applications and future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' We also present  a research platform that could be used to investigate many  interesting digital twin research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' TODAY’S TECHNOLOGY ADVANCES LEAD TO  REAL-TIME DIGITAL TWINS  The vision of building real-time digital twins is motivated  by the recent advances in 3D maps, multi-modal sensing, ray  tracing, and machine/deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Next, we briefly discuss  these four key enablers for real-time digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Precise 3D Maps: 3D maps contain information about the  positions, shapes, orientations, and materials of the commu-  nication devices and other objects in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The  current application trends of AR/XR, autonomous driving,  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='11283v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='SP]  26 Jan 2023  2  Radar  LiDAR  Camera  IMU  Position  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  High-Fidelity Sensing  Real-Time Ray Tracing  ML/DL Advances  Precise 3D Map  Real World  Digital Replica  Precise Real-Time  3D Map  Channel Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Geometric Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  (Position/Mobility)  Desicions  Configurations  Operations  Twin  Construction  & Update  Real-Time  Ray Tracing  Task Solving  Vehicle  Radar  Vehicle  LiDAR  Surveillance  Camera  User  Position  Digital Twin Guided  Beamforming  User  IMU  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This figure presents the general idea of the digital twin with the four key enablers: precise 3D map, high-fidelity sensing, real-time ray tracing, and  ML/DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The digital twin is constructed based on a real-time 3D map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' By performing real-time ray tracing, the digital twin infers channel information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The  channel and geometry information can facilitate various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  and metaverse technologies created an increasing demand  for precise 3D map data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' In response to that, the 3D map  data collection, processing, and management capabilities have  been significantly advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Vehicle, airborne, and satellite  3D imaginary sensors are now used to collect and build  very accurate 3D maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Further, the growing computational  and database resources are making it possible to process and  manage large-scale 3D maps, even at the scale of the full world  like Nvidia OmniVerse [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Thanks to all these developments,  precise 3D maps are becoming more affordable and accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  High-Fidelity Sensing: Recent trends in sensing-aided  communication,  integrated  sensing  and  communication  (ISAC), and internet-of-things (IoT) tend to deploy multi-  modal sensors, such as cameras, radars, LiDARs, positioning,  at the infrastructure, user equipment, and IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' These  distributed sensors can complement each other since they have  different observing angles and different types of information,  such as position, shape, and mobility measures about the  various stationary and moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This sensing capability  can be further enhanced by leveraging the recent advances in  multi-modal data fusion [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' As a result, it is becoming more  feasible to acquire high-fidelity sensing information about the  surrounding environment in nearly real-time, which is a key  enabler for the envisioned digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Real-Time Ray Tracing: Ray tracing simulators attempt  to trace the wireless signal propagation paths between trans-  mit and receive antennas, and generate the parameters, such  as the angles of arrival/departure and complex path gains,  of these propagation paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' A main limitation of the ray  tracing simulators is that they typically require considerable  computational overhead, and hence, incur high latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The  significant advances in parallel computing hardware and ray-  tracing computational approaches over the last two decades,  however, are enabling real-time ray-tracing for both wireless  and optical signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This means that, given precise real-time  3D maps, the wireless channels between the (possibly mobile)  transmitters and receiver can potentially soon be computed in  the digital twins in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Advances in Machine/Deep Learning: Machine and deep  learning have demonstrated powerful capabilities in extracting  features, approximating complex functions, and solving non-  trivial optimization problems in wireless communication and  sensing/perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' In the digital twins, these advances in  machine/deep learning can be utilized to (i) enhance the  quality and reduce the cost of building precise 3D maps, (ii)  improve the efficiency of the multi-modal sensing in terms  of sensory data processing, transferring, sharing, and fusion  [10], and (iii) advance the ray tracing accuracy and reduce its  latency and computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  All these advances in precise 3D maps, high-fidelity sens-  ing, real-time ray tracing, and machine/deep learning are  making it more feasible to realize real-time digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  In particular, the precise 3D map and high-fidelity sensing  complement each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' while the 3D maps mainly contain  information about the static objects in the environment, the  high-fidelity sensing can augment these maps with information  about the dynamic objects in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Conducting ray tracing  on these real-time 3D maps leads to real-time wireless digital  twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Further, the advances in machine/deep learning can  be utilized to enhance the various aspects of these real-time  digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The real-time digital twins open opportunities for  novel capabilities and applications in wireless communication  3  Level 1: Local Information and Individual Decisions  Level 2: Shared Information and Individual Decisions  Level 3: Shared Information and Joint Decisions  Local/Partial Digital Twin  of Vehicle 1  Local/Partial Digital Twin  of Vehicle 2  Vehicles 1 and 2  Equipped with Multi-modal Sensors  Device 1 Local Decisions  (Ray-Tracing, Beam Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=')  Device 2 Local Decisions  (Ray-Tracing, Beam Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=')  Device 1 Local Decisions  (Ray-Tracing, Beam Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=')  Device 2 Local Decisions  (Ray-Tracing, Beam Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=')  Joint Decisions  (Ray-Tracing, Beam Pred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=')  Sensing Information is Projected to A Shared Digital Twin  Sensing Information is Projected to A Shared Digital Twin  V1  V1  V1  V2  V2  V2  Vehicle 1 Sensing Information is  Projected to A Local Digital Twin  Vehicle 2 Sensing Information is  Projected to A Local Digital Twin  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This figure presents our vision of how the digital twin system will operate in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' In particular, it shows three different operating modes  (i) local information and individual decision, (ii) shared information and individual decision, and (iii) shared information and joint decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  and sensing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' These digital twins can, for instance,  be leveraged to compute channel or channel covariance in-  formation, predict LOS link blockages, proactively predict  handover and traffic steering, and even predict application-  specific caching requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' TRUE DIGITAL TWINS THAT KEEP LEARNING  There are different approaches of how such a real-time  digital twin could be leveraged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' In this section, we first discuss  two main approaches where digital twins are either only used  for training machine learning models or for making real-time  decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Then, we present our vision for true digital twins  that keep learning and improving over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Digital Twins for Training ML Models: With the pre-  cise 3D maps and accurate ray-tracing simulators, we can  build high-fidelity and site-specific synthetic datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' These  datasets could be utilized to train machine learning models  for the various wireless communication and sensing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' In  particular, these synthetic datasets could be generated in large-  scale and with high variance, which is hard and expensive to  collect in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The machine learning models that are  trained on these site-specific synthetic datasets could then be  deployed to make inference/decisions for the physical world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Further, these models could be refined using limited real-world  datasets for better and more robust performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This approach  relaxes the latency requirements as the digital twins are not  used for real-time decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The drawback, however, is that  these ML models are not benefiting from the global real-time  sensing and awareness that the real-time digital twins have,  which may limit their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Digital Twins for Real-Time Decisions: Another approach  is to use these real-time digital twins to directly make real-time  or near real-time decisions for the physical-world communi-  cation and sensing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' For example, an FDD massive  MIMO basestation can use the real-time digital twin to predict  the downlink channel or, at least, the dominant subspace of  this channel, which saves large channel training and feedback  overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Mobile users may also use this digital twin, for  instance, to predict if their LOS links are going to be blocked  by a moving scatterer or whether they need to switch to other  beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This approach can, therefore, benefit from the real-time  nature of the digital twin and the richer awareness about the  surrounding environment in making real-time decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The  drawback, however, is that the decisions that are solely made  based on the digital twin will be very sensitive to the modeling  accuracy, which challenges the robustness of these decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  True Digital Twins: The previous two approaches have a  clear trade-off between the ability to benefit from the real-  time awareness of the digital twins to make efficient decisions  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=', in terms of wireless resources and mobility support)  and the ability to ensure the robustness of these decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  This motivates what we call true digital twins that can be  thought of as machine learning models themselves that keep  learning and improving their approximation of the physical  world, and hence their decisions, over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' In particular,  these digital twins could leverage learning agents and use  prior decisions and feedback to enhance the modeling accuracy  of the 3D maps, the processing and integration of the multi-  modal sensing data, and the approximation fidelity of the ray  tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' These true digital twins can, therefore, be used to  make decisions that are both real-time and accurate for the  various wireless communication and sensing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  4  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' THREE DIGITAL TWIN LEVELS  Constructing and utilizing digital twins requires interaction  with the devices that are contributing to the digital twins  with sensing data and leveraging the digital twins in making  decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This, however, raises questions about the required  level of coordination between these devices for both the  sensing and communication tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' For that, we envision that  digital twins will evolve through three main digital twin  levels of coordination as the computation, synchronization,  and communication capabilities develop over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Next, we  briefly present these operating modes (levels) and highlight  how they could function in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Local information and individual decision: We envision  the first level of the digital twin to incorporate local sensing  and individual decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' In particular, each device equipped  with one or more sensors will be expected to collect its  local sensing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The acquired local data can then  be projected onto a 3D map to generate a real-time digital  twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This, in general, will help generate a partial view of the  entire map, resulting in a localized digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This local real-  time digital twin can then be utilized to facilitate the sensing  and wireless communication decision-making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' An  advantage to such a local approach is that the limited sensing  information captured by each device can be processed locally  (on the device or edge), reducing the downstream processing  and decision-making latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' However, the partial nature of the  generated digital twin limits the scope of the decision-making  capabilities of the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' For example, enabling advanced  applications, such as future blockage prediction and handoff,  may require access to a more global real-time digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Shared information and individual decision: The pro-  posed first level of the digital twin is limited by the local  view of the wireless environment, which results in a partial  digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Generating an accurate real-time digital twin  and reaping its benefits requires a global overview of the  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Fusing the sensing information collected across  different devices in the wireless environment at any given  time can be a promising way to achieve this vision of a  comprehensive digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' As such, this idea forms the basis  of our vision of the second level of the digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Similar to  the first level, each device is expected to collect sensing data  of its surrounding environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' However, here, we propose  to fuse the information collected across devices to generate a  detailed and thorough digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The fusion operation can  be performed at the edge or cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Even though information  sharing across devices (or at the edge/cloud) is enabled,  each device is envisioned to undertake its own sensing and  communication decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This is due to the high computation,  synchronization, and communication requirement for joint and  globally-optimized decisions for all the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Shared information and joint/cooperative decision: With  the increased computation, synchronization, and communica-  tion capabilities of future devices, we envision that digital  twins will evolve into a more global form where devices can  also coordinate and jointly optimize their sensing and commu-  nication decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Therefore, in this third level of the digital  twins, devices are expected to share and fuse their sensing  information, mostly at the edge, similar to the second level,  to form a global and comprehensive real-time digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  In addition, the sensing and communication decisions will be  jointly optimized either in a central or distributed way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This  is expected to enhance both the sensing and communication  performance of future wireless systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' These three digital  twin levels are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' 2, clarifying the differences  in the information-sharing and decision-making approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' APPLICATIONS  The proposed digital twin leverages precise 3D maps and  high-fidelity sensing to capture real-time information about the  geometry and materials of the static and dynamic objects in  the environment and to construct a real-time digital replica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' By  performing real-time ray tracing on this digital replica, we can  infer various information about the communication channels,  such as the propagation path parameters (path loss, delays,  angles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' ), channel and covariance information, link quality,  and blockage status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Moreover, by utilizing the temporal and  spatial consistency of the channels, the digital twin can predict  future channel information in dynamic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The real-  time and future channel predictions can be leveraged to make  real-time and proactive decisions that can potentially improve  the communication system operations in the physical, access,  network, and application layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Physical Layer: The physical layer operation has, by defini-  tion, a clear dependancy on the wireless channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Many phys-  ical layer tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=', MIMO precoding and link adaptation,  directly rely on partial or full knowledge about the communi-  cation channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' However, the channel acquisition is typically  associated with high overhead, especially in large-dimensional  systems, which degrades the overall system efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Real-  time digital twins open novel opportunities to revolutionize the  channel acquisition process: When the real-time 3D maps and  ray tracing computations are sufficiently accurate, the digital  twin could be directly used to accurately infer the channels,  reducing or even eliminating the channel acquisition overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Further, in scenarios where the approximation is not sufficient  to accurately estimate the full channels, the digital twin may  still be used to predict the dominant channel subspaces and  reduce the channel acquisition overhead [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' These  digital twins can also be used to estimate the signal-to-noise  ratio (SNR) of the communication links and improve the  modulation and coding scheme selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Another interesting  physical layer application of the digital twin is to generate a  massive amount of data for a given site in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This  site-specific data can then be utilized to optimize the traffic  beam sets and the channel feedback compression codebooks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Access Layer: Current and future communication systems,  especially at higher frequencies, employ large antenna arrays  and highly-directional beams to achieve sufficient SNR and  realize high data rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Aligning these beams, however, typi-  cally requires large beam training overhead that scales with  the number of antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The real-time and future channel  information predicted by the digital twin can facilitate the  initial access (initial beam alignment) and beam management  for these systems and reduce the beam training overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  5  Cloud  Edge  Ray-Tracing  3D Map  AI/ML  D1: Sensing Data  D2: Sensing Data  Dn: Sensing Data  Physical World  Digital World  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' : MIMO Precoding  Channel Prediction  Link Adaptation  Initial Access  Beam Management  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' : Proactive Blockage Prediction  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' : Service Migration  Proactive Network Configuration  Occlusion  Will Lead to Link Disruption  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='g: Proactive Caching  Flexible Application-Specific Decisions  Physical Layer  Access Layer  Network Layer  Application Layer  Massive MIMO  Basestation  User  Propagation Paths  Precoding Matrix  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This figure presents some example communication applications that digital twins can enable across the different layers, such as the physical layer,  MAC layer, network layer, and application layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' For example, in the MAC layer, the digital twin can proactively predict future blockages and initiate hand-off  by tracking the mobility pattern of the different objects in the environment over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Further, due to the increased penetration loss, high frequency  communication links, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=', in mmWave, experience sudden  disturbance due to blockages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' By tracking the motion of the  user and other objects in the environment, the digital twins can  proactively predict the occurance and duration of incoming  blockages before they happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This enables seamless han-  dover control and improve the network reliability and latency  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Digital twins can alsp enhance the access layer  resource allocation, user scheduling, MU-MIMO user pairing,  and interference management, among many other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Network Layer: In prior work, digital twins have been  used to optimize the network layer operations and applications  such as device and traffic monitoring, resource allocation,  edge computing, and cyber security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' We refer to this type  of digital twins as the network-level digital twins since they  typically focus on monitoring network status and modeling  the network-level entities, services, and dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Differently,  the envisioned physical-level digital twins can provide fine-  grained information about the communication links, which  can also be used to improve the accuracy of the radio access  network (RAN) modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This accurate RAN modeling can  enhance the efficiency and reliability of various network-level  operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Further, the physical-level digital twins can provide  real-time and future information about the user position and  mobility characteristics, which could be leveraged to improve  several edge computing operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' For instance, when a user  is predicted to move out of the service area of the in-use edge  server, proactive service migration can be triggered to improve  the service quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The physical-level digital twins can also  work cooperatively and in an integrated manner with the  network-level digital twins to enhance the end-user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Application Layer: New emerging applications, such as au-  tonomous driving and AR/VR, pose more stringent reliability  and latency requirements to advanced communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  The 6G is envisioned to support 9-nine reliability with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='1ms  latency for mission and safety-critical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Moreover,  different applications often have diverse requirements for data  throughput, reliability, and latency, thus different strategies  when handling link instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The digital twin can simulate  the physical signal propagation and channels, which can offer  fine-grained information on the communication link quality,  both in real-time and proactively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This link quality information  provides more flexibility for making efficient application-layer  decisions in a way that respects this application diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' For  example, if a video streaming application knows ahead of time  about the communication link disruption and blockage status,  this application can pre-load a certain portion of the video,  considering the duration of the cut-off, and achieve a seam-  less user experience with efficient usage of communication  resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' While the digital twin provides low-level and fine-  grained information about the communication links, how to  efficiently utilize this information is still to be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' DIGITAL TWIN RESEARCH PLATFORM  Here, we present a digital twin research platform based on  the DeepSense 6G [13] and the DeepVerse 6G [14] datasets  that can be used to investigate various digital twin research  directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Next, we briefly present these digital twin datasets  and show how they can be used to investigate an example  application of digital twin-assisted beam prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  GPSPosition,Orientation  Camera  User  LiDAR  Radar  Future BlockageFuture Blockage  Prediction  Prediction Algorithm  InputSequenceEdgeServerEdges  Cloud Server  Service MigrationServerUE Movement6  Digital Twin Datasets: The digital twins rely on co-existing  real-world data and high-fidelity synthetic data generated using  accurate 3D maps and ray tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' For that, the DeepSense 6G  and the DeepVerse 6G datasets are well-suited to facilitate the  research, development, and application of digital twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The  DeepSense 6G is a large-scale multi-modal sensing and com-  munication dataset collected in various real-world scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  the DeepVerse 6G is a synthetic dataset that can simulate high-  fidelity multi-modal sensing and communication data from ray  tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Combining real-world scenarios and their synthetic  replicas from the two datasets, we present a digital twin  research platform to investigate its efficacy, limitations, use  cases, and applications in real-world systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Example Application: In [15], the digital twin is used to  first infer the channels with real-time 3D maps (containing  user positions) and ray tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Then, these channels can be  used to infer optimal beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' However, the high computational  complexity of ray tracing may result in high latency, thus  limiting real-time applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' For that, ML models can be  employed to approximate the ray tracing simulation with  accelerated computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Furthermore, ML can be leveraged  to learn the mapping function from 3D maps to the optimal  beam in an end-to-end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This can potentially improve  the computational efficiency since the ML has the flexibility  to not model the unnecessary ray tracing for a specific com-  munication problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' When the ML is trained solely using the  digital replica, it is limited by the impairments in the 3D map  and ray tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' A small amount of real-world data can be  utilized to fine-tune the ML models (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' transfer learning) and  to even transcend the digital replica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Experimental Results: Scenario 1 of DeepSense is adopted  as the real-world scenario, and the digital replica of this  scenario is constructed and simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' An ML model is first  trained on the position and wireless beam data from the digital  replica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Then it is fine-tuned on the real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The top-2  beam prediction accuracy is obtained by testing the model on  unseen real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' After training on 200 digital replica  data points, the ML model can achieve a high top-2 accuracy  of 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Note that this digital twin approach does not need  any real-world training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Moreover, with transfer learning,  a small number of real-world data points (less than 20) can  quickly improve the ML model to go beyond the digital replica  and achieve near-optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Next, the digital replica  impairment is explicitly modeled by adopting a uniform beam  steering codebook that is different from the beam codebook of  the communication hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The uniform codebook leads to  lower performance when fine-tuned on a very limited amount  of real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' However, when more than 20 real-world  data points are used for fine-tuning, the performance of the two  codebooks becomes very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The mismatches between the  real world and the digital replica can be efficiently calibrated  by a small amount of real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' FUTURE RESEARCH DIRECTIONS  Digital twins are envisioned to be an integral component  of 6G and beyond wireless communication systems and have  recently piqued the interest of both the industry and academia  0  20  40  60  80  100  Number of Real Data Points Used for Training  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='9  Top-2 Accuracy Tested on Real-World Data  Trained on real-world data  Transfer learining (measured codebook)  Transfer learining (unifrom codebook)  0  5  10  15  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='96  Transfer learning  Pre-trained on 200  synthetic data points  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' This figure shows the top-2 accuracy performance by first training the  NN on the synthetic data generated by the digital twin and then fine-tuning  it on real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' However, realizing this vision and exploring the true  potential of digital twins requires overcoming the fundamental  challenges and thoroughly investigating some of the important  aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Next, we present some open research directions that  need to be investigated toward enabling digital twin-aided  next-generation wireless communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Impact of Digital Twin on Communication Tasks: De-  ploying digital twin-aided solutions in the real world will re-  quire revisiting all the problem statements, such as LOS/NLOS  beam prediction, blockage prediction, hand-off, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=', to evalu-  ate the efficacy of these solutions accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' In particular, it  is necessary to investigate how much gain in performance can  be achieved by utilizing the real-time digital twins compared  to conventional and sensing-aided approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' To that end,  the recently published DeepVerse 6G synthetic dataset (with  multi-modal sensing and communication data generated from  3D models and ray-tracing) was created to mimic the real-  world scenarios of the DeepSense 6G dataset, effectively  making them a digital twin of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' These two datasets  combined can enable the development and evaluation of digital  twin-aided applications in real-world communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Communication-Sensing Trade-Off: In order to gener-  ate an accurate real-time digital twin with minimal latency,  the sensing data collected across different devices, in some  cases, need to be transferred quickly to a central unit for  further processing (digital twin levels 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The data  transfer rate is dependent on the available bandwidth of the  communication system itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' As the amount of sensing data  increases (for example, with the increase in sensing modalities  or the number of devices), so does the requirement for com-  munication bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' While access to diverse and detailed  sensing information can help generate a more accurate digital  twin, which improves the performance of the communication  system, transferring that sensing data will consume communi-  cation resources, potentially offsetting any gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' A promising  solution to overcome this challenge is to first process the  7  sensory data locally and extract relevant features, and then  transfer the low-dimensional extracted features across devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Investigating this trade-off and the possible solutions is an  interesting open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Sensing Fusion and Coordination: As presented in Sec-  tion IV, in the third level of the digital twin, devices can  communicate and coordinate to make joint sensing and com-  munication decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' For instance, multiple devices in the  same location will capture similar sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Most of this  data may be redundant, and transferring this data over the  limited communication bandwidth for further processing will  only increase the computational cost overhead without signif-  icantly improving the generated digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' One solution to  improve resource utilization can be to sense and transfer the  optimum data required to generate a complete and accurate  digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Realizing such a solution will require the devices  to communicate with each other and adopt efficient protocols  for cooperative sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' These challenges highlight the need  for further investigation in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  Central and Distributed Decision: Generating an accurate  and complete real-time digital twin necessitates the sharing  and transferring of sensing data across devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Given access  to this digital twin of the real world, the next question is,  how will this replica of the real world be utilized to facilitate  the sensing and communication decision-making process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' For  instance, a user can make individual decisions in a distributed  manner or adopt a centralized way that fosters a collaborative  decision-making environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Both these approaches have  their advantages and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' For example, the individual  distributed decisions are mostly made on the edge devices,  which will minimize any latency involved with data transfer  back and forth from the cloud devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' However, they generally  require computation-intensive machine learning-based models,  which will further increase the overhead in these resource-  constrained edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' Developing a robust and efficient  solution that can be deployed in the real world requires a  detailed investigation of these different possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' CONCLUSION  This article presented and discussed a vision for future  wireless communication and sensing systems that would lever-  age precise 3D maps, distributed multi-modal sensing, effi-  cient ray-tracing computations, and advanced machine/deep  learning to construct, update, and utilize real-time digital  twins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' As computations, communication, and synchronization  capabilities evolve over time, we expect devices to gradu-  ally coordinate in building and updating global digital twins  using their sensing information and potentially make joint  sensing and communication decisions for the physical, access,  network, and application layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
+page_content=' The article also presented  a research platform, based on the real-world DeepSense 6G  dataset and its digital replica, DeepVerse 6G, to investigate  various digital twin research directions, and showed how to  use this platform for one example application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FIT4oBgHgl3EQfgSvJ/content/2301.11283v1.pdf'}
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+Data assimilation finite element method for the linearized
+Navier-Stokes equations with higher order polynomial
+approximation
+Erik Burman∗ and Deepika Garg† and Janosch Preuss‡
+January 16, 2023
+Abstract
+In this article, we design and analyze an arbitrary-order stabilized finite element
+method to approximate the unique continuation problem for laminar steady flow de-
+scribed by the linearized incompressible Navier–Stokes equation. We derive quantitative
+local error estimates for the velocity, which account for noise level and polynomial degree,
+using the stability of the continuous problem in the form of a conditional stability esti-
+mate. Numerical examples illustrate the performances of the method with respect to the
+polynomial order and perturbations in the data. We observe that the higher order poly-
+nomials may be efficient for ill-posed problems, but are also more sensitive for problems
+with poor stability due to the ill-conditioning of the system.
+Key words: linearized Navier–Stokes’ equations, data assimilation, stabilized finite element
+methods, error estimates
+1
+Introduction
+The question of how to assimilate measured data into large-scale computations of flow problems
+is receiving increasing attention from the computational mathematics community [27, 22, 7, 35,
+26, 3]. There are several different situations where such data assimilation problems as can be
+seen in the above examples. One situation is when the data necessary to make the flow problem
+well-posed is lacking, for instance, when the data on the boundary of the the domain is unknown;
+instead, measurements are available in some subset of the bulk domain or boundary to make up
+for this shortfall. In such a case, the problem is typically ill-posed, and numerical simulations
+are significantly more challenging to perform than when handling well-posed flow problems. Ill-
+posed problems usually come up in inverse problems and data assimilation. Traditionally, these
+∗Department
+of
+Mathematics,
+University
+College
+London,
+London,
+UK–WC1E
+6BT,
+UK.;
+[email protected]
+†Department
+of
+Mathematics,
+University
+College
+London,
+London,
+UK–WC1E
+6BT,
+UK.;
+[email protected]
+‡Department
+of
+Mathematics,
+University
+College
+London,
+London,
+UK–WC1E
+6BT,
+UK.;
+[email protected]
+1
+arXiv:2301.05600v1  [math.NA]  13 Jan 2023
+
+ill-posed problems have been solved by regularizing at the continuous level, using e.g. Tikhonov
+regularization [37] or quasi-reversibility [33]. The regularized problem is well-posed and may
+be discretized using any appropriate numerical technique. Then, the regularization parameter
+must be tuned to the optimal value for the noise in the data. There is considerable literature
+of research on Tikhonov regularization and inverse problems, and we suggest the reader to [31]
+and its references for an overview of computational approaches employing this strategy. The
+quasi-reversibility methods relevant to the current study may be found in [10, 11, 21, 12].
+The goal of the current contribution is to develop a finite element approach directly ap-
+plied to the ill-posed variational data assimilation form. Regularization is then introduced at
+the discrete level utilizing stabilized finite element methods that allow for a comprehensive
+analysis employing conditional stability estimates. The idea is presented in [14] for standard
+H1-conforming finite element methods. Ill-posed problems are analyzed in [15], and in [17],
+the technique is extended to nonconforming approximations. In both cases, low-order approxi-
+mation spaces are considered. The error analysis requires the availability of sharp conditional
+stability estimates for the continuous problem. The estimates are conditional in the sense that a
+particular a priori bound must be assumed to hold for the solution, and the continuity provided
+in this bound is often merely H¨older [32]. In the literature, these estimates are referred to as
+quantitative uniqueness results and employ theoretical methods such as Carleman estimates
+or three-ball estimates [1, 30]. Error bounds derived using conditional stability estimates can
+be optimal because they reflect the approximation order of the finite element space and the
+stability of the ill-posed problem. In particular, when applied to a well-posed problem, the
+finite element method recovers optimal convergence.
+The ill-posed problem that we consider here is the unique continuation problem. The unique
+continuation problem for the Stokes equations was initially studied in [25]. The analysis of the
+stability properties of ill-posed problems based on the Navier–Stokes equations is a very active
+field of research, and we refer to the works [4, 5, 6, 9, 28, 29, 34] for recent results.
+This study aims to determine whether using high-order methods in the primal-dual stabilized
+Galerkin methods is as helpful in the ill-posed case as in the well-posed situation. Inspired by
+the approach proposed in [8] for the lowest-order finite element discretization of the unique
+continuation problem subject to the Navier-Stokes equations, here we generalize the method to
+arbitrary polynomial orders and investigate the benefits of using higher-order polynomials in
+numerical experiments.
+The rest of the paper is organized as follows. In section 2, we introduce the considered
+inverse problem and some related stability estimates. In section 3, we describe the proposed
+stabilized finite element approximation of the data assimilation problem and state the local error
+estimate. The numerical analysis of the method is carried out in section 4. Finally, section
+5 presents a series of numerical examples which illustrate the performance of the proposed
+method.
+2
+The linearized Navier–Stokes problem
+Let Ω be an open polygonal (polyhedral) domain in Rd, d = 2, 3. Let (U, P) be the solution of
+the stationary incompressible Navier–Stokes equations and consider some perturbation (u, p)
+of this base flow. If the quadratic term is ignored, the linearized Navier–Stokes equations for
+2
+
+(u, p) can be written
+L(u, p) = f;
+in Ω,
+(1)
+∇ · u = 0
+in Ω,
+(2)
+where
+L(u, p) = (U · ∇)u + (u · ∇)U − ν∆u + ∇p.
+Here, ν is a diffusion coefficient. We assume that U belongs to [W 1,∞(Ω)]d and that (u, p)
+satisfies the regularity
+(u, p) ∈ [H2(Ω)]d × H1(Ω).
+For this problem, we assume that measurements on u are available in some subdomain
+ωM ⊂ Ω having a nonempty interior and our purpose is to reconstruct a fluid flow perturbation
+of u for system (1)–(2) based on the measurements of velocity.
+Now, we will present some useful notations. Consider the following spaces:
+V := [H1(Ω)]d, V0 := [H1
+0(Ω)]d, L0 := L2
+0(Ω), and L := L2(Ω)
+where L2
+0(Ω) = {p ∈ L2(Ω) :
+�
+Ω p = 0}. We also define the norms, for k = 1 or d,
+∥·∥L := ∥·∥[L2(Ω)]k , ∥·∥V := ∥·∥[H1(Ω)]k , ∥·∥V ′
+0 := ∥·∥[H−1(Ω)]k .
+Observe that in the definitions, we employ the same notation for k = 1 and k = d. For any
+subdomain X ⊂ Ω, we set
+|v|X :=
+��
+X
+|v|2
+� 1
+2
+, ∀ v ∈ [L2(X)]d.
+Next, define the bilinear forms as: for all (u, v) ∈ V × V
+a(u, v) :=
+�
+Ω
+((U · ∇)u + (u · ∇)U) · v + ν
+�
+Ω
+∇u : ∇v,
+(3)
+where H : G := �d
+i,j=1 Hi,jGi,j and, for all (p, v) ∈ L × V
+b(p, v) : =
+�
+Ω
+p∇ · v,
+(4)
+l(v) : =
+�
+Ω
+f · v.
+(5)
+The weak form of the inverse problem can be expressed as: f ∈ V
+′
+0, u|ωM being given, find
+(u, p) ∈ V × L0 such that
+u = q in ωM
+(6)
+and
+a(u, v) − b(p, v) + b(r, u) = ⟨f, v⟩V ′
+0 ,V0,
+∀ (v, r) ∈ V0 × L.
+(7)
+3
+
+Here, q ∈ [H1(ωM)]d corresponds to the exact fluid velocity on ωM, i.e. q is a solution to the
+linearized Navier-Stokes’ equations in ωM and has an extension u to all of Ω. Below in the
+finite element method we will assume that we do not have access to q, but only some measured
+velocities uM = q + δu. So uM corresponds to the exact velocity polluted by a small noise
+δu ∈ [L2(ωM)]d.
+Consider the linearized Navier–Stokes problem with a non-zero velocity divergence
+L(u, p) = f;
+in Ω,
+(8)
+∇ · u = g
+in Ω.
+(9)
+We assume that if the boundary conditions of system (8)–(9) are homogeneous Dirichlet bound-
+ary conditions, then it is well-posed. More precisely, we make the following assumption:
+Assumption A. For all f ∈ V ′
+0 and g ∈ L0 we assume that system (8)–(9) admits a unique
+weak solution (u, p) ∈ V0 × L0 and that there exists a constant CS > 0 depending only on U, ν
+and Ω such that
+∥u∥V + ∥p∥L ≤ CS(∥f∥V ′
+0 + ∥g∥L).
+(10)
+Furthermore, if ∥∇U∥[L∞(Ω)]d×d is small enough, then the Lax–Milgram lemma implies that
+Assumption A holds. The assumption of smallness on ∇U is a sufficient condition, there are
+reasons to believe that Assumption A holds in more general cases.
+In the homogeneous case (which corresponds to f = 0 in (1)–(2) or to f = 0 and g = 0 in
+(8)–(9)), a solution (u, p) satisfies a three-balls inequality which only involves the L2 norm of the
+velocity. This three-balls inequality result is stated in [34] (with their notations, A corresponds
+to U and B to ∇U).
+Theorem 2.1. (Conditional stability for the linearized Navier–Stokes problem). Let f ∈ V ′
+0,
+ωM ⊂ Ω and g ∈ L be given. For all B ⊂⊂ Ω, there exist C > 0 and 0 < τ < 1 such that
+|u|B ≤ C(∥f∥V ′
+0 + ∥g∥L + ∥u∥L)1−τ(∥f∥V ′
+0 + ∥g∥L + |u|ωM)τ,
+(11)
+for all (u, p) ∈ [H1(Ω)]d × H1(Ω) solution of (8)–(9).
+Proof. For the proof we refer the reader to [8, Appendix A].
+Theorem 2.1 provides a conditional stability result for ill-posed problems [1] in the sense that,
+for this estimate to be helpful, it must be accompanied by an a priori bound on the solution on
+the global domain (due to the presence of ∥u∥L on the right-hand side). Specifically, Theorem
+2.1 implies that a solution (u, p) in [H1(Ω)]d × H1(Ω) of problem (6) and (7), must be unique.
+For the pressure uniqueness holds up to a constant. Moreover, in inequality (11), the exponent
+τ depends on the dimension d, the size of the measure domain ωM and the distance between
+the target domain B and the boundary of the computational domain Ω.
+Moreover, let f ∈ [L2(Ω)]d and we introduce the operator A defined on (V × L0) × (V0 × L)
+by
+A((u, p), (v, r)) := a(u, v) − b(p, v) + b(r, u)
+(12)
+where a and b are respectively defined by (3) and (4). Thus, we look for (u, p) ∈ V × L0 such
+that
+A((u, p), (v, r)) = l(v) ∀(v, r) ∈ V0 × L
+(13)
+and (6) holds.
+4
+
+3
+Stabilized finite element approximation
+In this section, we first introduce a discretization of problem (13) using a standard finite ele-
+ment method. Then, the discrete inverse problem is reformulated as a constrained minimization
+problem in the discrete space where the regularization of the cost functional is achieved through
+stabilization terms. Finally, the estimation of the error between the exact continuous solution
+and the discrete solution of our minimization problem is stated in Theorem 4.2 which corre-
+sponds to our main theoretical result.
+Let {Th}h be a family of affine, simplicial meshes of Ω. For simplicity, the family {Th}h
+is supposed to be quasi-uniform. Mesh faces are collected in the set Fh which is split into
+the set of interior faces, Fint
+h , and of boundary faces, Fext
+h
+. For a smooth enough function v
+that is possibly double-valued at F ∈ Fint
+h
+with F = ∂T − ∩ ∂T +, we define its jump at F as
+[v] =: vT − − vT +, and we fix the unit normal vector to F, denoted by νF, as pointing from T −
+to T +. The arbitrariness in the sign of [v] is irrelevant in what follows.
+We next define a piecewise polynomial space as
+Pk(Th) :=
+�
+v ∈ L2(Ω) : v|T ∈ Pk(T)
+∀T ∈ Th
+�
+,
+where Pk(T), k ≥ 0, is the space of polynomials of degree at most k over the element T. Further,
+define a conforming finite element space as
+P c
+k(Th) :=
+�
+v ∈ H1(Ω) : v|T ∈ Pk(T) ∀ T ∈ Th
+�
+.
+Let V k
+h := [P c
+k(Th)]d, Wh := V0 ∩ V k1
+h , Q0
+h := L2
+0(Ω) ∩ P c
+k2(Th) and Qh := P c
+k3(Th). For the
+analysis below the polynomial degrees of the above spaces may be chosen as k ≥ 1, k1 ≥ 1,
+k2 ∈ {max{1, k − 1}, k} and k3 ≥ 1 and the convergence order will be given in terms of k.
+To make the notation more compact we introduce the composite spaces Vh := V k
+h × Q0
+h and
+Wh := Wh×Qh. We may then write the finite element approximation of (13): Find (uh, ph) ∈ Vh
+such that
+A((uh, ph), (vh, qh)) = l(vh),
+(14)
+for all (vh, qh) ∈ Wh.
+Let us introduce the measurement bilinear form to take into account the measurements on
+ωM given by (6).
+m(u, u) := |u|2
+ωM = γMξ−1
+�
+ωM
+u2,
+(15)
+where ξ = max(ν, ∥U∥[L∞(Ω)]d×d h) and γM > 0 will correspond to a free parameter representing
+the relative confidence in the measurements. The objective is then to minimize the functional
+1
+2m(uM − uh, uM − uh)
+(16)
+under the constraint that (uh, ph) satisfies (14).
+We now introduce the following discrete Lagrangian for ((uh, ph), (zh, yh)) ∈ Vh × Wh,
+Lh((uh, ph), (zh, yh)) := 1
+2m(uh − uM, uh − uM) + A((uh, ph), (zh, yh)) − l(zh).
+(17)
+5
+
+If we differentiate with respect to (uh, ph) and (zh, yh), we get the following optimality system:
+Find (uh, ph) ∈ Vh and (zh, yh) ∈ Wh such that
+A((uh, ph), (wh, xh)) = l(wh),
+(18)
+A((vh, qh), (zh, yh)) + m(uh, vh) = m(uM, vh),
+(19)
+for all (vh, qh) ∈ Vh and (wh, xh) ∈ Wh.
+However, the discrete Lagrangian associated to
+this problem leads to an optimality system which is ill-posed. To regularize it, we introduce
+stabilization operators that will convexify the problem with respect to the direct variables uh, ph
+and the adjoint variables zh, yh. We introduce Su : Vh × Vh → R, S∗
+u : Wh × Wh → R, Sp :
+Q0
+h × Q0
+h → R and S∗
+p : Qh × Qh → R. The choice of stabilization terms will be discussed later.
+For compactness, we introduce the primal and dual stabilizers: for all (uh, ph), (vh, qh) ∈ Vh
+Sh((uh, ph), (vh, qh)) = Sg((uh, ph), (vh, qh)) + ˜Sh((uh, ph), (vh, qh)),
+Sg((uh, ph), (vh, qh)) = γGLS
+�
+T∈Th
+�
+T
+h2
+Tξ−1
+T L(uh, ph)L(vh, qh) dx,
+(20)
+˜Sh((uh, ph), (vh, qh)) = α(h2k∇uh, ∇vh) + γu
+�
+F∈Fint
+h
+�
+F
+hFξF[∇uh · n][∇vh · n] ds
++ γdiv
+�
+Ω
+ξT(∇ · uh)(∇ · vh) dx,
+(21)
+where ξT = max(ν, ∥U∥[L∞(Ω)]d×d hT), ξF = max(ν, ∥U∥[L∞(Ω)]d×d hF) and γGLS, α, γu, and γdiv
+are positive user-defined parameters. And for all (zh, yh), (wh, xh) ∈ Wh
+S∗
+h((zh, yh), (wh, xh)) = S∗
+u(zh, wh) + S∗
+p(yh, xh),
+S∗
+u(zh, wh) = γ∗
+u
+�
+Ω
+∇zh : ∇wh dx,
+(22)
+S∗
+p(yh, xh) = γ∗
+p
+�
+Ω
+yhxh dx,
+(23)
+where γ∗
+u and γ∗
+p are positive user-defined parameters. Let us make some comments on these
+stabilization terms. The stabilization of the direct velocity acts on fluctuations of the discrete
+solution through a penalty on the jump of the solution gradient over element faces and has no
+equivalent on the continuous level. The form Sg(·, ·) is a Galerkin least squares stabilization. Let
+us mention that there is some freedom in the choice of dual stabilization, e.g. set S∗
+p(yh, xh) =
+γ∗
+p
+�
+Ω ∇yh∇xh dx. We will only detail the analysis for the first choice (23) below. We refer the
+reader to [14, 16] for a more general discussion of the possible stabilization operators.
+We may then write the discrete Lagrangian Lh : Vh × Wh → R, for all (uh, ph) ∈ Vh and
+(zh, yh) ∈ Wh.
+Lh((uh, ph), (zh, yh)) := 1
+2m(uh − uM, uh − uM) + A((uh, ph), (zh, yh)) − l(zh)
++1
+2Sg((uh − u, ph − p), (vh, qh)) + ˜Sh((uh, ph), (vh, qh)) − 1
+2S∗
+h((zh, yh), (zh, yh))
+(24)
+6
+
+If we differentiate with respect to (uh, ph) and (zh, yh), we get the following optimality system:
+Find (uh, ph) ∈ Vh and (zh, yh) ∈ Wh such that
+A((uh, ph), (wh, xh)) − S∗
+h((zh, yh), (wh, xh)) = l(wh),
+(25)
+A((vh, qh), (zh, yh)) + Sh((uh, ph), (vh, qh)) + m(uh, vh) = m(uM, vh)
++ γGLS
+�
+T∈Th
+�
+T
+fh2
+Tξ−1
+T L(vh, qh) dx,
+(26)
+for all (vh, qh) ∈ Vh and (wh, xh) ∈ Wh.
+4
+Stability and Error Analysis
+To prove the stability of our formulations, we need the following result.
+Lemma 4.1. There exists Cp such that for all vh ∈ Vh there holds
+∥vh∥H1(Ω) ≤ Cp(∥vh∥ωM + ∥∇vh∥L).
+(27)
+Proof. The following Poincar´e inequality is well known [24, lemma B.63]. If f : H1(Ω) → R is
+a linear functional that is non-zero for constant functions then
+∥v∥H1(Ω) ≤ Cp(|f(v)| + ∥∇v∥L),
+∀v ∈ H1(Ω).
+For instance, we may take
+f(v) =
+�
+ωM
+v dx ≤ C|v|ωM.
+As an immediate consequence we have the bound (27).
+Let us prove that the discrete problem is well-posed. We can write the discrete formulation
+in a more compact form. Let (uh, ph) = Uh, (zh, yh) = Zh, (vh, qh) = Xh and (wh, xh) = Yh.
+G((Uh, Zh), (Xh, Yh)) = Ah(Uh, Yh) − S∗
+h(Zh, Yh) + Ah(Xh, Zh) + Sh(Uh, Xh) + γM(uh, vh)ωM.
+(28)
+We define the norm on ([H2(Ω)]d + Vh) × (H1(Ω) + Qh)
+|||(Uh, Zh)|||2 := Sh(Uh, Uh) + γM|uh|2
+ωM + S∗
+h(Zh, Zh).
+(29)
+|||(Uh, Zh)||| defines a norm, since γM > 0, α > 0 and thanks to the Poincar´e inequality (27).
+The following result demonstrates the stability of the system (25)–(26).
+Theorem 4.1. The discrete bilinear form (28) satisfies the following inf-sup condition for some
+positive constant γ, independent of h:
+inf
+(Uh,Zh)∈Vh×Wh
+sup
+(Xh,Yh)∈Vh×Wh
+G((Uh, Zh), (Xh, Yh))
+|||(Uh, Zh)||| |||(Xh, Yh)||| ≥ γ.
+7
+
+Proof. In order to prove the stability result, it is enough to choose some (Xh, Yh) ∈ Vh×Wh
+for any arbitrary (Uh, Zh) ∈ Vh × Wh, such that
+G((Uh, Zh), (Xh, Yh))
+|||(Xh, Yh)|||
+≥ γ |||(Uh, Zh)||| > 0.
+First, consider the bilinear form in (28) with (Xh, Yh) = (Uh, −Zh):
+G((Uh, Zh), (Uh, −Zh)) = S∗
+h(Zh, Zh) + Sh(Uh, Uh) + γM(uh, uh)ωM
+= S∗
+h(Zh, Zh) + Sh(Uh, Uh) + γM|uh|2
+ωM.
+G((Uh, Zh), (Uh, −Zh)) ≥ |||(Uh, Zh)|||2 .
+(30)
+and
+|||(Uh, −Zh)||| ≤ |||(Uh, Zh)||| .
+(31)
+Finally, by dividing (30) by (31), we get the result.
+According to the Babuˇska–Neˇcas–Brezzi theorem (see [24]), the square linear system defined
+by (25)–(26) admits a unique solution for all h > 0.
+4.1
+Error Analysis
+Now recall the following technical results of finite element analysis.
+Lemma 4.2. Trace inequality [23]: Suppose F denotes an edge of T ∈ Th. For vh ∈ Pk(Th),
+there holds
+∥vh∥L2(F) ≤ Ch−1/2
+T
+∥vh∥L2(T).
+(32)
+Lemma 4.3. Inverse inequality [23]: Let v ∈ Pk(Th), for all k ≥ 0. Then,
+∥∇v∥L2(T) ≤ Ch−1
+T ∥v∥L2(T) .
+(33)
+Lemma 4.4. Let Ih : L2(Ω) → P c
+k(Th) be the Cl´ement interpolation. The following approxima-
+tion estimates hold for the interpolation operator Ih, see [24],
+∥Ihv∥L ≤ C ∥v∥L , ∀v ∈ L
+∥∇Ihv∥L ≤ C ∥∇v∥L , ∀v ∈ H1(Ω),
+(34)
+∥(v − Ihv)∥L + h ∥∇(v − Ihv)∥L ≤ Cht ∥v∥Ht(Ω) , for all v ∈ Ht(Ω), 1 ≤ t ≤ k + 1,
+(35)
+� �
+T∈Th
+∥∆(v − Ihv)∥2
+L2(T)
+�1/2
+≤ Cht−2 ∥v∥Ht(Ω) , for all v ∈ Ht(Ω), 2 ≤ t ≤ k + 1,
+(36)
+�
+� �
+F∈Fint
+h
+∥v − Ihv∥2
+L2(F)
+�
+�
+1/2
+≤ Cht−1/2 ∥v∥Ht(Ω) , for all v ∈ Ht(Ω), t ≤ k + 1,
+(37)
+�
+� �
+F∈Fint
+h
+∥∇(v − Ihv)∥2
+L2(F)
+�
+�
+1/2
+≤ Cht−3/2 ∥v∥Ht(Ω) , for all 1 ≤ v ∈ Ht(Ω), 2 ≤ t ≤ k + 1.
+(38)
+The same bound holds for interpolation of vector-valued functions, Ih : [L2(Ω)]d → V k
+h and for
+interpolation on Wh where homogeneous boundary conditions are imposed.
+8
+
+Using the above bounds to the componentwise extension of Ih to vectorial functions, we
+deduce the following approximation bound.
+Corollary 4.1. It holds for (u, p) ∈ [Hk+1(Ω)]d × Hk(Ω),
+� �
+T∈Th
+∥L(Ihu − u, Ihp − p)∥2
+L2(T)
+� 1
+2
+≤ Chk−1 �
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)
+�
+.
+In this section, we will present and prove several technical results.
+First observe that
+the formulation (25)–(26) is weakly consistent in the sense that we have a modified Galerkin
+orthogonality relation with respect to the scalar product associated to A:
+Lemma 4.5. (Consistency). Let (u, p) satisfy (1) and (uh, ph) be a solution of (25)–(26). Then
+there holds
+A((u − uh, p − ph), (wh, xh)) = −S∗
+h((zh, yh), (wh, xh)),
+∀(wh, xh) ∈ Wh.
+(39)
+Proof. The result follows by taking the difference between (13) and (25).
+Lemma 4.6. Let (u, p) ∈ [Hk+1(Ω)]d × L2
+0
+� Hk(Ω). Then,
+|||(u − Ihu, p − Ihp)||| ≤ Chk �
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)
+�
+.
+(40)
+Proof. The approximation bounds can be deduced using the component-wise extension of
+Ih to vector functions.
+Lemma 4.7. (Continuity). Let (u, p) ∈ [Hk+1(Ω)]d × L2
+0
+� Hk(Ω). Then,
+A((u − Ihu, p − Ihp), (zh, yh)) ≤ Chk �
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)
+�
+S∗
+h((zh, yh), (zh, yh))
+1
+2,
+(41)
+for all (zh, yh) ∈ Wh.
+Proof. Let us derive the estimate (41). Using the definition of A(·, ·) :
+A((Ihu − u, Ihp − p), (zh, yh)) = a(Ihu − u, zh) − b(Ihp − p, zh) + b(yh, Ihu − u).
+(42)
+Consider the first term on the right hand side of (42). Using the Cauchy–Schwarz inequality
+and Poincar´e inequality,
+a(Ihu − u, zh) ≤ C ∥U∥[W 1,∞]d (∥u − Ihu∥L + ∥∇(u − Ihu)∥L)(∥zh∥L + ∥∇zh∥L)
+≤ C ∥U∥[W 1,∞]d hk ∥u∥[Hk+1(Ω)]d S∗
+h((zh, 0), (zh, 0))
+1
+2.
+The second term of (42) can be handled as:
+b(Ihp − p, zh) ≤ ∥p − Ihp∥L ∥∇ · zh∥L
+≤ Chk ∥p∥Hk(Ω) S∗
+h((zh, 0), (zh, 0))
+1
+2.
+The last term of (42) can be handled as:
+b(yh, Ihu − u) ≤ ∥∇ · (u − Ihu)∥L ∥yh∥L
+≤ Chk ∥u∥k+1 S∗
+h((0, yh), (0, yh))
+1
+2.
+Finally, the result follows by combining all the above estimates.
+9
+
+Lemma 4.8. We assume that the solution (u, p) ∈ [Hk+1(Ω)]d × L2
+0 ∩ Hk(Ω) and we consider
+(uh, ph) ∈ Vh and (zh, qh) ∈ Wh the discrete solution of (25)–(26). Then there holds,
+|||(u − uh, p − ph), (zh, qh)||| ≤ C
+�
+hk(∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)) + γ
+1
+2m|δu|ωM
+�
+.
+(43)
+Proof. We introduce the discrete errors ζh = Ihu − uh, ηh = Ihp − ph. By this way,
+|||(u − uh, p − ph), (zh, qh)||| ≤ |||(u − Ihu, p − Ihp), (0, 0)||| + |||(ζh, ηh), (zh, qh)||| .
+(44)
+The first term of (44) can be handled by using the Lemma 4.6. Consider the second term of
+(44)
+|||(ζh, ηh), (zh, qh)|||2 = S∗
+h((zh, qh), (zh, qh)) + Sh((ζh, ηh), (ζh, ηh)) + γM|ζh|2
+ωM.
+To estimate the right-hand side, we notice that, using the second equation of (26) with (vh, qh) =
+(ζh, ηh)
+Sh((ζh,ηh), (ζh, ηh)) + γM|ζh|2
+ωM − A((ζh, ηh), (zh, qh))
+= Sh(Ihu, Ihp), (ζh, ηh)) + m(Ihu − u, ζh) − m(δu, ζh) − γGLS
+�
+T∈Th
+�
+T
+fh2
+Tξ−1
+T L(ζh, ηh) dx.
+(45)
+Using Lemma 4.5, we obtained
+A((u − Ihu, p − Ihp), (zh, qh)) + A((ζh, ηh), (zh, qh)) = −S∗
+h((zh, qh), (zh, qh)).
+(46)
+Adding (45) and (46),
+S∗
+h((zh, qh), (zh, qh)) + Sh((ζh, ηh), (ζh, ηh)) + γM|ζh|2
+ωM
+= A((Ihu − u, Ihp − p), (zh, yh))
+�
+��
+�
++ Sh(Ihu, Ihp), (ζh, ηh)) − γGLS
+�
+T∈Th
+�
+T
+h2
+Tξ−1
+T L(u, p)L(ζh, ηh))
+�
+��
+�
+dx
++ m(Ihu − u, ζh) − m(δu, ζh)
+�
+��
+�
+= (1) + (2) + (3).
+(47)
+We bound the terms (1)–(3) term by term. The first term is handled by using Lemma 4.7
+A((Ihu − u, Ihp − p), (zh, qh)) ≤ Chk �
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)
+�
+S∗
+h((zh, 0), (zh, 0))
+1
+2.
+(48)
+Consider the second term on the right hand side of (47)
+Sh(Ihu, Ihp), (ζh, ηh)) − γGLS
+�
+T∈Th
+�
+T
+h2
+Tξ−1
+T L(u, p)L(ζh, ηh) dx
+= (h2k∇Ihuh, ∇ζh) + γGLS
+�
+T∈Th
+�
+T
+h2
+Tξ−1
+T L(Ihu, Ihp)L(ζh, ηh) dx
+− γGLS
+�
+T∈Th
+�
+T
+h2
+Tξ−1
+T L(u, p)L(ζh, ηh) dx + γu
+�
+F∈Fint
+h
+�
+F
+hFξF[∇Ihuh · n][∇ζh · n] ds
++ γdiv
+�
+Ω
+ξT(∇ · Ihuh)(∇ · ζh) dx.
+(49)
+10
+
+We now estimate the terms on the right hand side of (49). Using the H1-stability of Ih, the
+first term of (49) can be handled as:
+(h2k∇Ihu, ∇ζh) ≤ Chk ∥u∥[H1(Ω)]d Sh((ζh, ηh), (ζh, ηh))
+1
+2.
+Consider the next two terms of (49). Using the Cauchy– Schwarz inequality and Corollary 4.1
+we obtain
+γGLS
+�
+T∈Th
+�
+T
+h2
+Tξ−1
+T L(Ihu, Ihp)L(ζh, ηh) dx − γGLS
+�
+T∈τh
+�
+T
+h2
+Tξ−1
+T L(u, p)L(ζh, ηh) dx
+=γGLS
+�
+T∈Th
+�
+T
+h2
+Tξ−1
+T L(Ihu − u, Ihp − p)L(ζh, ηh) dx
+≤
+�
+γGLS
+�
+T∈Th
+�
+T
+h2
+Tξ−1
+T L(Ihu − u, Ihp − p)2 dx
+� 1
+2 �
+γGLS
+�
+T∈Th
+�
+T
+h2
+Tξ−1
+T L(ζh, ηh)2 dx
+� 1
+2
+≤ Chk �
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)
+�
+Sh((ζh, ηh), (ζh, ηh))
+1
+2.
+The next term of (49) can be handled by using the Cauchy–Schwarz inequality and the estimate
+(38),
+γu
+�
+F∈Fint
+h
+�
+F
+hFξF[∇Ihuh · n][∇ζh · n] ds
+≤
+�
+�γu
+�
+F∈Fint
+h
+�
+F
+hFξF[∇Ihuh · n]2 ds
+�
+�
+1
+2 �
+�γu
+�
+F∈Fint
+h
+�
+F
+hFξF[∇ζh · n]2 ds
+�
+�
+1
+2
+≤
+�
+�γu
+�
+F∈Fint
+h
+�
+F
+hFξF[∇(Ihu − u) · n]2 ds
+�
+�
+1
+2
+Sh((ζh, ηh), (ζh, ηh))
+1
+2
+≤ Chk ∥u∥[Hk+1(Ω)]d Sh((ζh, ηh), (ζh, ηh))
+1
+2.
+Put together (49) leads to
+Sh(Ihu, Ihp), (ζh, ηh)) − γGLS
+�
+T∈Th
+�
+T
+h2
+Tξ−1
+T L(u, p)L(ζh, ηh) dx
+≤ Chk �
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)
+�
+Sh((ζh, ηh), (ζh, ηh))
+1
+2.
+The last term can be handled as:
+|m(Ihu − u, ζh) − m(δu, ζh)| ≤ C(|Ihu − u|ωM + |δu|ωM)γm|ζh|ωM
+≤ C(hk+1 ∥u∥[Hk+1(Ω)]d + γ
+1
+2m|δu|ωM)γ
+1
+2m|ζh|ωM.
+Put together (47) leads to
+|||(ζh, ηh), (zh, qh)|||2 ≤ C
+�
+hk�
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)
+�
++ γ
+1
+2m|δu|ωM
+�
+|||(ζh, ηh), (zh, qh)|||
+⇒ |||(ζh, ηh), (zh, qh)||| ≤ C(hk�
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)
+�
++ γ
+1
+2m|δu|ωM).
+The combination of the above estimates concludes the claim.
+11
+
+Corollary 4.2. Under the same assumptions as for Lemma 4.8, there holds
+∥u − uh∥V ≤ C
+�
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω) + h−kγ
+1
+2m|δu|ωM
+�
+,
+(50)
+and
+∥uh∥V ≤ C
+�
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω) + γ
+1
+2mh−k|δu|ωM
+�
+.
+(51)
+Proof. Using Lemma 4.8, we see that
+∥u − uh∥V = h−k ��hk(u − uh)
+��
+V ≤Ch−k(Sh(u − uh, u − uh) + |u − uh|2
+ωM)
+≤Ch−k�
+hk(∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)) + γ
+1
+2m|δu|ωM
+�
+≤C
+�
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω) + h−kγ
+1
+2m|δu|ωM
+�
+.
+The estimate (51) is immediate by using the triangle inequality and the estimate (50).
+The following theorem is the main theoretical result of the paper and states an error estimate
+for this method.
+Theorem 4.2. Let f ∈ [L2(Ω)]d and uM = u|ωM + δu be given. We assume that (u, p) ∈
+[Hk+1(Ω)]2 × L2
+0 ∩ Hk(Ω) is the solution of (13), and consider (uh, ph) ∈ Vh and (zh, yh) ∈ Wh
+the discrete solution of (25)–(26). Then for all B ⊂⊂ Ω there exists τ ∈ (0, 1) such that
+|u − uh|B ≤ Chkτ�
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω) + h−k|δu|ωM
+�
+.
+(52)
+Proof.
+Let us first consider the weak formulation of the problem satisfied by (ζ, η) =
+(u − uh, p − ph).
+A((ζ, η), (w, r)) = (f, w)L − A((uh, ph), (w, r)).
+We introduce uh and ph being fixed. The linear forms rf and rg on V0 and L respectively defined
+by: For all w ∈ V0 and r ∈ L
+⟨rf, w⟩V ′
+0,V + (rg, r)L := (f, w)L − A((uh, ph), (w, r)).
+(53)
+It follows that (ζ, η) is the solution of (8)–(9) with f and g in the right hand sides replaced
+respectively by rf and rg. Applying now corollary 2.1, we directly get
+|ζ|B ≤ C(∥rf∥V ′
+0 + ∥rg∥L + ∥ζ∥L)1−τ(∥rf∥V ′
+0 + ∥rg∥L + |ζ|ωM)τ.
+(54)
+Using (25), we can write the residuals: for all (wh, qh) ∈ Wh
+< rf, w >V ′
+0,V +(rg, r)L := (f, w − wh)L − A((uh, ph), (w − wh, r − qh)) − S∗
+h((zh, yh), (wh, qh)).
+(55)
+We take wh = Ihw and qh = Ihr in (55). Now, let us estimate the terms on the right hand
+side of (55). Consider the first two terms of (55)
+(f, w − wh)L−A((uh, ph), (w − wh, r − qh))
+= (f, w − wh)L − (a(uh, w − wh) − b(ph, w − wh)) + b(r − qh, uh)).
+(56)
+12
+
+Applying an integration by parts to the first two terms of (56) and using Lemma 4.8,
+(f, w − wh)L−(a(uh, w − wh) − b(ph, w − wh))
+= |
+�
+T∈Th
+�
+T
+L(u, p)(w − wh) dx −
+�
+T∈Th
+�
+T
+L(uh, ph)(w − wh) dx|
++
+�
+F∈Fint
+h
+�
+F
+|[∇(u − uh) · n]||(w − wh)| ds
+= |
+�
+T∈Th
+�
+T
+L(u − uh, p − ph)(w − wh) dx| +
+�
+F∈Fint
+h
+�
+F
+|[∇(u − uh) · n]||(w − wh)| ds
+≤ C
+� �
+T∈Th
+h2
+Tξ−1
+T ∥L(u − uh, p − ph)∥2
+L2(T)
+� 1
+2
+h−1 ∥w − wh∥L
++ C
+�
+� �
+F∈Fint
+h
+�
+F
+hFξF[∇(u − uh) · n]2 ds
+�
+�
+1
+2 �
+� �
+F∈Fint
+h
+�
+F
+h−1
+F ξ−1
+F (w − wh)2 ds
+�
+�
+1
+2
+≤ C |||(u − uh, p − ph), (zh, qh)||| h−1 ∥w − wh∥L
++ C
+�
+� �
+F∈Fint
+h
+�
+F
+hFξF[∇(u − uh) · n]2 ds
+�
+�
+1
+2 �
+� �
+F∈Fint
+h
+�
+F
+h−1
+F ξ−1
+F (w − wh)2 ds
+�
+�
+1
+2
+≤ C(hk ∥u∥[Hk+1(Ω)]d + hk ∥p∥Hk(Ω)) ∥w∥[H1(Ω)]d + hk ∥u∥[Hk+1(Ω)]d ∥w∥[H1(Ω)]d
+≤ Chk(∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)) ∥w∥[H1(Ω)]d .
+(57)
+The last term is handled using the Cauchy–Schwarz inequality and Lemma 4.8,
+b(r − qh, uh) ≤ ∥r − qh∥L ∥∇ · uh∥L
+≤ ∥r∥L ∥∇ · uh∥L
+≤ C(hk(∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)) + γ1/2
+m |δu|ωM) ∥r∥L .
+(58)
+Applying the above bounds in (56) leads to
+(f, w − wh)L−A((uh, ph), (w − wh, r − qh))
+(59)
+≤ C(hk(∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)) + γ1/2
+m |δu|ωM)(∥w∥[H1(Ω)]d + ∥r∥L).
+The last term of (55) is handled using Lemma 4.8
+S∗
+h((zh, yh), (wh, qh)) ≤ S∗
+h((zh, yh), (zh, yh))
+1
+2S∗
+h((wh, qh), (wh, qh))
+1
+2
+≤ C
+�
+hk�
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)
+�
++ γ1/2
+m |δu|ωM
+�
+(∥w∥[H1(Ω)]d + ∥r∥L).
+(60)
+As a consequence we can bound the quantity defined in (55) by
+⟨rf,w⟩V ′
+0,V + (rg, r)L
+≤ C(hk �
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)
+�
++ γ1/2
+m |δu|ωM)(∥w∥[H1(Ω)]d + ∥r∥L).
+(61)
+13
+
+Since this bound holds for all w ∈ V0 and r ∈ L, we conclude that
+∥rf∥V ′
+0 + ∥rg∥L ≤ C
+�
+hk�
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)
+�
++ γ1/2
+m |δu|ωM
+�
+.
+(62)
+Using the Poincar´e inequality (27), we have the bound
+∥ζ∥L ≤ C(|ζ|ωM + ∥∇ζ∥L) ≤ Ch−k �
+|hkζ|ωM +
+��hk∇ζ
+��
+L
+�
+≤ Ch−k |||(ζ, 0), (0, 0)|||
+≤ Ch−k�
+hk�
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)
+�
++ γ1/2
+m |δu|ωM
+�
+≤ C
+�
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω) + γ1/2
+m h−k|δu|ωM
+�
+.
+Thus, we can bound the terms in the right-hand side of (54) in the following way:
+∥rf∥V ′
+0 + ∥rg∥L + ∥ζ∥L ≤ C
+�
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω) + γ1/2
+m h−k|δu|ωM
+�
+.
+(63)
+And
+∥rf∥V ′
+0 + ∥rg∥L + |ζ|ωM ≤ Chk�
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω) + γ1/2
+m |δu|ωM
+�
+.
+(64)
+Using these two bounds in (54), we conclude that
+|ζ|B ≤ (∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω) + γ1/2
+m h−k|δu|ωM)1−τ(hk�
+∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω)
+�
++ γ1/2
+m |δu|ωM)τ
+≤ Chτk(∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω) + γ1/2
+m h−k|δu|ωM).
+which completes the proof of the theorem.
+5
+Numerical simulations
+In this section, we use several two-dimensional numerical examples to apply the methodology
+described in section 3. All experiments have been implemented using the open-source computing
+platform FEniCSx [36, 2]. A docker image to reproduce the numerical results is available at
+https://doi.org/10.5281/zenodo.7442458. The free parameters in (25)–(26) are set to
+α = γu = γdiv = γGLS = γ∗
+u = γ∗
+p = 10−1, γM = 1000.
+in all the numerical examples. In the first example we will verify the convergence orders for
+different polynomial orders using equal order interpolation k for all variables. Then we consider
+the same test case using the minimal polynomial order that results in the same error bounds,
+k1 = 1, k2 = max{k − 1, 1} and k3 = 1. Finally, we study the robustness of the error estimate
+with respect to the viscosity for a configuration where the target subdomain B is strictly
+downwind the data subdomain ωM, so that every point B is on a streamline intersecting ωM.
+14
+
+5.1
+Convergence study: Stokes example
+To demonstrate the convergence behaviour of the method introduced in section 3, we take the
+test case for the Stokes problem from [18]. Let Ω = [0, 1]2 be the unit square. We consider the
+velocity and pressure fields given by
+u(x, y) = (20xy3, 5x4 − 5y4)
+p(x, y) = 60x2y − 20y3 − 5.
+It is simple to demonstrate that (u, p) is a solution to the homogeneous Stokes problem with
+ν = 1, corresponding to the system (1)–(2) with U = 0 and f = 0. As a result, we consider
+(25)–(26) with U = 0 and f = 0. Two different geometric settings are considered: one in which
+the data is continued in the convex geometry, inside the convex hull of ωM, and one in which the
+solution is continued in the non-convex geometry, outside the convex hull of ωM. The convex
+geometry represented by Fig 1(a) is given as:
+ωM = Ω \ (0.1, 0.9) × (0.25, 1),
+B = Ω \ (0.1, 0.9) × (0.95, 1),
+and the non-convex geometry represented by fig 1(b) is given as:
+ωM = {(x, y) : 0.25 ≤ x ≤ 0.75, 0.05 ≤ y ≤ 0.5},
+B = {(x, y) : 0.125 ≤ x ≤ 0.875, 0.05 ≤ y ≤ 0.95}.
+(a) Convex geometry
+(b) Non-convex geometry
+Figure 1: Sketch of the domains used for computations in section 5.1.
+.
+We begin by performing the computation using unperturbed data. The relative L2- norm
+errors ∥u − uh∥[L2(B)]d / ∥u∥[L2(B)]d are computed in the subdomain B. In addition, we present
+the history of convergence of the residual quantity for velocity stabilization:
+�
+�γu
+�
+F∈Fint
+h
+�
+F
+hF[(∇uh − ∇Ihuh) · n]2 ds
+�
+�
+1
+2
+.
+15
+
+B
+wMB
+wMFig.
+2 displays the velocity, pressure errors and residual quantity in the convex and non-
+convex geometry.
+Filled squares, circles and triangles represent the velocity errors; dashed
+lines represent the pressure error, and the plain thin lines represent the residual. The expected
+order of convergence is observed for the residual in Lemma 4.8. The local velocity error behaves
+consistently with the convergence rates obtained in Theorem 4.2. We can also see in Fig. 2 that
+the higher order polynomials are more satisfactory for ill-posed problems. Next, we proceed
+with the numerical verification of the above method with data perturbation.
+Consider the
+perturbed data
+uM = u|ωM + δu,
+with random perturbations
+|δu|ωM = O(hk−θ),
+for some θ ∈ N0 available for implementing our method. According to Theorem 4.2, we have
+the estimate
+|u − uh|B �� Chkτ−θ(∥u∥[Hk+1(Ω)]d + ∥p∥Hk(Ω) + 1),
+(65)
+consequently, convergence requires the condition kτ − θ > 0. Figs 3–4 present the convergence
+history of the velocity, pressure and residual quantities with the data perturbation in the convex
+and non-convex geometry, respectively. The effect of different values of θ for relative L2-error
+are studied in Figs 3–4. The relative error for θ = 0 is displayed in Figs. 3(a) and 4(a). In
+both cases, the results are in agreement with the Theorem 4.2. As stated in (65), the p = 1
+polynomial approximation may diverge for θ = 1, which is confirmed by Fig. 3(b). Next, the
+method p = 2 converges linearly, whereas p = 3 still manages to converge, albeit at a slower
+rate. As shown in the Fig. 3(c), this result is consistent with the fact that for θ = 2, convergence
+is no longer observed for any p ≤ 3. Similar convergence results are observed in the non-convex
+domain as shown in Fig. 4. The results of Figs. 3–4 indicate that for the convex geometry
+τ ≈ 1 and for the non-convex geometry τ ≈ 2
+3. In Fig. 5-7 the same results are presented for
+the case where the minimal polynomial order is considered that is k1 = 1, k2 = max{k − 1, 1},
+k3 = 1. The results are very similar and we conclude that for these numerical examples there
+is no disadvantage in taking the smallest possible dual space.
+10−2
+10−1
+10−7
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+h
+k = 1
+k = 2
+k = 3
+10−2
+10−1
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+h
+O(h1)
+O(h2)
+O(h3)
+(a) Errors for convex geometry in Fig. 1(a)
+(b) Errors for non-convex geometry in Fig. 1(b)
+Figure 2: Relative error for geometrical setup displayed in Fig. 1.
+16
+
+10−2
+10−1
+10−7
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+h
+k = 1
+k = 2
+k = 3
+(a) θ = 0
+10−2
+10−1
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+h
+O(h)
+O(h2)
+O(h3)
+(b) θ = 1
+10−2
+10−1
+10−4
+10−3
+10−2
+10−1
+100
+101
+102
+h
+(c) θ = 2
+Figure 3:
+Relative error for geometrical setup Fig. 1(a) in terms of the strength of the data
+perturbation.
+10−2
+10−1
+10−4
+10−3
+10−2
+10−1
+100
+h
+k = 1
+k = 2
+k = 3
+(a) θ = 0
+10−2
+10−1
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+h
+O(h)
+O(h2)
+O(h3)
+(b) θ = 1
+10−2
+10−1
+10−2
+10−1
+100
+101
+102
+h
+(c) θ = 2
+Figure 4:
+Relative error for geometrical setup Fig. 1(b) in terms of the strength of the data
+perturbation.
+10−2
+10−1
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+h
+k = 2
+k = 3
+10−2
+10−1
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+h
+O(h1)
+O(h2)
+O(h3)
+(a) Errors for convex geometry in Fig. 1(a)
+(b) Errors for non-convex geometry in Fig. 1(b)
+Figure 5: Relative error with using the minimal polynomial order for geometrical setup displayed
+in Fig. 1.
+5.2
+Convergence study with varying viscosity
+In this subsection, we consider the flow of a viscous Newtonian fluid between two solid bound-
+aries at y = H, −H driven by a constant pressure gradient. The source term f is chosen such
+17
+
+10−2
+10−1
+10−5
+10−4
+10−3
+10−2
+10−1
+h
+k = 2
+k = 3
+(a) θ = 0
+10−1.8
+10−1.6
+10−1.4
+10−1.2
+10−1
+10−0.8
+10−4
+10−3
+10−2
+10−1
+100
+h
+O(h)
+O(h2)
+(b) θ = 1
+10−1.8
+10−1.6
+10−1.4
+10−1.2
+10−1
+10−0.8
+10−3
+10−2
+10−1
+100
+101
+h
+(c) θ = 2
+Figure 6:
+Relative error with using the minimal polynomial order for geometrical setup Fig.
+1(a) in terms of the strength of the data perturbation.
+10−2
+10−1
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+h
+k = 2
+k = 3
+(a) θ = 0
+10−1.6
+10−1.4
+10−1.2
+10−1
+10−0.8
+10−4
+10−3
+10−2
+10−1
+100
+101
+h
+O(h)
+O(h2)
+O(h3)
+(b) θ = 1
+10−1.6
+10−1.4
+10−1.2
+10−1
+10−0.8
+10−2
+10−1
+100
+101
+102
+h
+(c) θ = 2
+Figure 7:
+Relative error with using the minimal polynomial order for geometrical setup Fig.
+1(b) in terms of the strength of the data perturbation.
+that the solution of the plane Poiseuille flow
+u(x, y) = U(x, y) =
+� P
+2µ(H2 − y2), 0
+�
+,
+p(x, y) =
+�1
+2 − x
+�
+P,
+satisfies the model problem. We demonstrate the performance of the numerical method for
+varying viscosity in a domain where the target subdomain is aligned with the flow, shown in
+Figure 8, and defined by
+ωM = (0.0, 0.2) × (0.2, 0.8),
+B = (0.2, 0.8) × (0.45, 0.55).
+(66)
+As in the previous section, we have examined the convergence of the method by performing
+numerical tests on both unperturbed and perturbed data. We vary the viscosity between ν = 1
+and ν = 0.
+Observe that since no boundary conditions are imposed nothing needs to be
+changed in the formulation in the singular limit. Also note that the choice of ξT and ξF in
+(20)-(21) mimicks the choice for the stabilized method for (the well-posed) Oseen’s problem
+used to improve robustness in the high Reynolds limit. Also with reference to high Reynolds
+computations for the well-posed case we here consider equal order interpolation for all fields.
+We wish to explore if the results on stability for the unique continuation for convection–
+diffusion equations in the limit of small diffusivity [19, 20] carry over to the case of incompressible
+18
+
+flow.
+The key observation there was that for smooth solutions to the convection–diffusion
+equation the method had H¨older stable error estimates when diffusion dominates, similar to
+the analysis above, but in the convection dominated regime the stability in a subdomain slightly
+smaller than that spanned by the characteristics intersecting the data zone is Lipschitz. In that
+zone the convergence for the ill-posed problem coincides with that of the well-posed problem for
+piecewise affine approximation. As a means to study the effect of incompressibility we compare
+with the case where in addition to u in ωM, p is also provided as data in Ω. The proposed method
+can be modified to accommodate this case by including 1
+2 ∥ph − p∥Ω, as an additional term in
+the Lagrangian (17). Note that when the pressure is added the velocity pressure coupling is
+strongly reduced. The relative L2-errors for running the same problem as above are displayed
+in Fig. 9. Left side plots of Fig. 9 show the results without adding any additional pressure
+term, and right side plots of Fig. 9 display the results by including the pressure data. We
+observe that the results with pressure information are consistently better than those without.
+In particular for high order polynomials and high Reynolds number the information on the
+pressure appears to provide a very strong enhancement of the stability. Further, the effect of
+data perturbations for different values of the viscosity coefficient is studied with and without the
+pressure augmentation, see Figs. 10–11. We observe that if a priori information on the pressure
+is added and viscosity is reduced the convergence order for the relative L2-error increases. This
+is consistent with the results of [19, 20]. If the pressure is not added however we do not observe
+this effect and it appears from these computational examples that we can not expect the result
+from [20] to hold for linearized incompressible flow.
+In Fig. 10-11 we present the results under perturbations of data. These results show that
+the robustness under perturbations is also substantially enhanced if the pressure is known,
+indicating that the pressure velocity coupling introduces a strong sensitivity to perturbations.
+6
+Conclusions
+We have introduced a finite element data assimilation method for the linearized Navier-Stokes’
+equation. We proved the natural extension of the error estimates of [8] valid for piecewise
+affine approximation to the case of arbitrary polynomial orders.
+The expected increase in
+convergence rate was obtained, but the estimates also show that the sensitivity of the system
+to perturbations in data increase. The theoretical results were validated on some academic test
+cases. The main observations are that high order approximation for the ill-posed linearized
+Navier-Stokes’ equations pays off, at least for sufficiently clean data. The spaces for the dual
+variables on the other hand can be chosen with piecewise affine approximation without loss
+of accuracy of the approximation. A study where the viscosity was varied showed that the
+incompressibility condition and the associated velocity-pressure coupling severely compromise
+the convective Lipschitz stability that is known to hold in the zone in the domain defined by
+points on the characteristics intersecting the data zone. If additional data in the form of global
+pressure measurements were added the results improved and were similar to the those of the
+scalar convection–diffusion equation.
+Future work will focus on the nonlinear case and the possibility of enhancing stability by
+adding knowledge of some other variable than the pressure, such as for example a passive tracer
+as in scalar image velocimetry [13].
+19
+
+Figure 8: Data set ωM and error measurement regions (B).
+Acknowledgment
+This research was funded by EPSRC grants EP/T033126/1 and EP/V050400/1.
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+Fritz John.
+23
+
+10−2
+10−1
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+h
+O(h)
+O(h2)
+O(h3)
+(a) without pressure, ν = 100
+10−2
+10−1
+10−8
+10−6
+10−4
+10−2
+100
+h
+k = 1
+k = 2
+k = 3
+(b) with pressure, ν = 100
+10−2
+10−1
+10−4
+10−3
+10−2
+10−1
+100
+h
+(c) without pressure, ν = 10−2
+10−2
+10−1
+10−8
+10−6
+10−4
+10−2
+100
+h
+(d) with pressure, ν = 10−2
+10−2
+10−1
+10−4
+10−3
+10−2
+10−1
+100
+h
+(e) without pressure, ν = 10−4
+10−2
+10−1
+10−8
+10−6
+10−4
+10−2
+100
+h
+(f) with pressure, ν = 10−4
+10−2
+10−1
+10−4
+10−3
+10−2
+10−1
+100
+h
+(g) without pressure, ν = 0
+10−2
+10−1
+10−8
+10−6
+10−4
+10−2
+100
+h
+(h) with pressure, ν = 0
+Figure 9: Relative error for geometrical setup Fig. 8.
+24
+
+10−2
+10−1
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+h
+O(h)
+O(h2)
+O(h3)
+(a) without pressure θ = 0, ν = 100
+10−2
+10−1
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+h
+k = 1
+k = 2
+k = 3
+(b) with pressure θ = 0, ν = 100
+10−2
+10−1
+10−4
+10−3
+10−2
+10−1
+100
+101
+h
+(c) without pressure θ = 0, ν = 10−2
+10−2
+10−1
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+h
+(d) with pressure θ = 0, ν = 10−2
+10−2
+10−1
+10−4
+10−3
+10−2
+10−1
+100
+101
+h
+(e) without pressure θ = 0, ν = 10−4
+10−2
+10−1
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+h
+(f) with pressure θ = 0, ν = 10−4
+10−2
+10−1
+10−4
+10−3
+10−2
+10−1
+100
+101
+h
+(g) without pressure θ = 0, ν = 0
+10−2
+10−1
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+101
+h
+(h) with pressure θ = 0, ν = 0
+Figure 10:
+Relative errors in terms of the strength of the data perturbation for geometrical
+setup displayed in Fig. 8.
+25
+
+10−2
+10−1
+10−2
+10−1
+100
+101
+h
+(a) without pressure θ = 1, ν = 1
+10−2
+10−1
+10−4
+10−3
+10−2
+10−1
+100
+101
+h
+(b) with pressure θ = 1, ν = 1
+10−2
+10−1
+10−3
+10−2
+10−1
+100
+101
+102
+103
+h
+(c) without pressure θ = 2, ν = 1
+10−2
+10−1
+10−3
+10−2
+10−1
+100
+101
+102
+103
+h
+(d) with pressure θ = 2, ν = 1
+Figure 11:
+Relative errors in terms of the strength of the data perturbation for geometrical
+setup displayed in Fig. 8.
+26
+

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@@ -0,0 +1,2205 @@
+Characterization of attractors for non-autonomous
+Lotka-Volterra cooperative systems
+Juan Garcia-Fuentes1, Jos´e A. Langa1, Piotr Kalita1,2, Antonio Su´arez1
+1Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de
+Sevilla, Campus Reina Mercedes, 41012, Sevilla, Spain
+2Faculty of Mathematics and Computer Science, Jagiellonian University, ul.
+�Lojasiewicza 6, 30-348 Krakow, Poland
+E-mail addresses: J.G.-F.: [email protected] J.A.L.: [email protected] P. K.:
+[email protected] A.S.: [email protected]
+January 12, 2023
+Abstract
+In this paper we study the long time behaviour of cooperative n- dimensional non-auto-
+nomous Lotka–Volterra systems from population dynamics. We provide conditions to obtain
+extinction of a given subset of species. We also give sufficient conditions to get existence of
+a globally stable global solution with one species extinct, or with of all species extinct except
+one. Moreover, we obtain the structure of the forwards non-autonomous attractor in one, two,
+and three dimensions by constructing the heteroclinic connections between the globally stable
+solution and the semistables ones in cases of permanence and extinction of species.
+1
+Introduction
+The Lotka–Volterra system is a classical model of population dynamics representing the behavior
+of species in ecosystems. This model consists of the following n-dimensional system of differential
+equations, in which the unknown ui corresponds to the density of individuals of the i-th species:
+u1
+i “ ui
+¨
+˚
+˚
+˝aiptq ´ biiptqui ´
+n
+ÿ
+j“1
+j‰i
+bijptquj
+˛
+‹‹‚
+for i P t1, ..., nu.
+(1)
+ai determines the intrinsic growth rate of the species i, and bij represents the interaction between
+the species i and j. In particular bii stands for the intrinsic competition of the same species i.
+If bij is negative, then the species j contributes positively to the growth of the species i, and we
+1
+
+1
+INTRODUCTION
+speak about a cooperative system. Alternatively, if bij is positive, j contributes negatively to the
+growth of i which means that all species compete with the others.
+When these coefficients are constant, the system is autonomous. In such case if the matrix
+B “ pbijqn
+i,j“1 is Volterra–Lyapunov stable (see, for instance, [Tak96]), the full characterization of
+the asymptotic behavior of the solutions of the system is well known. In particular, it is possible to
+give conditions on the coefficients paiqn
+i“1 which guarantee the existence of a globally asymptotically
+stable equilibrium, in which all the species are present, or, on the contrary, conditions for this stable
+equilibrium to possess one or more species extinct. These observations allow to determine the full
+structure of the global attractor associated to (1) [Tak96, Gue17, HS22, AKL22]. However, the
+situation is completely different in the non-autonomous case, i.e., when coefficients depend on
+time, and the corresponding theory is not fully developed, in particular the one related to the
+detailed description of their global attractors. In the present paper we study the non-autonomous
+cooperative Lotka-Volterra system and we give conditions to describe its long time behaviour,
+including the characterization of their attractors in low dimensional cases.
+We briefly recall the most important results obtained for this non-autonomous version of the
+problem.
+In the works of Gopalsamy [Gop86a, Gop86b] the author treat periodic and almost
+periodic functions aiptq and bijptq in competitive framework, i.e.
+bijptq ě 0 for all t P R and
+i ‰ j. Conditions are given to ensure the existence of globally stable solutions with all components
+strictly positive for all time t P R. Ahmad and Lazer develop the study also for the competitive
+case, relaxing the requirement for periodicity or almost periodicity of coefficients, and still obtaining
+the existence of a unique globally stable strictly positive permanent solutions, i.e. with all species
+present [AL95, AL00], or with one of the species extinct [Ahm93, AL98]. On the other hand,
+Redheffer in [Red96] finds some general conditions on the coefficients to get a globally stable
+solution with no restriction on the sign for the interspecific parameters, but only for the case of
+permanence of the system.
+In the first part of this paper we obtain the result, not considered before, providing criteria for
+the extinction of any given subcommunity of species in the cooperative case. The main difficulty
+of the proof was to obtain the decay of the density of species despite the possible cooperation from
+the other species. In Lemma 4.1, we obtain that under the diagonal dominance condition (H) for
+the non-autonomous matrix B “ pbijqn
+i,j“1 and conditions (B), which links matrix B with intrinsic
+growth rates of individual species a “ paiqn
+i“1, densities for a given subcommunity of species decay
+jointly to zero. Condition (B) is a variation of the conditions in [Red96] although we allow some
+of the growth rates aiptq to be also negative. We also prove the existence of a stable solution for
+two particular cases of extinction in the n-dimensional species community: in which all species
+except one become extinct and in the case where only one species becomes extinct. These results
+are contained in Sections 4.3 and 4.4, respectively.
+The feature of the system (1) is, that by setting some of the variables to zero, one obtains,
+for each subset of the community t1, . . . , nu, a smaller system, modeling the dynamics of this sub-
+community, embedded in the original system. Hence, the above results of Redheffer, Ahmad and
+Lazer as well as the ones in our Section 4 allow us to construct the asymptotically stable solu-
+tions for the full community as well as all its subcommunities. Here a natural question appears,
+motivated by recent developments in the theory of attractors for non-autonomous dynamical sys-
+tems [KR11, CLR13, BCL20]: how to describe all solutions which are bounded both in the past
+and future, and how are they related to each other in terms of attraction when time tends to
+plus or minus infinity. This question naturally leads to the study of the geometrical structure of
+non-autonomous attractors, which consist of the globally asymptotically stable solutions, which
+2
+
+2
+NON-AUTONOMOUS COOPERATIVE LOTKA–VOLTERRA SYSTEM AND
+EXISTENCE OF GLOBAL SOLUTIONS
+were the main interest to previous researchers analysing (1), asymptotically stable solutions for
+the subcommunities, and heteroclinic connections between them. By the theory of exponential
+dichotomies for the linearized systems (see Section 2), based in [BP15] for dichotomies for upper
+triangular systems, and the non-autonomous versions of the Hadamard–Perron theorem on the
+existence of local stable and unstable manifolds [CLR13, KR11] we are able to construct the het-
+eroclinic connections between various stable and semi-stable (i.e. stable for the subsystems of the
+original systems) solutions. We also prove that all other solutions must be unbounded in the past.
+Thus, we show in Sections 5, 6, and 7 the complete structure of the non-autonomous attractor
+for various cases of extinction and persistence, for one, two, and three dimensions, respectively.
+Observe that the detailed description of a non-autonomous attractor is always a very difficult task,
+and we have just a few very recent papers on this subject (such as [CRLO23]). However, thanks
+to the previous results, we construct the full structure of the forward (and pullback) attractor for
+a non-autonomous problem governed by a system differential equations.
+2
+Non-autonomous cooperative Lotka–Volterra system and
+existence of global solutions
+2.1
+Lotka–Volterra system
+We use the following convention for the functions defined on the real line: aL “ inftPR aptq,
+aU “ suptPR aptq. Moreover, for two such functions a, b we will say that a ą b if aptq ą bptq for
+t P R and likewise we define the non-sharp inequalities a ě b.
+We will consider the following n-dimensional cooperative Lotka–Volterra system.
+u1
+i “ ui
+ˆ
+aiptq ´
+n
+ÿ
+j“1
+bijptquj
+˙
+,
+(LV-n)
+for i “ 1, ..., n, where aiptq and bijptq are continuous real-valued functions. Assume that the system
+is cooperative, i.e. bijptq ď 0 for i ‰ j, and biiptq ą 0 for all t P R, and for i “ 1, ..., n. Moreover,
+we make the following standing assumption of the column diagonal dominance
+˜
+cibii `
+n
+ÿ
+j“1,j‰i
+cjbij
+¸L
+ě δ ą 0
+for all i “ 1, ..., n,
+(H)
+for some positive constants tciun
+i“1.
+2.2
+Existence of global attracting solution. Case of permanence
+We recall the result in [Red96] which provides conditions for the existence of a positive solution
+u˚ “ pu˚
+1, ..., u˚
+nq, isolated from zero and infinity, which is forwards globally attractive for all
+trajectories with positive initial data upt0q ą 0.
+We need the following assumption: there exist positive vectors d ą 0 and d ą 0 such that
+¯dibii ď ai ď dibii `
+ÿ
+j‰i
+djbij
+for all i “ 1, ..., n.
+(A)
+The following result comes from Redheffer ([Red96, Theorem 1 (ii), (v), and (vi)]):
+3
+
+2
+NON-AUTONOMOUS COOPERATIVE LOTKA–VOLTERRA SYSTEM AND
+EXISTENCE OF GLOBAL SOLUTIONS
+Theorem 2.1. Assume (A) and (H). Then there exists a unique complete positive trajectory u˚
+separated from zero and infinity. This trajectory satisfies ¯di ď u˚
+i ď di for every t P R. Moreover,
+for every solution u of (LV-n) such that upt0q ą 0 the following convergence holds
+lim
+tÑ8 |uptq ´ u˚ptq| Ñ 0.
+The following result establishes Lipschitz continuous dependence on initial data for forward
+bounded solutions.
+Lemma 2.2. Let two solutions u and v with the initial data at t0 be bounded for t ě t0. Assume
+that |ai|, |bij| are bounded by a constant independent of time. Then there exists 0 ă κ (depending
+on the bounds on the coefficients and both solutions) such that
+|upt0 ` tq ´ vpt0 ` tq| ď eκt|upt0q ´ vpt0q| for t ě 0.
+Proof. We skip the proof which follows in a standard way from the Gronwall lemma and estimation
+of the difference of two solutions.
+If the solutions are separated from zero, a stronger result holds under the diagonal dominance
+assumption (H). The following lemma is proved in [Red96, Lemma 8].
+Lemma 2.3. Assume (H) and let u, v be two solutions of (LV-n) such that there exist positive
+constants σ1, σ2 ą 0 with σ1 ď uipτq, vipτq ď σ2 for every τ ě t0 and i P t1, . . . , nu. Then
+|upt0 ` tq ´ vpt0 ` tq| ď σ2
+σ1
+|upt0q ´ vpt0q|e´δσ1t,
+for every t ě 0.
+2.3
+Existence of global solution. Case of extinction
+We provide a generalized version of the above condition (A) for which there exists the global
+solution consisting of zeros on a subset of indexes denoted by J and far from zero and infinity
+on the remaining subset of indexes, denoted by I. Namely, we replace (A) with the following
+hypothesis
+0 ă ¯dibii ď ai ď dibii `
+ÿ
+j‰i,jPI
+djbij
+for all i P I Ă t1, ..., nu,
+(AI)
+for positive vectors pdjqjPI and pdjqjPI.
+As an application of Theorem 2.1 we have the next result. If the indexes are subdivided into
+two disjoint sets I and J, it is possible to assume that zero is a solution on the set J. Observe
+that (H) holds for any subset of indexes.
+Theorem 2.4. If (H) and (AI) hold, then there exists a complete trajectory u˚ of the (LV-n)
+such that u˚
+i ” 0 for i P J “ t1, . . . , nuzI and di ď u˚
+i ď di for i P I. For every solution u of
+(LV-n) such that uipt0q ą 0 for i P I and uipt0q “ 0 for i P J the following convergence holds
+lim
+tÑ8 |uptq ´ u˚ptq| Ñ 0.
+4
+
+3
+EXPONENTIAL DICHOTOMIES AND LINEARIZATION
+Remark 1. If J is nonempty we call the solution semitrivial, as it is equal to zero on some
+coordinates. Observe that if (A) holds then (AI) holds for every set of indexes I. In the same way,
+if (AI) holds, then the same condition is satisfied on any subset of I. Note that while Theorem 2.1
+gives the convergence of solutions with nonzero initial data for all coordinates, in above theorem
+we only have convergence for nonzero initial data on coordinates indexed by I. We will provide
+conditions which guarantee that solutions with all nonzero initial data also converge towards the
+semitrivial solution with some of the species (indexed by J) extinct.
+3
+Exponential dichotomies and linearization
+3.1
+Exponential dichotomies for upper triangular systems
+We will consider the following linear nonautonomous system of ODE’s
+x1ptq “ Dptqxptq,
+(2)
+where D : R Ñ Rnˆn is continuous and bounded, i.e. suptPR }Dptq} ď M, where M is a positive
+constant. The solutions of this system of ODE’s, with the initial condition xpt0q “ x0 are given by
+the multiplication of the initial data by the following fundamental matrix xptq “ MDpt, t0qx0. This
+fundamental matrix is invertible and if t ă t0, then MDpt, t0q “ MDpt0, tq´1. When t0 “ 0, we
+use the notation MDpt, 0q “ MDptq. Then MDpt, sq “ MDptqMDpsq´1. Moreover the fundamental
+matrix satisfies the following variational ODE M 1
+Dptq “ DptqMDptq for every t P R with the initial
+data MDp0q “ I.
+Definition 3.1. We say that the system (2) has an exponential dichotomy on an interval I with
+projection P : I Ñ Rnˆn, constant k ě 1 and exponents α, β ą 0 if the fundamental matrix satisfies
+the invariance property
+PptqMDpt, sq “ MDpt, sqPpsq for all t, s P I,
+(3)
+and we have the inequalities
+}MDpt, sqPpsq} ď ke´αpt´sq for all s ď t P I,
+(4)
+}MDpt, sqpI ´ Ppsqq} ď keβpt´sq for all t ď s P I.
+(5)
+Typically I “ R, I “ r0, 8q or I “ p´8, 0s. In each of these cases if we substitute s “ 0 in
+(3) we obtain PptqMDptq “ MDptqPp0q, and hence Pptq “ MDptqPp0qMDptq´1, so we can recover
+Pptq from Pp0q and the fundamental matrix. We denote Pp0q “ P. Substituting s “ t in (4) we
+obtain }Pptq} ď k. This means that if system (2) has an exponential dichotomy, then }Pptq} is
+bounded uniformly with respect to t P I, and hence the moduli of all entries of the matrix Pptq
+must be also bounded. If I “ R then the projection P is given uniquely, cf. [Cop78, page 19]. If
+I “ R´, then P in the definition of the dichotomy can be replaced by any other projection with
+the same kernel, cf. [BF20, Proposition 2], and if I “ R`, then P can be replaced by any other
+projection with the same range. If the equation has the dichotomy on R` and R´ with the same
+projection P and exponents then it also has the dichotomy on R with the same projection and
+exponents, cf. [BF20, Proposition 1] or [Cop78, page 19].
+5
+
+3
+EXPONENTIAL DICHOTOMIES AND LINEARIZATION
+We recall the results of [BP15] on upper triangular systems. We will consider the problem
+governed by the system
+ˆ
+xptq
+yptq
+˙1
+“
+ˆ
+Aptq
+Cptq
+0
+Bptq
+˙ ˆ
+xptq
+yptq
+˙
+,
+(6)
+assuming that Aptq, Bptq, Cptq are bounded and continuous. We discuss the relation between the
+fact that the smaller systems governed by the matrices given by the diagonal blocks x1ptq “ Aptqxptq
+and y1ptq “ Bptqyptq have the dichotomies, with the fact that (6) has the exponential dichotomy.
+We cite the following result, cf [BP15, Corollary 1]
+Corollary 3.2. Assume that the linear systems x1ptq “ Aptqxptq and y1ptq “ Bptqyptq have ex-
+ponential dichotomies on R, where Aptq P Rdˆd and Bptq P Rpn´dqˆpn´dq, and Cptq is piecewise
+continuous and bounded d ˆ pn ´ dq matrix. Then (6) has the exponential dichotomy on R.
+The projection P for (6) can be constructed in the following way.
+First we construct the
+projections for the dichotomies on R´ and R` in the following way, cf. [BP15, Theorem 1]
+P ` “
+ˆ
+P A
+L`P B
+0
+P B
+˙
+on R`
+P ´ “
+ˆ
+P A
+L´pIn´d ´ P Bq
+0
+P B
+˙
+on R´,
+(7)
+where P A and P B are the projections for systems with Aptq and Bptq respectively, and L`, L´
+are the linking operators given by
+L` “ ´
+ż 8
+0
+MApsq´1pId ´ P ApsqqCpsqMBpsq ds,
+L´ “
+ż 0
+´8
+MApsqP ApsqCpsqMBpsq´1 ds.
+Next, we construct the unique projection P, such that its kernel coincides with the kernel of P ´
+and its range with the range of P `. This projection gives the dichotomy on R.
+3.2
+Linearization of nonautonomous Lotka–Volterra system
+Assume that pu˚
+1ptq, u˚
+2ptq, . . . , u˚
+kptq, 0, . . . , 0q is a solution of (LV-n), such that u˚
+i are separated
+from zero and infinity for i P t1, . . . , ku. We consider the linearized system around this solution.
+Denoting u˚
+j “ 0 for j P tk ` 1, . . . , nu and wi “ ui ´ u˚
+i we obtain the system of differential
+equations
+w1
+iptq “ aiptqwiptq ´ u˚
+i ptq
+n
+ÿ
+j“1
+bijptqwjptq ´ wiptq
+kÿ
+j“1
+bijptqu˚
+j ptq ´
+n
+ÿ
+j“1
+bijptqwjptqwiptq.
+(8)
+The above system can be written as w1ptq “ Mptqwptq ` Rpwptq, tq, where R is a remainder,
+quadratic with respect to w. Dropping this quadratic term we get the linearized system v1ptq “
+Mptqvptq, where the unknown is now denoted by v. The system has the following block diagonal
+form
+v1ptq “
+ˆ
+Aptq
+Cptq
+0
+Bptq
+˙
+vptq,
+(9)
+6
+
+4
+ASYMPTOTIC STABILITY FOR SOLUTIONS WITH EXTINCTION
+where Aptq is k ˆ k matrix with entries given by aiiptq “ aiptq ´ řk
+j“1 bijptqu˚
+j ptq ´ biiptqu˚
+i ptq
+and aijptq “ ´bijptqu˚
+i ptq for j ‰ i. The matrix Bptq is an pn ´ kq ˆ pn ´ kq diagonal matrix
+with ciiptq “ aiptq ´ řk
+j“1 bijptqu˚
+j ptq. Finally, Cptq in an k ˆ pn ´ kq matrix with entries given by
+cijptq “ ´u˚
+i ptqbijptq.
+The following Lemma, which has been proved in [AL98, Lemma 3.6], states that the system
+with the matrix Aptq always has the exponential dichotomy with P “ I.
+Lemma 3.3. Assume the diagonal dominance condition (H). Let u˚ptq be a positive solution
+of (LV-n) separated from zero and infinity, and let Aptq “ paijptqq be an n ˆ n matrix of the
+system linearized around u˚ptq, that is aiiptq “ aiptq ´ řn
+j“1 bijptqu˚
+j ptq ´ biiptqu˚
+i ptq and aijptq “
+´bijptqu˚
+i ptq for j ‰ i. Let be MAptq will be a fundamental matrix of the system v1ptq “ Aptqvptq,
+that is
+M 1
+Aptq “ AptqMAptq,
+MAp0q “ I.
+There exist constants K ą 0 and γ ą 0 such that
+}MApt, sq} “ }MAptqM ´1
+A psq} ď Ke´γpt´sq
+for ´8 ă s ď t ă 8.
+Of course the above lemma remains valid if the dimension n is replaced with smaller numer k,
+then the linear stability holds in the k dimensional subspace.
+4
+Asymptotic stability for solutions with extinction
+4.1
+Extinction conditions for a given subset of species
+In this section we present the assumptions on ai and bij which enable the possibility of splitting the
+set t1, . . . , nu into the sum of two disjoint sets of indexes I and J. For any initial data upt0q ą 0
+the species indexed by J go to extinction, that is, uiptq Ñ 0 when t Ñ 8 for i P J and uiptq is
+separated from zero forward in time when i P I. We impose the following condition.
+#
+bii ¯di ` ε ď ai ď biidi ` řn
+j“1,j‰i bijpdj ` θcjq ´ ε for i P I
+ai ď řn
+j“1,j‰i bijpdj ` θcjq ´ ε for i P J
+(B)
+for some positive vectors d, ¯d ą 0, and positive numbers ε, θ ą 0. The constants cj appear in (H).
+Note that if i P J, then aU
+i ď ´ε, and if i P I then aL
+i ě ε. Hence ai is negative and separated
+from zero for i P J and positive and separated from zero for i P I.
+The proof of the following result is the generalization to n dimensions of the result of Ahmad
+[Ahm93, Lemma 2], where the two dimensional problem is considered and both sets I and J consist
+of a single index.
+Lemma 4.1. If (B) and (H) hold then for every solution u of (LV-n) such that upt0q ą 0 there
+exists t˚ ą t0 such that for every t ě t˚
+• ¯di ă uiptq ă di for i P I,
+• uiptq ă di for i P J.
+7
+
+4
+ASYMPTOTIC STABILITY FOR SOLUTIONS WITH EXTINCTION
+Proof. Step 1. We first prove that for every i P I there exists ti “ tpiq such that uiptq ą ¯di for
+every t ě ti. First suppose that there exists t ě t0 such that uiptq ą di. We prove that uipsq ą di
+for every s ě t. Indeed assume that uipsq ď di for some s ą t. Define r “ mintr ě t : uiprq “ diu.
+On one hand it must be 9uiprq ď 0. On the other hand
+u1
+iprq “ di
+˜
+aiprq ´ biiprqdi ´
+n
+ÿ
+j“1,j‰i
+bijprqujprq
+¸
+ě diε,
+a contradiction. Now assume that there exists t ą t0 such that uipsq ď ¯di for every s ě t. Then
+we would have
+u1
+ipsq
+uipsq “ aipsq ´ biipsquipsq ´
+n
+ÿ
+j“1,j‰i
+bijpsqujpsq ą aipsq ´ biipsq ¯di ě ε ą 0
+for every s ě t. This implies that uipsq Ñ 8 as s Ñ 8, so we have a contradiction.
+Step 2. Define the sets
+Ak “ r0, d1 ` kθc1q ˆ r0, d2 ` kθc2q ˆ ... ˆ r0, dn ` kθcnq for k ě 0.
+(10)
+We first prove that if u0 P Ak, then uptq P Ak for all t ě t0, i.e. the sets Ak are positively invariant.
+Assume that this is not the case, i.e. upt0q P Ak and for some t ě t0 uptq R Ak. Then there must
+exist s ą t0 and an index i such that uipsq “ di ` kθci and 9uipsq ě 0, and ujpsq ď dj ` kθcj for
+j ‰ i. We calculate
+aipsq ě biipsquipsq `
+n
+ÿ
+j“1,j‰i
+bijpsqujpsq “ biipsqcikθ ` biipsqdi `
+n
+ÿ
+j“1,j‰i
+bijpsqujpsq
+ě biipsqcikθ ` biipsqdi `
+n
+ÿ
+j“1,j‰i
+bijpsqdj `
+n
+ÿ
+j“1,j‰i
+bijpsqcjkθ
+“ biipsqdi `
+n
+ÿ
+j“1,j‰i
+bijpsqdj ` θk
+˜ n
+ÿ
+j“1
+cjbijpsq
+¸
+ą biipsqdi `
+n
+ÿ
+j“1,j‰i
+bijpsqdj ą biipsqdi `
+n
+ÿ
+j“1,j‰i
+bijpsqpdj ` θcjq.
+Now if i P I, we directly get the contradiction with (B), and if i P J, then the above computation
+implies that
+aipsq ą
+n
+ÿ
+j“1,j‰i
+bijpsqpdj ` θcjq,
+and we arrive at the contradiction with (B), too.
+Step 3. Now let k ě 0. We prove that if uptq P Ak`1 then there exists s ą t such that
+upsq P Ak. To this end, it is enough to show, that if uptq P Ak`1zAk, then there exists s ą t such
+that upsq P Ak.
+8
+
+4
+ASYMPTOTIC STABILITY FOR SOLUTIONS WITH EXTINCTION
+Step 3.1. We first prove that if ujptq ă dj ` pk ` 1qθcj for every j P t1, . . . , nu and every
+t ě t0, then for every i P t1, . . . , nu there exists s ą t0 such that uipsq ă di ` kθci. To this end
+suppose that uiptq ě di ` kθci for all t ě t0. Then for every t ě t0
+uiptq “ uipt0q exp
+˜ż t
+t0
+aiprq ´ biiprquiprq ´
+n
+ÿ
+j“1,j‰i
+bijprqujprq dr
+¸
+ď uipt0q exp
+˜ż t
+t0
+aiprq ´ biiprqdi ´ biiprqcjkθ ´
+n
+ÿ
+j“1,j‰i
+bijprqpdj ` pk ` 1qθcjq dr
+¸
+“ uipt0q exp
+˜ż t
+t0
+aiprq ´ biiprqdi ´
+n
+ÿ
+j“1,j‰i
+bijprqpdj ` θcjq ´ kθ
+n
+ÿ
+j“1
+bijprqcj dr
+¸
+ď uipt0q exp
+˜ż t
+t0
+aiprq ´ biiprqdi ´
+n
+ÿ
+j“1,j‰i
+bijprqpdj ` θcjq dr
+¸
+If i P I, then, by (B), uiptq ď uipt0q expp´εpt ´ t0qq, a contradiction with uiptq ě di ` kθci for
+every t ě t0. If i P J, then as biiprqdi ą 0, it follows by (B) that also uiptq ď uipt0q expp´εpt ´ t0qq
+and we arrive at the same contradiction.
+Step 3.2. We now show that if for some t and for some i we have uiptq ă di ` kθci and for
+every s ě t and every j P t1, . . . , nu we have ujpsq ă dj ` pk ` 1qθcj, then for every s ě t we also
+have uipsq ă di ` kθci. Assume the contrary, then there exists t˚ ą t such that uipt˚q “ di ` kθci
+and 9uipt˚q ě 0. This means that
+aipt˚q ě biipt˚quipt˚q `
+n
+ÿ
+j“1,j‰i
+bijpt˚qujpt˚q
+ě biipt˚qdi ` biipt˚qkθci `
+n
+ÿ
+j“1,j‰i
+bijpt˚qdj `
+n
+ÿ
+j“1,j‰i
+bijpt˚qpk ` 1qθcj
+“ biipt˚qdi ` kθ
+n
+ÿ
+i“1
+biipt˚qci `
+n
+ÿ
+j“1,j‰i
+bijpt˚qpdj ` θcjq
+ě biipt˚qdi `
+n
+ÿ
+j“1,j‰i
+bijpt˚qpdj ` θcjq.
+Now if i P I we immediately get the contradiction with (B), and if i P J the last estimate implies
+that aipt˚q ě řn
+j“1,j‰i bijpt˚qpdj ` θcjq which also, by (B) leads to a contradiction which ends the
+proof of Step 3.
+Step 4. For every initial condition u0 “ pu1pt0q, u2pt0q, ..., unpt0qq ą 0 there exists some k ě 0
+such that u0 P Ak, so, repeating the process above, we know that there exists some t˚ such that
+upt˚q P A0, and then
+uiptq ă di
+for every i “ 1, ..., N and every t ě t˚.
+Remark 2. Redheffer in [Red96, Lemma 5] obtains a similar result. However, note that we do
+9
+
+4
+ASYMPTOTIC STABILITY FOR SOLUTIONS WITH EXTINCTION
+not need that
+n
+ÿ
+j“1
+bijptqdj ą 0 for i P J.
+The diagonal dominance condition (H) holds with constants cj, but it does not have to hold with
+the constants dj, by which bij and aj relate.
+Lemma 4.2. Assume (H) and (B).
+Let u “ pu1, u2, ..., unq be solution of (LV-n) such that
+uipt0q ą 0 for i “ 1, .., n. If i P J then ui Ñ 0 when t Ñ 8.
+Proof. From Lemma 4.1 it follows that there exists t˚ ě t0 such that for every t ě t˚, we have
+uiptq ă di for i “ 1, ..., n. Then, if only k P J
+ˆukptq “ ˆukpt˚q exp
+˜ż t
+t˚ akpsq ´
+n
+ÿ
+j“1
+bkjpsqujpsqds
+¸
+ď dk exp
+˜ż t
+t˚ akpsq ´
+n
+ÿ
+j“1,j‰k
+bkjpsqdj ds
+¸
+ď dk exp p´εpt ´ t˚qq Ñ 0,
+whence ukptq Ñ 0 as t Ñ 8.
+4.2
+Backward unboundedness of solutions in case of extinction
+In this subsection we prove that if the set J is not empty, then any solution with all positive initial
+data must be backward unbounded.
+Lemma 4.3. Assume (H) and (B) and let J be nonempty. Every solution u of (LV-n) with the
+initial condition upt0q ą 0 is backward unbounded.
+Proof. We take a solution u with the initial condition upt0q P A0, where A0 is given by (10) with
+k “ 0. We take i P J, then
+u1
+i “ ui
+ˆ
+ai ´ biiui ´
+ÿ
+j‰i
+bijuj
+˙
+ď ui
+ˆ
+ai ´
+ÿ
+j‰i
+bijdj
+˙
+´ biiu2
+i ď ui
+ˆ ÿ
+j‰i
+bijcjθ ´ ε
+˙
+´ biiu2
+i ď ´εui.
+Since A0 is positively invariant, the straightforward application of the Gronwall lemma yields
+together with the fact that upt0q P A0
+di ě uipt0q ě uiptqeεpt´t0q.
+We deduce that there exists t˚
+0 such that upt˚
+0q R A0, otherwise we would get a contradiction by
+passing with t0 Ñ ´8.
+Now we suppose that the solution has the initial condition upt0q P Ak`1zAk. We are going to
+prove that there exists t˚ ď t0 such that, upt˚q P Ak`2zAk`1, and then we would get the backward
+unboundedness of the solution by iteration. For contradiction assume that for every t ď t0 we have
+uptq P Ak`1. Then, as every set in family tAkukě0 is positively invariant, for every t ď t0 uptq R Ak,
+and, arguing as in Step 3.2 of Lemma 4.1 there exists i P t1, 2, ..., nu such that uiptq ě di ` kθci
+for every t ď t0.
+10
+
+4
+ASYMPTOTIC STABILITY FOR SOLUTIONS WITH EXTINCTION
+Suppose that i P I. We prove that it must be 9uiptq ď ´ε for every t ď t0. Indeed, if there
+exists t1 ď t0 such that the opposite inequality holds, then at that time
+ai ą biiui`
+ÿ
+j‰i
+bijuj´ε ě biidi`biikθci`
+ÿ
+j‰i
+pbijdj`bijkθcj`bijθcjq´ε ě biidi`
+ÿ
+j‰i
+bijpdj`θcjq´ε
+and we get a contradiction with (B). Then, there exists t˚ ď t0 such that uipt˚q ě di ` pk ` 2qθci
+so it can not be uptq P Ak`1 for all the time in the past, a contradiction.
+Analogous argument with i P J yields the inequality
+ai ą
+ÿ
+j‰i
+bijdj ` bijθcj ´ ε,
+which also leads to a contradicion.
+4.3
+The case of extinction of all species except one
+We have given conditions which guarantee the extinction of given a subset of species.
+In this
+section we study the asymptotic behavior of the persistent species indexed by I. We prove that,
+as time goes to infinity, the quantities of these species tend to the unique separated from zero and
+infinity solutions of the subsystem given by the equations with index I. We start from the situation
+when only one species persists. Hence, we provide the conditions over the vector aptq so that the
+trajectory pu˚
+1, 0, ..., 0q is the global attractive solution of the system (LV-n). The function u˚
+1 is
+the solution of the logistic equation
+9u1ptq “ u1ptqpa1ptq ´ b11ptqu1ptqq
+separated from zero and infinity, which, by Theorem 2.1 is unique under assumptions (B1) below
+(which implies (B) for one dimensional system) and (H). Hence, we need to impose (H) and the
+version of (B) with I “ t1u and J “ t2, . . . , nu, namely we assume that there exists a number
+d1 ą 0, a vector d ą 0, and two numbers ε, θ ą 0 such that,
+#
+b11 ¯d1 ` ε ď a1 ď b11d1 ` řn
+j“2 b1jpdj ` cjθq ´ ε
+ai ď řn
+j“1,j‰i bijpdj ` cjθq ´ ε for i “ 2, ..., n
+(B1)
+We skip the proof of the following result which exactly follows the lines of the proof of [Ahm93,
+Lemma 4], where, however, only two species, one that persists and one that decays, are considered.
+Lemma 4.4. Assume (B1) and (H). Let be pu1, u2, ..., unq a solution of (LV-n) such that ¯d1 ă
+u1pt0q ă d1 and 0 ă uipt0q ă di for every i “ 2, ..., n, then u˚
+1 ´ u1 Ñ 0 as time tends to infinity.
+Finally, combining the above lemma with Lemma 4.1 and Lemma 4.2 we have the following
+result on the asymptotic behavior.
+Theorem 4.5. Supposse the hypothesis (H) and (B1) and let |b1j|U ă 8 for j “ t2, ..., nu, if
+pu1, u2, ..., unq is a solution with the initial conditions uipt0q ą 0 for i “ 1, ..., n, then ui Ñ 0 for
+i “ 2, ..., n and u1 ´ u˚
+1 Ñ 0 as time tends to infinity.
+Proof. By Lemma 4.1, we can choose t˚ as the initial time, where ¯d1 ă u1pt˚q ă d1 and 0 ă
+uipt˚q ă di for i “ 2, ..., n. Then, by Lemma 4.4, we have that u1ptq ´ u˚
+1ptq Ñ 0 and by Lemma
+4.2 we deduce uiptq Ñ 0 when t Ñ 8 for i “ 2, ..., n.
+11
+
+4
+ASYMPTOTIC STABILITY FOR SOLUTIONS WITH EXTINCTION
+4.4
+The case of extinction of one species
+We continue with the analysis of the case when only one species goes to extinction.
+Hence,
+we provide the conditions on the coefficients aiptq, bijptq which guarantee that the trajectory
+pu˚
+1, u˚
+2, ..., u˚
+n´1, 0q is the global attractive solution when t Ñ 8 for (LV-n), and where pu˚
+1, u˚
+2, ..., u˚
+n´1q
+is the solution of the Lotka–Volterra pn ´ 1q-dimensional system obtained by removing the last
+variable.
+We need to impose again (H) and the version of (B) with I “ t1, ..., n ´ 1u and J “ tnu,
+namely the existence of vectors d, ¯d P Rn and parameters θ, ε ą 0 such that
+#
+bii ¯di ` ε ď ai ď biidi ` řn
+j“1;j‰i bijpdj ` cjθq ´ ε
+for
+i “ 1, ..., n ´ 1
+an ď řn´1
+j“1 bnjpdj ` cjθq ´ ε
+(B2)
+for every t P R.
+Lemma 4.6. Assume (H) and (B2). Let be ¯u be a solution of an pn ´ 1q-dimensional Lotka–
+Volterra system obtained by setting the last variable to zero (un “ 0), with the initial data ¯upt0q ą 0.
+There exists a unique solution u˚ of pn ´ 1q-dimensional system, which is separated from zero and
+infinity and such that
+|¯uptq ´ u˚ptq| Ñ 0
+when
+t Ñ 8
+and
+¯di ď u˚
+i ptq ď di
+for i “ 1, ..., n ´ 1, and for all t P R.
+Proof. We use Theorem 2.1 for dimension pn ´ 1q.
+Lemma 4.7. Assume (H) and (B2) and let |bin|U ă 8 for i P t1, . . . , n ´ 1u.
+Let u˚ “
+pu˚
+1, u˚
+2, ..., u˚
+n´1q be the solution of pn ´ 1q-dimensional Lotka–Volterra system given by Lemma
+4.6 and let ˆu “ pu˚
+1, u˚
+2, ..., u˚
+n´1, 0q. There exists a constant δ ą 0 such that if for some t0 P R we
+have |ˆupt0q ´ u0| ă δ, then limtÑ8 |ˆuptq ´ uptq| “ 0 where u solves (LV-n) with upt0q “ u0.
+Proof. We start by studying the system linearized around ˆu “ pu˚
+1, ..., u˚
+n´1, 0q. We write
+wptq “ uptq ´ ˆuptq
+where uptq is a solution with the initial condition in a neighbourhood of ˆu. Then we obtain
+w1ptq “ Mptqwptq ` Rpwptq, tq,
+where Mptq and Rpw, tq are as in (8). The linearized system has the form (9) with Cptq being an
+pn ´ 1q ˆ 1 which is bounded from the bound on bin. Moreover Bptq is 1 ˆ 1 matrix with its entry
+given by anptq´řn´1
+j“1 bn,jptqu˚
+j ptq. We observe that the one dimensional system v1
+n “ Bptqvn has an
+exponential dichotomy with projection Pptq “ 1 since by (B2), it holds Bptq ď ´ε ă 0. Moreover,
+by Lemma 3.3, the system pv1, ..., vn´1q1 “ Aptqpv1, ..., vn´1q has an exponential dichotomy with
+Pptq “ Ipn´1qˆpn´1q, the pn ´ 1q ˆ pn ´ 1q identity.
+Hence, by Corollary 3.2 and results of [BP15] recalled in Section 3 the linearized system (9)
+admits an exponential dichotomy with the projection given by P “ Inˆn. Is easy to see that the
+time dependent projection Pptq is given by Pptq “ Mpt, 0qPMp0, tq “ Inˆn. So
+|vptq| ď K|vpsq|e´γpt´sq
+for all
+t, s P R,
+12
+
+5
+STRUCTURE OF THE ATTRACTOR FOR NON-AUTONOMOUS LOGISTIC
+EQUATION
+for some constants K, γ ą 0, and by [Cop65, page 70] we get the local asymptotic stability of pu
+given in the assertion of the lemma.
+In the next result we establish the global asymptotic stability. We skip the proof because it
+exactly follows the lines of [AL98, Theorem 2.3].
+Theorem 4.8. Assume (H) and (B2) and let |bij|U ă 8 and |ai|U ă 8 for i, j P t1, . . . , nu. Let
+be u a solution of (LV-n) such that upt0q ą 0. Then
+lim
+tÑ8 |uptq ´ ˆuptq| Ñ 0
+where ˆu “ pu˚
+1, ..., u˚
+n´1, 0q, pu˚
+1, ..., u˚
+n´1q being the globally asymptotically stable positive solution
+solution of pn ´ 1q-dimensional system given in Lemma 4.6.
+5
+Structure of the attractor for non-autonomous logistic
+equation
+We start from the analysis for the non-autonomous problem in one dimension.
+Although the
+results of this section are mostly known, we need them for the study of problems in two and three
+dimensions. Thus, the aim of this chapter is the study of
+u1ptq “ uptqpaptq ´ bptquptqq,
+(11)
+where a, b P CpRq, b ą 0.
+5.1
+Permanence
+We need the assumption
+dbptq ď aptq ď dbptq and bL ą 0,
+(12)
+with constants d, d ą 0. The next result follows from [Red96, Theorem 1. (ii)], cited above as
+Theorem 2.1.
+Lemma 5.1. The function uptq “ 0 for t P R is a solution of (11). Moreover, there exists a
+function d ď u˚ptq ď d for t P R which is a solution to (11). This is the unique complete solution
+separated away from zero and infinity.
+Next lemma provides the characterization of the asymptotic behavior of solutions to (11) other
+than u˚.
+Lemma 5.2. If u : R Ñ R is a solution to (11) with upt0q ě 0 then exactly one of the four
+possibilities below holds:
+(a) uptq “ 0 for every t P R,
+(b) u “ u˚,
+(c) if upt0q P p0, u˚pt0qq then limtÑ´8 uptq “ 0 and limtÑ8 u˚ptq ´ uptq “ 0,
+13
+
+5
+STRUCTURE OF THE ATTRACTOR FOR NON-AUTONOMOUS LOGISTIC
+EQUATION
+(d) if upt0q ą u˚pt0q then limtÑ´8 uptq “ 8 and limtÑ8 uptq ´ u˚ptq “ 0.
+Proof. If we take the initial data upt0q P p0, u˚pt0qq then it is clear that uptq P p0, u˚ptqq for every
+t. If inftPR uptq ą 0, then by [Red96, Theorem 1 (v)] we get a contradiction because the solution
+bounded away from zero must be unique. So we suppose that inftPR uptq “ 0. If there exists a time
+t1 P R such that upt1q “ 0, then u “ 0 by uniqueness of solution. Then there exists a decreasing
+sequence ttnu such that tn Ñ ´8, uptnq Ñ 0 and uptnq ă d. If for some t P ptn`1, tnq, we have
+uptq ą uptnq then there exists t˚ P rt, tnq such that upt˚q ą uptnq and on pt˚, tnq the function u is
+strictly less that d. Hence
+0 ą uptnq ´ upt˚q “
+ż tn
+t˚ upsqpapsq ´ bpsqupsqq ds ě
+ż tn
+t˚ upsqpapsq ´ bpsqdqds ě 0,
+a contradiction, and hence limtÑ´8 uptq “ 0. Convergence to u˚ as t Ñ 8 follows from [Red96,
+Theorem 1 (vi)]. If upt0q ą u˚pt0q then analogously as for the case (c) we have the convergence at
+`8 and the existence of a decreasing sequence tn Ñ ´8 such that uptnq Ñ 8 and uptnq ą d. If
+uptq ă uptnq for t P ptn`1, tnq then there exists t˚ P rt, tnq such that upt˚q ă uptnq and u is strictly
+greater than d on rt˚, tnq hence, analogously as in the case (c), limtÑ´8 uptq “ 8.
+We finish this section by recalling two results on one dimensional problem useful in further
+analysis. We first recall a result that the linearized problem has the exponential dichotomy with
+its only direction being exponentially stable.
+Lemma 5.3. There exist δ ą 0 and M ą 0 such that if w : R Ñ R solves the following one-
+dimensional problem linearized around u˚
+w1ptq “ wptqpaptq ´ 2bptqu˚ptqq,
+then we have
+|wptq| ď M|wpt0q|e´δpt´t0q for every t ě t0.
+Proof. The result is a direct application of Lemma 3.3 which in turn follows from [AL98, Lemma
+3.6].
+Finally we establish the invariance and monotonicity result for the solution of one-dimensional
+problem (11).
+Lemma 5.4. If upt0q ě d then uptq ě d for every t ě t0 and if upt0q ď d then uptq ď d for every
+t ě t0. If 0 ă upt0q ă d then uptq ě upt0q for every t ě t0 and if upt0q ą d then uptq ď upt0q for
+every t ě t0.
+Proof. The first part follows from [Red96, Theorem 1. (i)]. For the second part it is enough to
+prove that if 0 ă uptq ă d then u1ptq ą 0. Indeed if upt0q P p0, dq
+u1ptq “ uptq paptq ´ bptquptqq ą uptq
+ˆ
+aptq ´ bptq
+´a
+b
+¯L˙
+“ uptqbptq
+ˆaptq
+bptq ´
+´a
+b
+¯L˙
+ě 0.
+In turn, if uptq ą d, then
+u1ptq “ uptq paptq ´ bptquptqq ă uptq
+ˆ
+aptq ´ bptq
+´a
+b
+¯U˙
+“ uptqbptq
+ˆaptq
+bptq ´
+´a
+b
+¯U˙
+ď 0,
+and the proof is complete.
+14
+
+6
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 2-D SYSTEM
+5.2
+Extinction
+To get the criterion on the extinction of the single species we give a theorem that is a direct
+application of Lemma 4.2.
+Lemma 5.5. If aU ă 0 then for every solution of (11) with upt0q ą 0 we have limtÑ8 uptq “ 0
+and limtÑ´8 uptq “ 8.
+Proof. To get the assertion limtÑ´8 uptq “ 8 observe that since
+u1ptq
+uptq “ aptq ´ bptquptq,
+we deduce that
+lnpuptqq ´ lnpupt0qq “ aptq ´ bptquptq ď aU.
+Hence
+upt0q ě uptqe´aUpt´t0q,
+and the proof is complete.
+6
+Attractor for non-autonomous Lotka-Volterra 2-D system
+In Section 2 we recalled the result of [Red96] on the existence of complete trajectories which are
+bounded and separated from zero. In particular we have shown condition which guarantees the
+existence of such solution. This condition can be used for the whole n-dimensional system or for its
+subsystems obtained by setting some of the variables ui to zero. Using this theorem, combined with
+the results of Section 4, we present the conditions under which one can characterize the structure
+of the non-autonomous attractor for cooperative Lotka–Volterra problem in two dimensions. We
+will consider three cases depending on the globally asymptotically stable solution: either the one
+with both nonzero components, or the one with one nonzero component, or the one with both
+zeros. Hence we will consider the system
+#
+u1
+1 “ u1pa1ptq ´ b11ptqu1 ´ b12ptqu2q
+u1
+2 “ u2pa2ptq ´ b21ptqu1 ´ b22ptqu2q,
+(LV-2)
+with bL
+11, bL
+22 ą 0 and b12, b21 ď 0.
+We will use the notation Spt, τqu0 for τ P R and t ě τ to denote the process associated to
+(LV-2), i.e. the family of maps which assign to the initial data u0 at time τ the solution of (LV-2)
+at time t.
+6.1
+Structure of attractor for the case of permanence
+We will first study the case when there exists the complete solution bounded away from zero and
+infinity on both variables, which attracts all solutions with nonzero initial data as t Ñ 8. In such
+case, of the permanence of the two species, the structure of non-autonomous attractor is depicted
+in Figure 1.
+15
+
+6
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 2-D SYSTEM
+Figure 1: Four complete solutions and their connections for the two dimensional problem with
+globally asymptotically stable state with coexistence of both species. Black dot means the strictly
+positive function while white dot relates with the function identically equal to zero.
+Following Theorem 2.1 we have to impose the following conditions
+#
+pc1b11 ` c2b12qL ą 0,
+pc2b22 ` c1b21qL ą 0,
+#
+d1b11 ď a1 ď d1b11 ` d2b12,
+d2b22 ď a2 ď d1b21 ` d2b22,
+(13)
+with some constants d1, d2, d1, d2, c1, c2 ą 0. These bounds are exactly the conditions (A) and (H)
+for the two-dimensional case. We also assume that
+|b12|U, |b21|U, aU
+1 , aU
+2 ă 8.
+(14)
+Note that the above assumptions in particular imply that a1, a2 ą 0. The next result follows from
+[Red96, Theorem 1. (ii)], cited above as Theorem 2.1.
+Lemma 6.1. There exists a function u˚ “ pu˚
+1, u˚
+2q defined for t P R, the complete solution of
+(LV-2) such that ¯di ď ui ď di for i “ 1, 2. This is the unique complete trajectory bounded away from
+zero and infinity in both variables. Moreover, there exist functions ppu1, 0q, p0, pu2q and uptq “ p0, 0q
+defined for t P R that are complete solutions to (LV-2), such that ¯di ď pui ď di for i “ 1, 2.
+The next Lemma follows directly from Lemma 5.2.
+Lemma 6.2. If the function u : R Ñ R2 is a solution to (LV-2) with u1pt0q ě 0 and u2pt0q “ 0,
+then exactly one of the four possibilities below holds:
+(a) uptq “ p0, 0q for every t P R,
+(b) u “ ppu1, 0q,
+(c) if u1pt0q P p0, pu1pt0qq then limtÑ´8 u1ptq “ 0, limtÑ8 pu1ptq ´ u1ptq “ 0 and u2ptq “ 0 for
+t P R,
+16
+
+u1
+W2
+1
+u2
+u1
+W2
+u1
+W26
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 2-D SYSTEM
+(d) if u1pt0q ą pu1pt0q then limtÑ´8 u1ptq “ 8, limtÑ8 u1ptq ´ pu1ptq “ 0, and u2ptq “ 0 for
+t P R.
+The result analogous to Lemma 6.2 holds, and allows us to establish the connection from p0, 0q
+to p0, pu2q. In order to obtain the structure as in Fig. 1, it remains to establish connections from
+p0, pu2q, ppu1, 0q and p0, 0q to pu˚
+1, u˚
+2q.
+Theorem 6.3. There exists the trajectory of (LV-2) denoted by z “ pz1, z2q : R Ñ R2 such that
+lim
+sÑ´8 |pz1psq, z2psqq ´ ppu1psq, 0q| “ 0.
+and
+lim
+sÑ8 |pz1psq, z2psqq ´ pu˚
+1psq, u˚
+2psqq| “ 0.
+Analogous result holds for p0, pu2q.
+Proof. We prove that the solution ppu1, 0q is locally unstable, i.e. its non-autonomous unstable
+manifold
+W upppu1, 0qq “ tpt, pw1, w2qq : there exists a solution z : R Ñ R2 such that
+zptq “ pw1, w2q and
+lim
+sÑ´8 |zpsq ´ ppu1psq, 0q| “ 0u,
+is nonempty and intersects the interior of the positive quadrant. As we have seen in Section 2, cf
+Theorem 2.1, the permanent solution pu˚
+1psq, u˚
+2psqq attracts forward in time the solution with the
+initial data in this intersection. We start by proving the existence of the local unstable manifold.
+To this end we first study the system linearized around ppu1, 0q. We write
+wptq “ uptq ´ puptq
+where puptq “ ppu1ptq, 0q, and uptq is a solution with initial condition in a neighbourhood of pu. Then
+we obtain
+w1ptq “ Mptqwptq ` Rpwptq, tq
+where Mptq is the derivative of the vector field at puptq and Rptq is a remainder which is of higher
+(quadratic) order with respect to w. In the 2D case we have the following ODE
+w1ptq “
+ˆ
+a1ptq ´ 2b11ptqpu1ptq
+´b12ptqpu1ptq
+0
+a2ptq ´ b21ptqpu1ptq
+˙
+wptq `
+ˆ
+´b11ptqw1ptq2 ´ b12ptqw1ptqw2ptq
+´b21ptqw1ptqw2ptq ´ b22ptqw2ptq2
+˙
+(15)
+The linearized system has the form
+#
+v1ptq “
+˜
+a1ptq ´ 2b11ptqpu1ptq
+´b12ptqpu1ptq
+0
+a2ptq ´ b21ptqpu1ptq
+¸
+vptq “
+˜
+Aptq
+Cptq
+0
+Bptq
+¸
+vptq.
+(16)
+where Aptq “ a1ptq ´ 2b11ptqpu1ptq, Cptq “ ´b12ptqpu1ptq and Bptq “ a2ptq ´ b21ptqpu1ptq. We observe
+that the one dimensional system v1
+2 “ Bptqv2 has an exponential dichotomy with projection PBptq ”
+0 since by (13) it holds Bptq ě δ ą 0 and for every t ě s
+v2ptq “ v2psqe
+şt
+s a2prq´b21prqpu1prqds ñ |v2ptq| ě |v2psq|eδpt´sq.
+17
+
+6
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 2-D SYSTEM
+We also observe that by Lemma 5.3 the system v1
+1 “ Aptqv1 has an exponential dichotomy with
+PAptq “ I. Then, by results recalled in Section 3, since by (14) and Lemma 6.1 the function Cptq
+is bounded, the system (16) admits an exponential dichotomy. According to (7), the projections
+are given by
+P ` “
+ˆ
+1
+0
+0
+0
+˙
+on R` and P ´ “
+ˆ
+1
+L´
+0
+0
+˙
+on R´.
+The projection
+P “ P ´ “
+ˆ
+1
+L´
+0
+0
+˙
+has the same range as P ` and hence the system (16) has exponential dichotomy with P. For the
+time t P R the associated projection is given by Pptq “ Mpt, 0qPMp0, tq, where Mpt, τq is the
+fundamental matrix of (16), which yields, as both Mpt, 0q and Mp0, tq are upper triangular and
+inverse to each other,
+Pptq “
+ˆ
+1
+Lptq
+0
+0
+˙
+,
+for certain bounded function Lptq.
+In particular dim ker Pptq “ 1 and dim range Pptq “ 1.
+Moreover,
+rangepPptqq “
+"
+β
+ˆ
+1
+0
+˙
+: β P R
+*
+and rangepI ´ Pptqq “
+"
+α
+ˆ
+´Lptq
+1
+˙
+: α P R
+*
+.
+We will use [KR11, Theorem 6.10 and Exercise 6.11]. There exists a neighborhood U of zero in
+R2 and a continuous function Σ´ : R ˆ U Ñ R2, a local non-auntonomous unstable manifold, such
+that Σ´pt, xq “ Σ´pt, pI ´ Pptqqxq P rangepPptqq, Σ´pt, 0q “ 0, lim|x|Ñ0
+Σ´pt,xq
+|x|
+“ 0. Moreover, if
+for some t P R and x P rangepI ´ Pptqq X U we have y “ x ` Σ´pt, xq, then, provided |y| is small
+enough, the backward trajectory w : p´8, ts Ñ R2 of (15) with wptq “ y belongs to the graph of
+Σ´, i.e. Σ´pτ, pI ´ Ppτqqwpτqq “ Ppτqwpτq and satisfies |wpτq| ď Ceδpτ´tq for every τ ď t. The
+points in the graph of Σ´ have the form
+y “ x ` Σ´pt, xq “ α
+ˆ
+´Lptq
+1
+˙
+` Σ´
+ˆ
+t, α
+ˆ
+´Lptq
+1
+˙˙
+“
+¨
+˝Σ´
+ˆ
+t, α
+ˆ
+´Lptq
+1
+˙˙
+´ αLptq
+α
+˛
+‚.
+(17)
+Because Lptq is bounded, there exists a neighborhood of zero in R such that for α in this neighbor-
+hood, we have x “ α
+ˆ
+´Lptq
+1
+˙
+P U. We take small positive α. The second coordinate of y is equal
+to α and hence positive, while the first coordinate can be made arbitrarily small, so that after
+adding to the solution ppu1ptq, 0q, with pu1 separated away from zero, the first coordinate of the sum
+is also positive. Hence, for any t P R there exists the point with second coordinate positive and
+first arbitrarily small, at time t, which is backward exponentially attracted to zero, and in terms
+of the original system, there exists the point at any t P R with both coordinates positive which is
+backwards exponentially attracted to ppu1ptq, 0q. In view of [Red96, Theorem 1 (iii) and (vi)] this
+gives us the assertion of the Theorem.
+18
+
+6
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 2-D SYSTEM
+Theorem 6.4. There exists a trajectory of (LV-2) denoted by y “ py1, y2q : R Ñ R2 such that
+lim
+sÑ´8 |py1psq, y2psqq| “ 0.
+and
+lim
+sÑ8 |py1psq, y2psqq ´ pu˚
+1psq, u˚
+2psqq| “ 0.
+Proof. The linearized system at p0, 0q has the form
+#
+v1ptq “
+˜
+a1ptq
+0
+0
+a2ptq
+¸
+vptq,
+(18)
+which has the exponential dichotomy with Pptq ” 0. Thus, by [KR11, Theorem 6.10 and Exercise
+6.11] there exists a sufficiently small neighborhood of p0, 0q such that the constant function Σ´ :
+R ˆ R2 Ñ t0u is the local unstable manifold of p0, 0q and thus every sufficiently small positive
+initial condition is exponentially backwards attracted to zero. Since it must be forwards attracted
+to pu˚
+1psq, u˚
+2psqq, the proof is complete.
+Theorem 6.5. If a trajectory with a strictly positive initial consition is bounded both in the past
+and in the future then it must be one of trajectories described above.
+Proof. From [Red96, Theorem 1. (v)] we know that u˚ is the only solution on R that is bounded
+away from 0 and 8. Thus, if the complete trajectory u “ pu1, u2q is bounded, not coincides with
+u˚, and none of the two coordinates is zero, then for at least one of its coordinates, say u1, there
+must exist a decreasing sequence tn Ñ ´8 such that u1ptnq Ñ 0 and u1ptnq ă d1. We shall prove
+that limtÑ´8 u1ptq “ 0. We proceed analogously to the Lemma 5.2. If for some t P ptn`1, tnq,
+we have u1ptq ą u1ptnq then there exists t˚ P rt, tnq such that u1pt˚q ą u1ptnq and on pt˚, tnq the
+function u1 is strictly less that d1. Hence
+0 ą u1ptnq ´ u1pt˚q “
+ż tn
+t˚ u1psqpa1psq ´ b11psqu1psq ´ b12psqu2psqq ds
+ě
+ż tn
+t˚ u1psqpa1psq ´ b11psqd1qds ě 0,
+and we have the contradiction. Now we need to prove that if both u1 and u2 do not converge
+backwards to zero, say u1ptq Ñ 0 as t Ñ ´8 and u2ptq is separated from zero then it must be
+lim
+tÑ´8 |u2ptq ´ pu2ptq| “ 0.
+To this end assume that for some sequence tn Ñ ´8
+lim
+nÑ8 |u2ptnq ´ pu2ptnq| “ c ą 0.
+For t ě 0 we have
+|u2ptnq ´ pu2ptnq| ď |Sptn, tn ´ tqpu1ptn ´ tq, u2ptn ´ tqq ´ Sptn, tn ´ tqp0, pu2ptn ´ tqq|.
+19
+
+6
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 2-D SYSTEM
+Since u2ptq is separated from zero, denote K “ pu2qL and M “ pu2qU. For every x P rK, Ms
+|u2ptnq ´ pu2ptnq| ď |Sptn, tn ´ tqpu1ptn ´ tq, u2ptn ´ tqq ´ Sptn, tn ´ tqp0, xq|
+` |Sptn, tn ´ tqp0, xq ´ Sptn, tn ´ tqp0, pu2ptn ´ tqq|
+ď eκt|pu1ptn ´ tq, u2ptn ´ tqq ´ p0, xq| ` 2 maxtM, d2u2
+mintK, d2u
+e´δ mintK,d2ut,
+where we have used Lemma 2.2 and 2.3, and κ depends on the bounds on u1, u2 and the coefficients
+of the problem. Pick ε ą 0. We can find t ą 0 such that 2 maxtM,d2u2
+mintK,d2u e´δ mintK,d2ut ă ε
+2. The
+sequence u2ptn ´ tq has a convergent subsequence, let x be a limit of this subsequence and pick n
+large enough such that Ceκt|pu1ptn ´ tq, u2ptn ´ tqq ´ p0, xq| ă ε
+2. Hence for n large enough, on a
+subsequence,
+|u2ptnq ´ pu2ptnq| ă ε,
+a contradiction.
+To finish this section we summarize the results obtained above, and we construct the full image
+of the atractor depicted in Figure 1.
+Theorem 6.6. Assuming (13) the system (LV-2) has the following trajectories u : R Ñ R2 bounded
+both in the past and in the future:
+(a) uptq “ p0, 0q for t P R,
+(b) uptq “ ppu1ptq, 0q and uptq “ p0, pu2ptqq, corresponding to the unique solutions for one-dimensional
+subproblems bounded away from zero and infinity, given in Lemma 6.1.
+(c) Solutions of type uptq “ pu1ptq, 0q with initial condition 0 ă u1pt0q ă pu1pt0q, where limtÑ´8 u1ptq “
+0 and limtÑ8pu1ptq ´ pu1ptqq “ 0, given in Lemma 6.2. Analogously, uptq “ p0, u2ptqq with
+initial condition 0 ă u2pt0q ă pu2pt0q, where limtÑ´8 u2ptq “ 0 and limtÑ8pu2ptq´pu2ptqq “ 0.
+(i) uptq “ pu˚
+1ptq, u˚
+2ptqq the unique solution with both nonzero coordinates bounded away from
+zero and infinity given in Lemma 6.1.
+(j) Solutions of type uptq “ pu1ptq, u2ptqq such that limtÑ´8pu1ptq, u2ptqq “ p0, 0q and limtÑ8pu1ptq´
+u˚
+1ptq, u2ptq ´ u˚
+2ptqq, given in Theorem 6.4.
+(k) Solutions of type uptq “ pu1ptq, u2ptqq such that limtÑ´8pu1ptq, u2ptqq “ ppu1ptq, 0q and limtÑ8pu1ptq´
+u˚
+1ptq, u2ptq ´ u˚
+2ptqq, given in Theorem 6.3.
+Analogously, uptq “ pu1ptq, u2ptqq such that
+limtÑ´8pu1ptq, u2ptqq “ p0, pu2ptqq and limtÑ8pu1ptq ´ u˚
+1ptq, u2ptq ´ u˚
+2ptqq “ 0.
+Moreover any solution of (LV-2) which is bounded both in the past and in the future is one of the
+solutions described in items (a)-(k).
+20
+
+6
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 2-D SYSTEM
+6.2
+Structure of attractor for the case of extinction of one species
+Now we study is the extinction of one species. Hence, we consider (LV-2) with assumptions
+#
+pc1b11 ` c2b12qL ą 0,
+pc1b21 ` c2b22qL ą 0,
+#
+b11d1 ` ε ď a1 ď b11d1 ` b12pd2 ` θc2q ´ ε,
+a2 ď b21pd1 ` θc1q ´ ε,
+(19)
+with some constants d1, d2, c1, c2, d1 ą 0 and constants ε, θ ą 0. The first inequalities constitute
+the condition (H), while the second one in (B1) for the two dimensional case. We also assume
+that |b12|U ă 8.
+We prove that the dynamics of the problem is described by the diagram presented in Fig. 2.
+Indeed, it is sufficient to use Lemma 4.3, Theorem 4.5, Lemma 5.1, Lemma 5.2, and Lemma 5.5
+to state the following theorem.
+Theorem 6.7. Assuming (19) the system (LV-2) has the following trajectories u : R Ñ R2 bounded
+both in the past and in the future
+(a) uptq “ p0, 0q for t P R,
+(b) uptq “ ppu1ptq, 0q for t P R, where pu1 is the unique trajectory of u1
+1 “ u1pa1ptq ´ b11ptqu1ptqq
+separated from zero and infinity given by Lemma 5.1,
+(c) uptq “ pu1ptq, 0q where limtÑ´8 u1ptq “ 0 and limtÑ8pu1ptq ´ pu1ptqq “ 0. Any solution with
+the initial data pu1pt0q, 0q satisfying 0 ă u1pt0q ă pu1pt0q constitutes such trajectory.
+All solutions other than the ones named in (a)–(c) are backwards unbounded. Moreover,
+(d) Any solution uptq “ pu1ptq, u2ptqq with initial data u1pt0q ą 0 and u2pt0q ą 0 satisfies
+limtÑ8pu1ptq ´ pu1ptq, u2ptqq “ p0, 0q and limtÑ´8 |uptq| “ 8.
+(e) Any solution uptq “ p0, u2ptqq with u2pt0q ą 0 satisfies limtÑ8 u2ptq “ 0 and limtÑ´8 u2ptq “
+8,
+(f) Any solution uptq “ pu1ptq, 0q with u1pt0q ą pu1pt0q satisfies limtÑ8pu1ptq ´ pu1ptqq “ 0 and
+limtÑ´8 u1ptq “ 8.
+Figure 2: Two complete solutions and their connections for the two dimensional problem for which
+the state with existence of one species is globally asymptotically stable.
+21
+
+u1
+u2
+1
+u27
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 3-D SYSTEM
+6.3
+Extinction of both species
+The last situation is the extinction of both species. To obtain this case we make the following
+assumptions
+#
+pc1b11 ` c2b12qL ą 0,
+pc2b22 ` c1b21qL ą 0,
+#
+a1 ď b12pd2 ` θc2q ´ ε,
+a2 ď b21pd1 ` θc1q ´ ε,
+(20)
+with some constants d1, d2, c1, c2 ą 0 and ε, θ ą 0. We observe that the first inequalities is condition
+(H) and the second ones is condition pBq with J “ t1, 2u and I “ H.
+Theorem 6.8. Assuming (20) the only trajectory of system (LV-2) which is bounded both in the
+past and and the future is uptq “ p0, 0q for t P R. Moreover for every solution with initial data
+u1pt0q ą 0 or u2pt0q ą 0 we have limtÑ8pu1ptq, u2ptqq “ p0, 0q.
+Proof. The convergence limtÑ8pu1ptq, u2ptqq “ p0, 0q follows from Lemma 4.2. It remains to prove
+that any solution with initial data u1pt0q ą 0 or u2pt0q ą 0 is backward unbounded, done it in
+Lemma 4.3.
+7
+Attractor for non-autonomous Lotka-Volterra 3-D system
+In this section we will study different possibilities of the non-autonomous attractor structure for
+the following three-dimensional system.
+$
+’
+&
+’
+%
+u1
+1 “ u1pa1ptq ´ b11ptqu1 ´ b12ptqu2 ´ b13ptqu3q,
+u1
+2 “ u2pa2ptq ´ b21ptqu1 ´ b22ptqu2 ´ b23ptqu3q,
+u1
+3 “ u3pa3ptq ´ b31ptqu1 ´ b32ptqu2 ´ b33ptqu3q.
+(LV-3)
+with bL
+ii ą 0 and bij ď 0 for i, j P t1, 2, 3u. Remind that Spt, τq for t ě τ denotes the mapping that
+assigns to the initial condition taken at time τ the value of the solution at time t, now associated
+to (LV-3).
+7.1
+Structure of attractor for the case of permanence
+The first case, that of permanence, corresponds to the dynamics depicted in Fig. 3. Following
+Theorem 2.1 we have to impose the following conditions on the coefficients of the problem to obtain
+the permanence of the three species, as well as the admissibility of all intermediate transient states
+together with the existence of all appropriate connections.
+22
+
+7
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 3-D SYSTEM
+Figure 3: Eight complete solutions and their connections for the three dimensional problem with
+globally asymptotically stable state with coexistence for all the species.
+To this end, we assume that there exist constants ci ą 0, di ą 0 and di ą 0 for i P t1, 2, 3u,
+such that the following two systems of inequalities are satisfied.
+$
+’
+&
+’
+%
+pc1b11 ` c2b12 ` c3b13qL ą 0,
+pc2b22 ` c1b21 ` c3b23qL ą 0,
+pc3b33 ` c1b31 ` c2b32qL ą 0.
+$
+’
+&
+’
+%
+b11 ¯d1 ď a1 ď b11d1 ` b12d2 ` b13d3
+b22 ¯d2 ď a2 ď b21d1 ` b22d2 ` b23d3
+b33 ¯d3 ď a3 ď b31d1 ` b32d2 ` b33d3
+(21)
+The above inequalities are exactly conditions pAq and (H), respectively, for the three-dimensional
+case. Moreover we need to assume that
+|bij|U ă 8 and aU
+i ă 8 for every i, j P t1, 2, 3u.
+(22)
+As a direct consequence of Theorem 2.1, analogously as in the two-dimensional case, we have
+the following lemma.
+Lemma 7.1. There exists a function u˚ “ pu˚
+1, u˚
+2, u˚
+3q defined for t P R, the complete solution of
+(LV-3) such that ¯di ď u˚
+i ď di for i “ 1, 2, 3. This is a unique complete trajectory bounded away
+from zero and infinity in all the variables. Moreover, there exist functions ppu1, pu2, 0q, p0, pu2, pu3q,
+ppu1, 0, pu3q, p¯u1, 0, 0q, p0, ¯u2, 0q, p0, 0, ¯u3q, and uptq “ p0, 0, 0q defined for t P R that are complete
+solutions to (LV-3), such that ¯di ď pui, ¯ui ď di for i “ 1, 2, 3.
+To obtain the attractor structure we prove the existence of the connections between the tra-
+jectories obtained in the above theorem, which play the role of non-autonomous equilibria. Next
+theorem is our first result in this context.
+Theorem 7.2. There exists the trajectory of (LV-3) denoted by z “ pz1, z2, z3q : R Ñ R3 such
+that
+lim
+sÑ´8 |pz1psq, z2psq, z3psqq ´ ppu1psq, pu2psq, 0q| “ 0.
+and
+lim
+sÑ8 |pz1psq, z2psq, z3psqq ´ pu˚
+1psq, u˚
+2psq, u˚
+3psqq| “ 0.
+23
+
+1
+W2
+3
+u1
+u2
+3
+1
+u2
+3
+u1
+u2
+u31
+u1
+2
+3
+u1
+u2
+3
+u1
+u2
+u1
+u7
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 3-D SYSTEM
+Analogous result holds for p0, pu2, pu3q and ppu1, 0, pu3q. Furthermore, there exist a trajectory denoted
+by w “ pw1, w2, w3q : R Ñ R3 such that
+lim
+sÑ´8 |pw1psq, w2psq, w3psqq ´ p¯u1psq, 0, 0q| “ 0.
+and
+lim
+sÑ8 |pw1psq, w2psq, w3psqq ´ pu˚
+1psq, u˚
+2psq, u˚
+3psqq| “ 0.
+Analogous result holds for p0, ¯u2, 0q and p0, 0, ¯u3q.
+Proof. The proof is analogous as for the two-dimensional case. First, we need to prove that the
+solution ppu1, pu2, 0q is locally unstable, i.e. its non-autonomous unstable manifold
+W upppu1, pu2, 0qq “ tpt, pw1, w2, w3qq : there exists a solution z : R Ñ R3 such that
+zptq “ pw1, w2, w3q and
+lim
+sÑ´8 |zpsq ´ ppu1psq, pu2psq, 0q| “ 0u,
+is nonempty and intersects the interior of the positive quadrant. We linearize the system around
+ppu1psq, pu2psq, 0q. So we write
+wptq “ uptq ´ ˆuptq
+(23)
+where ˆuptq “ ppu1ptq, pu2ptq, 0q, and now the linearized system has the form
+v1ptq “
+ˆ
+Aptq
+Cptq
+0
+Bptq
+˙
+vptq.
+(24)
+where
+Aptq “
+ˆ
+a1ptq ´ 2b11ptqpu1ptq ´ b12ptqpu2ptq
+´b12ptqpu1ptq
+´b21ptqpu2ptq
+a2ptq ´ b21ptqpu1ptq ´ 2b22ptqpu2ptq
+˙
+,
+Cptq “
+ˆ
+´b13ptqpu1ptq
+´b23ptqpu2ptq
+˙
+and Bptq “ a3ptq ´ b31ptqpu1ptq ´ b32ptqpu2ptq. We observe that the one dimensional system v1
+3 “
+Bptqv3 has an exponential dichotomy with projection Pptq “ 0 since by (21) it holds Bptq ě δ ą 0.
+Now ppu1ptq, pu2ptqq is the solution of the two-dimensional system obtained by taking u3 ” 0, and
+separated from zero and infinity. Hence, by Lemma 3.3, the system pv1, v2q1 “ Aptqpv1, v2q has an
+exponential dichotomy with Pptq “ I2ˆ2.
+Analogously as in two-dimensional case, we follow the results of Section 3, and since by (22) and
+Lemma 7.1 the function Cptq is bounded, and the system (24) admits an exponential dichotomy.
+So by (7), the projections are given by
+P ` “
+¨
+˝
+1
+0
+0
+0
+1
+0
+0
+0
+0
+˛
+‚ on R` and P ´ “
+¨
+˝
+1
+0
+L´
+1
+0
+1
+L´
+2
+0
+0
+0
+˛
+‚ on R´.
+Now the projection
+P “ P ´ “
+¨
+˝
+1
+0
+L´
+1
+0
+1
+L´
+2
+0
+0
+0
+˛
+‚,
+24
+
+7
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 3-D SYSTEM
+has the same range as P ` and hence the system (24) has an exponential dichotomy with P on
+whole R, and
+Pptq “
+¨
+˝
+1
+0
+L1ptq
+0
+1
+L2ptq
+0
+0
+0
+˛
+‚,
+for certain bounded functions L1ptq and L2ptq. Note that rangepPptqq “ tpβ1, β2, 0q : β1, β2 P Ru,
+of dimension 2, is always the stable space in the linearization, while
+rangepI ´ Pptqq “
+$
+&
+%α
+¨
+˝
+´L1ptq
+´L2ptq
+1
+˛
+‚ : α P R
+,
+.
+- ,
+of dimension 1, is the time dependent unstable space in the linearization.
+Analogously as in the two-dimensional case we use the non-autonomous unstable manifold
+theorem, cf., [KR11, Theorem 6.10 and Exercise 6.11]. There exists a neighborhood U of zero
+in R3 and a function Σ´ : R ˆ U Ñ R3 such that Σ´pt, xq “ Σ´pt, pI ´ Pptqqxq P rangepPptqq,
+Σ´pt, 0q “ 0, lim|x|Ñ0
+Σ´pt,xq
+|x|
+“ 0. Moreover, if for some t P R and x P rangepI ´PptqqXU we have
+y “ x ` Σ´pt, xq, then, provided |y| is small enough, the backward trajectory w : p´8, ts Ñ R3
+of (23) with wptq “ y belongs to the graph of Σ´, i.e. Σ´pτ, pI ´ Ppτqqwpτqq “ Ppτqwpτq and
+satisfies |wpτq| ď Ceδpτ´tq for every τ ď t. Moreover for a small positive α the third coordinate of
+the point
+α
+¨
+˝
+´L1ptq
+´L2ptq
+1
+˛
+‚` Σ´
+¨
+˝t, α
+¨
+˝
+´L1ptq
+´L2ptq
+1
+˛
+‚
+˛
+‚
+is equal to α and hence positive, while the first two coordinates are small enough so that in the
+original coordinates they are also positive. Hence, for any t P R there exists the point with all the
+coordinates positive at time t which is backward exponentially attracted to zero. In terms of the
+original system (LV-3) in view of [Red96, Theorem 1 (iii) and (vi)] this gives us the assertion of
+the Theorem.
+The proof of local instability of p¯u1, 0, 0q is analogous. The system linearized around p¯u1, 0, 0q
+has the form
+v1ptq “
+ˆ
+Aptq
+Cptq
+0
+Bptq
+˙
+vptq.
+(25)
+where
+Aptq “
+`
+a1ptq ´ 2b11ptq¯u1ptq
+˘
+,
+Bptq “
+ˆ
+a2ptq ´ b21ptq¯u1ptq
+0
+0
+a3ptq ´ b31ptq¯u1ptq
+˙
+Cptq “
+`
+´b12ptq¯u1ptq
+´b13ptq¯u1ptq
+˘
+Now, the two dimensional system pv2ptq, v3ptqq1 “ Bptqpv2ptq, v3ptqq has an exponential dichotomy
+with projection Pptq “ 0 since by (21) both diagonal terms are positive and separated from zero.
+The system v1
+1 “ Aptqv1, by Theorem 6.3, has an exponential dichotomy with projection Pptq “ I,
+We follows the same process as the first case: the function Cptq is bounded, and the system (24)
+admits an exponential dichotomy, such that by (7) the projections are given by
+25
+
+7
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 3-D SYSTEM
+P ` “
+¨
+˝
+1
+0
+0
+0
+0
+0
+0
+0
+0
+˛
+‚ on R` and P ´ “
+¨
+˝
+1
+L´
+1
+L´
+2
+0
+0
+0
+0
+0
+0
+˛
+‚ on R´.
+Since the projection
+P “ P ´ “
+¨
+˝
+1
+L´
+1
+L´
+2
+0
+0
+0
+0
+0
+0
+˛
+‚,
+has the same range as P `, hence the system (24) has exponential dichotomy on R with the
+projection
+Pptq “
+¨
+˝
+1
+L1ptq
+L2ptq
+0
+0
+0
+0
+0
+0
+˛
+‚,
+for certain bounded functions L1ptqq and L2ptq. Note that in this case rangepPptqq “ tβp1, 0, 0q :
+β P Ru, is of dimension 1, and is always the stable space in the linearization, while
+rangepI ´ Pptqq “
+$
+&
+%α
+¨
+˝
+´L1ptq
+1
+0
+˛
+‚` β
+¨
+˝
+´L2ptq
+0
+1
+˛
+‚ : α, β P R
+,
+.
+- ,
+is a two dimensional space,and it is the time dependent unstable space in the linearization. The
+end of the proof is analogous to the first case. with these new spaces.
+Theorem 7.3. There exists a trajectory of (LV-3) denoted by y “ py1, y2, y3q : R Ñ R3 such that
+lim
+sÑ´8 |py1psq, y2psq, y3psq| “ 0.
+and
+lim
+sÑ8 |py1psq, y2psq, y3psqq ´ pu˚
+1psq, u˚
+2psq, u˚
+3psqq| “ 0.
+Proof. The proof is analogous the the proof in the two-dimensional case, cf. Theorem 6.4, with
+following linearized system at p0, 0, 0q
+$
+’
+&
+’
+%
+v1ptq “
+¨
+˚
+˝
+a1ptq
+0
+0
+0
+a2ptq
+0
+0
+0
+a3ptq
+˛
+‹‚vptq,
+(26)
+We prove now the result of the three-dimensional analogy of Theorem 6.5. The proof follows
+the same argument as in Theorem 6.5.
+Theorem 7.4. If a trajectory with a strictly positive initial condition is bounded both in the past
+and in the future then it must be one of trajectories described above.
+26
+
+7
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 3-D SYSTEM
+Proof. From [Red96, Theorem 1. (v)] we know that u˚ is the only solution on R that is bounded
+away from 0 and 8. Thus, if the complete trajectory u “ pu1, u2, u3q is bounded, not coincides
+with u˚, and none of the three coordinates is zero, then for at least one of its coordinates there
+must exist a decreasing sequence tn Ñ ´8 such that uiptnq Ñ 0 and uiptnq ă di. We consider two
+cases.
+Case 1. Two coordinates converge to zero backward and one is separated from zero. Assume
+that there exist two sequences tn Ñ ´8 and ˆtn Ñ ´8 such that u1ptnq Ñ 0, u2pˆtnq Ñ 0,
+u1ptnq ă d1 and u2pˆtnq ă d2 We first prove that limtÑ´8 uiptq “ 0 for i “ 1, 2. We proceed
+analogously as in Lemma 5.2 and Theorem 6.5. If for some t P ptn`1, tnq, we have u1ptq ą u1ptnq
+then there exists t˚ P rt, tnq such that u1pt˚q ą u1ptnq and on pt˚, tnq the function u1 is strictly
+less that d1. Hence
+0 ą u1ptnq ´ u1pt˚q “
+ż tn
+t˚ u1psqpa1psq ´ b11psqu1psq ´ b12psqu2psq ´ b13u3psqq ds ě
+ě
+ż tn
+t˚ u1psqpa1psq ´ b11psqd1qds ě 0,
+and we have the contradiction. The same argument allows us to get limtÑ´8 u2ptq “ 0.
+Now we need to prove that for u3ptq, that is separated from zero, it must be
+lim
+tÑ´8 |u3ptq ´ ¯u3ptq| “ 0.
+To this end assume that for some sequence tn Ñ ´8
+lim
+nÑ8 |u3ptnq ´ ¯u3ptnq| “ c ą 0.
+Now, since u3ptq is separated from zero, denoting K “ pu3qL and M “ pu3qU and using Lemma
+2.2 and Lemma 2.3, we obtain for every x P rK, Ms
+|u3ptnq ´ ¯u3ptnq| ď |Sptn, tn ´ tqpu1ptn ´ tq, u2ptn ´ tq, u3ptn ´ tqq ´ Sptn, tn ´ tqp0, 0, xq|
+` |Sptn, tn ´ tqp0, 0, xq ´ Sptn, tn ´ tqp0, 0, ¯u3ptn ´ tqq|
+ď eκt|pu1ptn ´ tq, u2ptn ´ tq, u3ptn ´ tqq ´ p0, 0, xq| ` 2 maxtM, d3u2
+mintK, d3u
+e´δ mintK,d3ut,
+with a constant κ depending on the bounds on u and the coefficients of the problem. Pick ε ą 0.
+We can find t ą 0 such that
+2 maxtM,d3u2
+mintK,d3u e´δ mintK,d3ut ă
+ε
+2.
+The sequence u3ptn ´ tq has a
+convergent subsequence, let x be a limit of this subsequence and pick n large enough such that
+eκt|pu1ptn ´ tq, u2ptn ´ tq, u3ptn ´ tqq ´ p0, 0, xq| ă ε
+2. Hence for n large enough, on a subsequence,
+|u3ptnq ´ ¯u3ptnq| ă ε,
+a contradiction.
+Case 2. Only one coordinate converges to zero backwards. We assume, for example, that there
+exists a sequence tn Ñ ´8 such that u1ptnq Ñ 0, then, analogously to the previous case, we
+can prove that limtÑ´8 u1ptq “ 0. We must prove that for u2ptq, and u3ptq, that are backward
+separated from zero, it must be
+lim
+tÑ´8 |pu2ptq, u3ptqq ´ ppu2ptq, pu3ptqq| “ 0.
+27
+
+7
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 3-D SYSTEM
+To this end assume that for some sequence tn Ñ ´8
+lim
+nÑ8 |pu2ptnq, u3ptnqq ´ ppu2ptnq, pu3ptnqq| “ c ą 0.
+(27)
+For every xi P ruL
+i , uU
+i s for i P t2, 3u, and using Lemma 2.3 with (H) and Lemma 2.2 we have for
+t ě 0
+|pu2ptnq, u3ptnqq ´ ppu2ptnq, pu3ptnqq|
+ď |Sptn, tn ´ tqpu1ptn ´ tq, u2ptn ´ tq, u3ptn ´ tqq ´ Sptn, tn ´ tqp0, x2, x3q|
+` |Sptn, tn ´ tqp0, x2, x3q ´ Sptn, tn ´ tqp0, pu2ptnq, pu3ptn ´ tqq|
+ď eκt|pu1ptn ´ tq, u2ptn ´ tq, u3ptn ´ tqq ´ p0, x2, x3q| ` 2σ2
+¯σ e´¯σδt,
+where κ, σ, ¯σ depend on the bounds on coefficients of the problem, constants appearing in (21)
+and on uL
+i , uU
+i . In particular [Red96, Lemma 2] implies that the solution starting from p0, x2, x3q
+at any time is separated from zero by a constant independent on initial time and the choice of
+initial data as long as xi P ruL
+i , uU
+i s. Pick ε ą 0. We can find t ą 0 such that 2σ2
+¯σ e´¯σδt ă ε
+2.
+The sequence pu2ptn ´ tq, u3ptn ´ tqq has a convergent subsequence, let px2, x3q be a limit of
+this subsequence and associated times are denoted by tptnu. We pick n large enough such that
+eκt|pu1ptn ´ tq, u2ptn ´ tq, u3ptn ´ tqq ´ p0, x2, x3q| ă ε
+2, which leads into a direct contradiction with
+(27).
+Finally, the full image of the attractor represented in Figure 3 is described in the following
+theorem with a summary of the results obtained in this section and using Lemma 5.2, Theorem
+6.3 and Theorem 6.4 for the remaining heterolinics connections.
+Theorem 7.5. Assuming (21) the system (LV-3) has the following solutions u : R Ñ R3 bounded
+both in the past and in the future:
+(a) uptq “ p0, 0, 0q for t P R,
+(b) uptq “ p¯u1ptq, 0, 0q, uptq “ p0, ¯u2ptq, 0q and uptq “ p0, 0, ¯u3q, given in Lemma 7.1.
+(c) Any solution uptq “ pu1ptq, 0, 0q with initial condition 0 ă u1pt0q ă ¯u1pt0q, where limtÑ´8 u1ptq “
+0 and limtÑ8pu1ptq ´ ¯u1ptqq “ 0.
+Analogously, uptq “ p0, u2ptq, 0q with initial condi-
+tion 0 ă u2pt0q ă ¯u2pt0q, where limtÑ´8 u2ptq “ 0 and limtÑ8pu2ptq ´ ¯u2ptqq “ 0, and
+uptq “ p0, 0, u3ptqq with initial condition 0 ă u3pt0q ă ¯u3pt0q, where limtÑ´8 u3ptq “ 0 and
+limtÑ8pu3ptq ´ ¯u3ptqq “ 0.
+(d) uptq “ ppu1ptq, pu2ptq, 0q, uptq “ ppu1ptq, 0, pu3ptqq and uptq “ p0, pu2ptq, pu3ptqq, given in Lemma
+7.1.
+(e) uptq “ pu1ptq, u2ptq, 0q where limtÑ´8pu1ptq, u2ptqq “ p0, 0q and limtÑ8ppu1ptq, u2ptqq ´
+ppu1ptq, pu2ptqqq “ 0.
+Analogously, uptq “ pu1ptq, 0, u3ptqq where limtÑ´8pu1ptq, u3ptqq “
+p0, 0q and limtÑ8ppu1ptq, u3ptqq ´ ppu1ptq, pu3ptqqq “ 0, and uptq “ p0, u2ptq, u3ptqq where
+limtÑ´8pu2ptq, u3ptqq “ p0, 0q and limtÑ8ppu2ptq, u3ptqq ´ ppu2ptq, pu3ptqqq “ 0.
+28
+
+7
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 3-D SYSTEM
+(f) uptq “ pu1ptq, u2ptq, 0q where limtÑ´8pu1ptq, u2ptqq “ p¯u1, 0q and limtÑ8ppu1ptq, u2ptqq ´
+ppu1ptq, pu2ptqqq “ 0.
+Analogously, uptq “ pu1ptq, u2ptq, 0q where limtÑ´8pu1ptq, u2ptqq “
+p0, ¯u2q and limtÑ8ppu1ptq, u2ptqq ´ ppu1ptq, pu2ptqqq “ 0.
+(g) uptq “ pu1ptq, 0, u3ptqq where limtÑ´8pu1ptq, u3ptqq “ p¯u1, 0q and limtÑ8ppu1ptq, u3ptqq ´
+ppu1ptq, pu3ptqqq “ 0.
+Analogously, uptq “ pu1ptq, 0, u3ptqq where limtÑ´8pu1ptq, u3ptqq “
+p0, ¯u3q and limtÑ8ppu1ptq, u3ptqq ´ ppu1ptq, pu3ptqqq “ 0.
+(h) uptq “ p0, u2ptq, u3ptqq where limtÑ´8pu2ptq, u3ptqq “ p¯u2, 0q and limtÑ8ppu2ptq, u3ptqq ´
+ppu2ptq, pu3ptqqq “ 0.
+Analogously, uptq “ p0, u2ptq, u3ptqq where limtÑ´8pu2ptq, u3ptqq “
+p0, ¯u3q and limtÑ8ppu2ptq, u3ptqq ´ ppu2ptq, pu3ptqqq “ 0.
+(i) uptq “ pu˚
+1ptq, u˚
+2ptq, u˚
+3ptqq bounded away from zero and infinity given in Lemma 7.1.
+(j) uptq “ pu1ptq, u2ptq, u3ptqq such that limtÑ´8pu1ptq, u2ptq, u3ptqq “ p0, 0, 0q and limtÑ8pu1ptq´
+u˚
+1ptq, u2ptq ´ u˚
+2ptq, u3ptq ´ u˚
+3ptqq “ 0, given in Theorem 7.3.
+(k) uptq “ pu1ptq, u2ptq, u3ptqq such that limtÑ´8pu1ptq, u2ptq, u3ptqq “ ppu1ptq, pu2ptq, 0q and limtÑ8pu1ptq´
+u˚
+1ptq, u2ptq´u˚
+2ptq, u3ptq´u˚
+3ptqq “ 0, given in Theorem 7.2. Analogously, uptq “ pu1ptq, u2ptq, u3ptqq
+such that limtÑ´8pu1ptq, u2ptq, u3ptqq “ ppu1ptq, 0, pu3ptqq and limtÑ8pu1ptq ´ u˚
+1ptq, u2ptq ´
+u˚
+2ptq, u3ptq´u˚
+3ptqq “ 0, and uptq “ pu1ptq, u2ptq, u3ptqq such that limtÑ´8pu1ptq, u2ptq, u3ptqq “
+p0, pu2ptq, pu3ptqq and limtÑ8pu1ptq ´ u˚
+1ptq, u2ptq ´ u˚
+2ptq, u3ptq ´ u˚
+3ptqq “ 0.
+(l) uptq “ pu1ptq, u2ptq, u3ptqq such that limtÑ´8pu1ptq, u2ptq, u3ptqq “ p¯u1ptq, 0, 0q and limtÑ8pu1ptq´
+u˚
+1ptq, u2ptq´u˚
+2ptq, u3ptq´u˚
+3ptqq, given in Theorem 7.2. Analogously, uptq “ pu1ptq, u2ptq, u3ptqq
+such that limtÑ´8pu1ptq, u2ptq, u3ptqq “ p0, ¯u2ptq, 0q and limtÑ8pu1ptq´u˚
+1ptq, u2ptq´u˚
+2ptq, u3ptq´
+u˚
+3ptqq “ 0, and uptq “ pu1ptq, u2ptq, u3ptqq such that limtÑ´8pu1ptq, u2ptq, u3ptqq “ p0, 0, ¯u3ptqq
+and limtÑ8pu1ptq ´ u˚
+1ptq, u2ptq ´ u˚
+2ptq, u3ptq ´ u˚
+3ptqq “ 0.
+Moreover, any solution of (LV-3) bounded both in the past and in the future is one of the solutions
+described in items (a)-(l).
+7.2
+Structure of attractor for the case of extinction of one species
+In case of the permanence of the two species we would have a scheme depicted in the next Figure.
+29
+
+7
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 3-D SYSTEM
+Figure 4: Four complete solutions and their connections for the three dimensional problem with
+globally asymptotically stable state with coexistence of two of the three species.
+Following Theorem 4.5 we have to impose the following conditions to a1, a2 and a3 to obtain
+the extinction of one species.
+$
+’
+&
+’
+%
+pc1b11 ` c2b12 ` c3b13qL ą 0,
+pc2b22 ` c1b21 ` c3b23qL ą 0,
+pc3b33 ` c1b31 ` c2b32qL ą 0,
+$
+’
+&
+’
+%
+b11 ¯d1 ` ε ď a1 ď b11d1 ` b12d2 ` b13d3 ` θpb12c2 ` b13c3q ´ ε,
+b22 ¯d2 ` ε ď a2 ď b21d1 ` b22d2 ` b23d3 ` θpb21c1 ` b23c3q ´ ε,
+a3 ď b31d1 ` b32d2 ` θpb31c1 ` b32c2q ´ ε,
+(28)
+with some constants di, ci ą 0 for i “ 1, 2, 3 , ¯dj ą 0 for j “ 1, 2 and ε, θ ą 0.
+The above
+inequalitites are the conditions (H) and (B2) for the three-dimensional case. Moreover we need
+to assume that |bij|U ă 8 and |ai|U ă 8 for every i P t1, 2u and j P t1, 2, 3u. Summarizing all
+previous results we can formulate the following theorem.
+Theorem 7.6. Assuming (28) the system (LV-3) has the following trajectories u : R Ñ R3 bounded
+both in the past and in the future
+(a) uptq “ p0, 0, 0q for t P R,
+(b) uptq “ p¯u1ptq, 0, 0q, uptq “ p0, ¯u2ptq, 0q for t P R, where ¯ui, for i “ 1, 2, is the unique trajectory
+of u1
+i “ uipaiptq ´ biiptquiptqq separated from zero and infinity given by Lemma 5.1,
+(c) uptq “ pu1ptq, 0, 0q where limtÑ´8 u1ptq “ 0 and limtÑ8pu1ptq ´ ¯u1ptqq “ 0. Any solution
+with the initial data pu1pt0q, 0, 0q satisfying 0 ă u1pt0q ă ¯u1pt0q constitutes such trajectory.
+Analogously with any solution with the initial data p0, u2pt0q, 0q satisfying 0 ă u2pt0q ă ¯u2pt0q,
+constitutes the trajectory that satisfies uptq “ p0, u2ptq, 0q where limtÑ´8 u2ptq “ 0 and
+limtÑ8pu2ptq ´ ¯u2ptqq “ 0.
+(d) uptq “ ppu1ptq, pu2ptq, 0q, where ppu1, pu2q, is the unique trajectory of the two dimensional system
+obatined taking u3 ” 0 separated from zero and infinity given by Lemma 6.1.
+30
+
+u1
+W2
+u3
+1
+u2
+u1
+2
+u3
+3
+2
+u3
+u17
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 3-D SYSTEM
+(e) uptq “ pu1ptq, u2ptq, 0q where limtÑ´8pu1ptq, u2ptqq “ p0, 0q and limtÑ8ppu1ptq, u2ptqq ´
+ppu1ptq, pu2ptqqq “ 0.
+(f) uptq “ pu1ptq, u2ptq, 0q where limtÑ´8pu1ptq, u2ptqq “ p¯u1, 0q and limtÑ8ppu1ptq, u2ptqq ´
+ppu1ptq, pu2ptqqq “ 0.
+Analogously, uptq “ pu1ptq, u2ptq, 0q where limtÑ´8pu1ptq, u2ptqq “
+p0, ¯u2q and limtÑ8ppu1ptq, u2ptqq ´ ppu1ptq, pu2ptqqq “ 0.
+If a solution u : R Ñ R3 is bounded then it must be on of the solutions of items (a)-(f). Moreover,
+(g) Any solution uptq “ pu1ptq, u2ptq, u3ptqq with initial data uipt0q ą 0 for i “ 1, 2, 3, satisfies
+limtÑ8pu1ptq ´ pu1ptq, u2ptq ´ pu2ptq, u3ptqq “ p0, 0, 0q and limtÑ´8 |uptq| “ 8.
+(h) Any solution uptq “ p0, 0, u3ptqq with u3pt0q ą 0 satisfies limtÑ8 u3ptq “ 0 and limtÑ´8 u3ptq “
+8,
+(i) Any solution uptq “ pu1ptq, 0, 0q with u1pt0q ą ¯u1pt0q satisfies limtÑ8pu1ptq ´ ¯u1ptqq “ 0
+and limtÑ´8 u1ptq “ 8. Analogously, any solution uptq “ p0, u2ptq, 0q with u2pt0q ą ¯u2pt0q
+satisfies limtÑ8pu2ptq ´ ¯u2ptqq “ 0 and limtÑ´8 u2ptq “ 8.
+(j) Any solution uptq “ pu1ptq, 0, u3ptqq with u1pt0q ą 0, u3pt0q ą 0 satisfies limtÑ8pu1ptq ´
+¯u1ptq, u3ptqq “ 0 and limtÑ´8 |pu1ptq, u3ptqq| “ 8. Analogously, any solution uptq “ p0, u2ptq, u3ptqq
+with u2pt0q ą 0, u3ptq ą 0 satisfies limtÑ8pu2ptq´¯u2ptq, u3ptqq “ 0 and limtÑ´8 |pu2ptq, u3ptqq| “
+8.
+7.3
+The case of extinction of two species
+In case of the extinction of the two species we would have a scheme as the next Figure.
+Figure 5: Two complete solutions and their connections for the three dimensional problem with
+globally asymptotically stable state with existence of one species.
+By Theorem 4.5 we have to impose the following conditions on the problem coefficients to
+obtain the case of the permanence of one species and extinction of the remaining two.
+31
+
+u1
+u2
+u3
+u3
+u1
+u27
+ATTRACTOR FOR NON-AUTONOMOUS LOTKA-VOLTERRA 3-D SYSTEM
+$
+’
+&
+’
+%
+pc1b11 ` c2b12 ` c3b13qL ą 0,
+pc2b22 ` c1b21 ` c3b23qL ą 0,
+pc3b33 ` c1b31 ` c2b32qL ą 0,
+$
+’
+&
+’
+%
+b11 ¯d1 ` ε ď a1 ď b11d1 ` b12d2 ` b13d3 ` θpb12c2 ` b13c3q ´ ε,
+a2 ď b21d1 ` b23d3 ` θpb21c1 ` b23c3q ´ ε,
+a3 ď b31d1 ` b32d2 ` θpb31c1 ` b32c2q ´ ε,
+(29)
+with some constants di, ci ą 0 for i “ 1, 2, 3 , ¯d1 ą 0 and ε, θ ą 0. The above inequalitites are the
+conditions (H) and (B1) for the three-dimensional case. We need to assume again that |b1j|U ă 8
+and |a1|U ă 8 for every j P t1, 2, 3u
+Theorem 7.7. Assuming (29) the system (LV-3) has the following trajectories u : R Ñ R3 bounded
+both in the past and in the future
+(a) uptq “ p0, 0, 0q for t P R,
+(b) uptq “ p¯u1ptq, 0, 0q, for t P R, where ¯u1 is the unique trajectory of u1
+1 “ u1pa1ptq´b11ptqu1ptqq
+separated from zero and infinity given by Lemma 5.1,
+(c) uptq “ pu1ptq, 0, 0q where limtÑ´8 u1ptq “ 0 and limtÑ8pu1ptq ´ ¯u1ptqq “ 0. Any solution
+with the initial data pu1pt0q, 0, 0q satisfying 0 ă u1pt0q ă ¯u1pt0q constitutes such trajectory.
+The solutions described in (a)-(c) are the only ones which are bounded both in the past and in the
+future. Moreover,
+(d) Any solution uptq “ pu1ptq, u2ptq, u3ptqq with initial data uipt0q ą 0 for i “ 1, 2, 3, satisfies
+limtÑ8pu1ptq ´ ¯u1ptq, u2ptq, u3ptqq “ p0, 0, 0q and limtÑ´8 |uptq| “ 8.
+(e) Any solution uptq “ p0, u2ptq, 0q with u2pt0q ą 0 satisfies limtÑ8 u2ptq “ 0 and limtÑ´8 u2ptq “
+8. Analogously, uptq “ p0, 0, u3ptqq with u3pt0q ą 0 satisfies limtÑ8 u3ptq “ 0 and limtÑ´8 u3ptq “
+8,
+(f) Any solution uptq “ pu1ptq, 0, 0q with u1pt0q ą ¯u1pt0q satisfies limtÑ8pu1ptq ´ ¯u1ptqq “ 0 and
+limtÑ´8 u1ptq “ 8.
+(g) Any solution p0, u2ptq, u3ptqq with u2pt0q ą 0, u3pt0q ą 0 satisfies limtÑ8pu2ptq, u3ptqq “ 0
+and limtÑ´8 |pu2ptq, u3ptqq| “ 8.
+(h) Any solution pu1ptq, 0, u3ptqq with u1pt0q ą 0 and u3pt0q ą 0 satisfies limtÑ8pu1ptq´¯u1ptq, 0, u3ptqq “
+0 and limtÑ´8 |pu1ptq, u3ptqq| “ 8. Analogously any solution pu1ptq, u2ptq, 0q with u1pt0q ą 0
+and u2pt0q ą 0 satisfies limtÑ8pu1ptq ´ ¯u1ptq, u2ptq, 0q “ 0 and limtÑ´8 |pu1ptq, u2ptqq| “ 8.
+7.4
+Extinction of all species
+The last situation is the extinction of all species.
+To obtain this case we make the following
+assumptions
+$
+’
+&
+’
+%
+pc1b11 ` c2b12 ` c3b13qL ą 0,
+pc2b22 ` c1b21 ` c3b23qL ą 0,
+pc3b33 ` c1b31 ` c2b32qL ą 0.
+$
+’
+&
+’
+%
+a1 ď b12d2 ` b13d3 ` θpb12c2 ` b23c3q ´ ε
+a2 ď b21d1 ` b23d3 ` θpb21c1 ` b23c3q ´ ε
+a3 ď b31d1 ` b32d2 ` θpb31c1 ` b32c2q ´ ε
+(30)
+32
+
+REFERENCES
+with some constants di, ci ą 0 for i P t1, 2, 3u, and ε, θ ą 0. We observe that the first inequalities
+is the condition (H) and the second ones is the condition pBq with J “ t1, 2, 3u and I “ H.
+Theorem 7.8. Assuming (30) the only trajectory of system (LV-3) which is bounded both in
+the past and and the future is uptq “ p0, 0, 0q for t P R.
+Moreover for every solution with
+initial data uipt0q ą 0 for any i P t1, 2, 3u, we have limtÑ8pu1ptq, u2ptq, u3ptqq “ p0, 0, 0q and
+limtÑ´8 |pu1ptq, u2ptq, u3ptqq| “ 8.
+Proof. The convergence limtÑ8pu1ptq, u2ptq, u3ptqq “ p0, 0, 0q follows from Lemma 4.2. The fact
+that any solution with initial data u1pt0q ą 0, u2pt0q or u3pt0q ą 0 is backward unbounded, follows
+from Lemma 4.3.
+Funding
+Work of JGF has been partially supported by the Spanish Ministerio de Ciencia, Innovaci´on y
+Universidades (MCIU), Agencia Estatal de Investigaci´on (AEI), Fondo Europeo de Desarrollo
+Regional (FEDER) under grant PRE2019-087385. JGF and JALR have been also supported by
+projects PGC2018-096540-B-I00 and PID2021-122991NB-C21. Work of PK has been supported
+by Polish National Agency for Academic Exchange (NAWA) within the Bekker Programme under
+Project No. PPN/BEK/2020/1/00265/U/00001, and by National Science Center (NCN) of Poland
+under Projects No. UMO-2016/22/A/ST1/00077 and DEC-2017/25/B/ST1/00302. AS has been
+partially supported by projects US-1381261 and PGC2018-098308-B-I00.
+References
+[Ahm93] S. Ahmad. On the nonautonomous Volterra–Lotka competition equations. Proceedings
+of the American Mathematical Society, 117:199–204, 1993.
+[AKL22] P. Almaraz, P. Kalita, and J.A. Langa.
+Structural stability of invasion graphs for
+generalized lotka–volterra systems. https://arxiv.org/abs/2209.09802v4, 2022.
+[AL95] S. Ahmad and A. Lazer. On the nonautonomous n-competing species problems. Ap-
+plicable Analysis., 57:309–323, 1995.
+[AL98] S. Ahmad and A. Lazer. Necessary and sufficient average growth in a Lotka–Volterra
+system. Nonlinear Analysis, 34:191–228, 1998.
+[AL00] S. Ahmad and A. Lazer. Average conditions for global asymptotic stability in an nonau-
+tonomous Lotka–Volterra system. Nonlinear Analysis, 40:37–49, 2000.
+[BCL20] M.C. Bortolan, A.N. Carvalho, and J.A. Langa. Attractors Under Autonomous and
+Non-autonomous Perturbations, volume 246 of Mathematical Surveys and Monographs.
+AMS, 2020.
+[BF20] F. Battelli and M. Feˇckan. On the exponents of exponential dichotomies. Mathematics,
+8:651, 2020.
+[BP15] F. Battelli and K.J. Palmer. Criteria for exponential dichotomy for triangular systems.
+Journal of Mathematical Analysis and Applications, 428:525–543, 2015.
+33
+
+REFERENCES
+[CLR13] A.N. Carvalho, J.A. Langa, and J.C. Robinson.
+Attractors for Infinite-dimensional
+Non-autonomous Dynamical Systems, volume 182 of Applied Mathematical Sciences.
+Springer-Verlag, 2013.
+[Cop65] W.A. Coppel. Stability and Asymptotic Behaviour of Differential Equations. Heath
+Mathematical Monographs. D.C.Heath, 1965.
+[Cop78] W.A. Coppel. Dichotomies in Stability Theory, volume 629 of Lecture Notes in Math-
+ematics. Springer-Verlag, 1978.
+[CRLO23] A.N. Carvalho, L.R.N. Rocha, J.A. Langa, and R. Obaya. Structure of non-autonomous
+attractors for a class of diffusively coupled ODE. Discrete and Continuous Dynamical
+Systems - Series B, 28:426–448, 2023.
+[Gop86a] K. Gopalsamy. Global asymptotic stability in a periodic Lotka–Volterra system. The
+ANZIAM Journal, 27:66–72, 1986.
+[Gop86b] K. Gopalsamy. Global asymptotic stability in an almost periodic Lotka–Volterra sys-
+tem. The ANZIAM Journal, 27:346–360, 1986.
+[Gue17] G.F. Guerrero. Din´amica de redes mutualistas en ecosistemas complejos. PhD thesis,
+Universidad de Sevilla, Sevilla, Espa˜na, 6 2017.
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+munity assembly into modern coexistence theory. Journal of Mathematical Biology,
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+Mathematical Surveys and Monographs. American Mathematical Society, 2011.
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+Nonautonomous Lotka–Volterra systems. I. J.Differential Equations,
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+Scientific Publishing, 1996.
+34
+

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+arXiv:2301.01565v1  [quant-ph]  4 Jan 2023
+General pseudo self-adjoint boundary conditions for a 1D KFG
+particle in a box
+Salvatore De Vincenzo1, ∗
+1The Institute for Fundamental Study (IF),
+Naresuan University, Phitsanulok 65000, Thailand
+(Dated: January 3, 2023)
+Abstract We consider a 1D Klein-Fock-Gordon particle in a finite interval, or box. We
+construct the most general set of pseudo self-adjoint boundary conditions for the Hamil-
+tonian operator that is present in the first order in time 1D Klein-Fock-Gordon wave equa-
+tion, or the 1D Feshbach-Villars wave equation. This set depends on four real parameters
+and can be written in terms of the one-component wavefunction for the second order in
+time 1D Klein-Fock-Gordon wave equation and its spatial derivative, both evaluated at
+the endpoints of the box. This set can also be made dependent on the two-component
+wavefunction for the 1D Feshbach-Villars equation and its spatial derivative, evaluated
+at the ends of the box; however, the set actually depends on these two column vectors
+each multiplied by the singular matrix that is present in the kinetic energy term of the
+Hamiltonian. As a consequence, this set of boundary conditions does not imply that the
+two-component wavefunction for the Feshbach-Villars equation and its spatial derivative,
+evaluated at the ends of the box, satisfy it by themselves. However, any specific bound-
+ary condition for the 1D Feshbach-Villars equation can be obtained from the respective
+boundary condition for the second order in time 1D Klein-Fock-Gordon wave equation
+and using a pair of relations that arise from the very definition of the two-component
+wavefunction for the 1D Feshbach-Villars equation. Our results can be extended to the
+problem of the 1D Klein-Fock-Gordon particle moving on the real line with a hole at one
+point.
+PACS numbers: 03.65.-w, 03.65.Ca, 03.65.Db, 03.65.Pm
+Keywords: Klein-Fock-Gordon wave equation in (1+1) dimensions; Feshbach-Villars wave equa-
+tion in (1+1) dimensions; pseudo-Hermitian operator; pseudo self-adjoint operator; boundary
+conditions
+
+2
+I.
+INTRODUCTION
+Let us write the one-dimensional (1D) Klein-Fock-Gordon (KFG) wave equation in Hamiltonian
+form,
+iℏ ∂
+∂tΨ = ˆhΨ,
+(1)
+where
+ˆh = − ℏ2
+2m (ˆτ3 + iˆτ2) ∂2
+∂x2 + mc2ˆτ3 + V (x)ˆ12,
+(2)
+is, let us say, the KFG Hamiltonian differential operator. Here, ˆτ3 = ˆσz and ˆτ2 = ˆσy are Pauli ma-
+trices and V (x) ∈ R is the external electric potential (ˆ12 is the 2×2 identity matrix). The operator
+ˆh acts on two-component column state vectors of the form Ψ = Ψ(x, t) = [ ψ1(x, t) ψ2(x, t) ]T
+(the symbol T represents the transpose of a matrix). Equation (1) with ˆh given in Eq. (2) is
+also called the 1D Feshbach-Villars (FV) wave equation [1–4].
+The 1D KFG wave equation in its standard form, or the second order in time KFG equation
+in one spatial dimension [5, 6] is given by
+�
+iℏ ∂
+∂t − V (x)
+�2
+ψ =
+�
+−ℏ2c2 ∂2
+∂x2 + (mc2)2
+�
+ψ,
+(3)
+where ψ = ψ(x, t) is a one-component state vector or one-component wavefunction.
+The relation between ψ and Ψ can be defined as follows:
+Ψ =
+
+ ψ1
+ψ2
+
+ = 1
+2
+
+ ψ + iτ
+� ∂
+∂t − V
+iℏ
+�
+ψ
+ψ − iτ
+� ∂
+∂t − V
+iℏ
+�
+ψ
+
+ ,
+(4)
+where τ ≡ ℏ/mc2. The Compton wavelength is precisely λC ≡ cτ; thus, τ is the time taken for
+a ray of light to travel the distance λC. The expression given in Eq. (3) is fully equivalent to Eq.
+(1) (with ˆh given in Eq. (2)) [3, 4]. Note that, from Eq. (4), the solution ψ of Eq. (3) depends
+only on the components of the column vector Ψ, namely,
+ψ = ψ1 + ψ2.
+(5)
+Additionally,
+�
+iℏ ∂
+∂tψ − V ψ
+�
+1
+mc2 = ψ1 − ψ2.
+(6)
+∗URL: https://orcid.org/0000-0002-5009-053X; Electronic address: [[email protected]]
+
+3
+All the results we have presented thus far are well known; however, to the best of our knowl-
+edge, there is one issue within the 1D KFG theory that has received little attention and that
+we can raise with the following questions: What are the boundary conditions that the 1D FV
+equation can support? In particular, what are the appropriate boundary conditions for this equa-
+tion in the problem of the 1D KFG particle inside an interval? For example, some unexpected
+boundary conditions for the solutions of the 1D FV wave equation in simple physical situations
+were presented in Refs. [7–9]. In general, the boundary conditions for solutions of the second
+order in time KFG wave equation appear to be similar to those supported by the Schrödinger
+wavefunction (see, for example, Refs. [7, 9–11]), but we do not have at our disposal a wave
+equation that could have boundary conditions similar to those of the 1D FV equation (the pres-
+ence of the singular matrix ˆτ3 + iˆτ2 in the Hamiltonian has much to do with this). In general,
+the physically acceptable boundary conditions for a wave equation must ensure that the corre-
+sponding Hamiltonian maintains its essential property, for example, that of being self-adjoint (if
+that is the case). In the case of the 1D FV equation, it is known that the Hamiltonian is a
+formally pseudo-Hermitian operator (or a formally pseudo self-adjoint operator) [1, 3], and we
+could find families of general boundary conditions that agree with the property of being a pseudo
+self-adjoint operator, namely, not just formally, i.e., not only without specifying the domain of
+the Hamiltonian (as is done in the case of Hamiltonians that are self-adjoint).
+The article is organized as follows. In Section II, we introduce the pseudo inner product for the
+two-component solutions of the 1D FV equation and briefly discuss its relation to other distinctive
+inner products of quantum mechanics.
+This pseudo inner product could also be considered
+the scalar product for the one-component solutions of the KFG equation in its standard form.
+Certainly, the pseudo inner product does not possess the property of positive definiteness, but
+can be independent of time.
+Thus, the corresponding pseudo norm can be a constant, and
+because this implies that the probability current density takes the same value at each end of
+the box, the Hamiltonian for this problem can be a pseudo-Hermitian operator. In fact, ˆh is
+formally pseudo-Hermitian, and we find in this section a general four-parameter set of boundary
+conditions that ensures that ˆh is indeed a pseudo-Hermitian operator. We write this set in terms
+of ψ and ∂ψ/∂x evaluated at the ends of the box. Here, we also consider the nonrelativistic
+approximation of the general set of boundary conditions and the results support the idea that this
+set is indeed the most general. In Section III, we write the general set of boundary conditions in
+terms of the values that Ψ and ∂Ψ/∂x take at the ends of the box. To be precise, the set must
+
+4
+be written in terms of the values that (ˆτ3 + iˆτ2)Ψ and (ˆτ3 + iˆτ2)(∂Ψ/∂x) take at the endpoints
+of the box, but (ˆτ3 + iˆτ2) is a singular matrix, i.e., it does not have an inverse. In Section IV
+(Appendix I), we check that the time derivative of the pseudo inner product of two solutions of
+the 1D FV equation with a potential other than zero, but expressed in terms of the respective
+solutions of the usual KFG equation with V ̸= 0, is proportional to a term evaluated at the ends
+of the box that also does not depend on the potential, i.e., it is a boundary term. In Section
+V (Appendix II), we show that the Hamiltonian operator ˆh for a 1D KFG particle in a box is in
+fact a pseudo self-adjoint operator; that is, the general matrix boundary condition ensures that
+the domains of ˆh and its generalized adjoint are equal. From the results shown in this section it
+follows that the boundary term that arose in Section IV (Appendix I) always vanishes, certainly
+for any boundary condition included in the general family of boundary conditions. Consequently,
+the value of the pseudo inner product in this problem is conserved. Finally, some concluding
+remarks are presented in Section VI.
+II.
+BOUNDARY CONDITIONS FOR THE 1D KFG PARTICLE IN A BOX I
+Let us consider a 1D KFG particle moving in the interval x ∈ Ω = [a, b], i.e., in a box. The
+corresponding Hamiltonian operator given in Eq. (2) acts on two-component column state vectors
+of the form Ψ = [ ψ1 ψ2 ]T and Φ = [ φ1 φ2 ]T, and the scalar product for these two state vectors
+must be defined as
+⟨⟨Ψ, Φ⟩⟩ ≡
+ˆ
+Ω
+dx Ψ†ˆτ3Φ
+(7)
+(the symbol † denotes the usual Hermitian conjugate, or the usual formal adjoint, of a matrix
+and an operator) [1–4]. Additionally, the square of the corresponding norm (or rather, pseudo
+norm) is ∥∥Ψ∥∥2 ≡ ⟨⟨Ψ, Ψ⟩⟩ =
+´
+Ω dx ̺, where ̺ = ̺(x, t) = Ψ†ˆτ3Ψ = |ψ1|2 − |ψ2|2 is the
+KFG probability density. Certainly, ̺ is not positive definite and calling it probability density
+is not absolutely correct (although it can be interpreted as a charge density) [1–4]. Note that
+the integral in (7) can also be identified with the usual scalar product in Dirac theory in (1+1)
+dimensions, namely, ⟨Ψ, Φ⟩D ≡
+´
+Ω dx Ψ†Φ, which is an inner product on the Hilbert space of
+two-component square-integrable wavefunctions, L2(Ω) ⊕ L2(Ω); therefore,
+⟨⟨Ψ, Φ⟩⟩ ≡ ⟨Ψ, ˆτ3Φ⟩D,
+(8)
+
+5
+and ⟨Ψ, Φ⟩D = ⟨⟨Ψ, ˆτ3Φ⟩⟩. Because ⟨⟨Ψ, Ψ⟩⟩ can be a negative quantity, the scalar product in
+Eq. (7) is an indefinite (or improper) inner product, or a pseudo inner product, on an infinite-
+dimensional complex vector space. In general, such a vector space itself is not necessarily a
+Hilbert space.
+Similarly, writing Ψ and Φ in the integrand in (7) in terms of their respective components,
+that is, using the relations that arise from Eq. (4) and other analogous relations for Φ (which are
+obtained from Eq. (4) by making the replacements Ψ → Φ, ψ1 → φ1, ψ2 → φ2 and ψ → φ),
+we obtain
+⟨⟨Ψ, Φ⟩⟩ =
+iℏ
+2mc2
+ˆ
+Ω
+dx
+�
+ψ∗φt − ψ∗
+t φ − 2V
+iℏ ψ∗φ
+�
+(9)
+(where the asterisk ∗ denotes the complex conjugate, and ψt ≡ ∂ψ/∂t, etc), or also,
+⟨⟨Ψ, Φ⟩⟩ =
+iℏ
+2mc2
+�
+⟨ψ, φt⟩S − ⟨ψt, φ⟩S − 2V
+iℏ ⟨ψ, φ⟩S
+�
+≡ ⟨ψ, φ⟩KFG,
+(10)
+where ⟨ψ, φ⟩KFG can be considered the scalar product for the one-component solutions of the
+KFG equation in Eq. (3) (see Appendix I). Note that ⟨ , ⟩S denotes the usual scalar product
+in the Schrödinger theory in one spatial dimension, namely, ⟨ψ, φ⟩S ≡
+´
+Ω dx ψ∗φ, which is an
+inner product on the Hilbert space of one-component square-integrable wavefunctions, L2(Ω).
+Certainly, ψ and ψt, and φ and φt, must belong to L2(Ω) to ensure that ⟨ψ, φ⟩KFG exists [12].
+It can be noted that there is an isomorphism between the vectorial space of the solutions ψ
+of the common KFG equation for the corresponding 1D particle, namely,
+��
+∂t − V
+iℏ
+�2
++ ˆd
+�
+ψ = 0
+(11)
+(Eq. (3)), where ˆd ≡ −c2∂xx + τ −2 (∂t ≡ ∂/∂t and ∂xx ≡ ∂2/∂x2, etc) and the vectorial space
+of the initial state vectors of the KFG equation in Hamiltonian form for this 1D particle, namely,
+Eq. (1) with ˆh given in Eq. (2) [13]. In effect, a possible initial state vector, for example, at
+t = 0, would have the form
+Ψ(0) =
+
+ ψ1(0)
+ψ2(0)
+
+ = 1
+2
+
+ ψ(0) + iτ
+�
+ψt(0) − V
+iℏψ(0)
+�
+ψ(0) − iτ
+�
+ψt(0) − V
+iℏψ(0)
+�
+
+ ,
+(12)
+that arises immediately from the relation given in Eq. (4). Thus, giving an initial state vector
+as Ψ(0) is equivalent to providing the initial data for the solution vector ψ, namely, ψ(0) and
+ψt(0). Incidentally, operators ˆd, which can act on the one-component state vectors ψ, and ˆh,
+
+6
+which can act on the two-component state vectors Ψ, are related as follows:
+ˆh = +ℏ
+2τ (ˆτ3 + iˆτ2) ˆd + ℏ
+2τ −1 (ˆτ3 − iˆτ2) + V (x)ˆ12.
+(13)
+Although the scalar product in Eqs. (7) and (10) does not possess the property of positive
+definiteness (i.e., ⟨⟨Ψ, Ψ⟩⟩ > 0), it is a time-independent scalar product. Indeed, using Eq. (3)
+for ψ and ψ∗, and for φ and φ∗, it can be demonstrated that the following relation is verified:
+d
+dt⟨⟨Ψ, Φ⟩⟩ = − iℏ
+2m [ ψ∗
+x φ − ψ∗φx ]|b
+a = d
+dt⟨ψ, φ⟩KFG,
+(14)
+where [ g ]|b
+a ≡ g(b, t)−g(a, t), and ψx ≡ ∂ψ/∂x, etc. This result is also valid when V is different
+from zero inside the box (see Appendix I). The term evaluated at the endpoints of the interval
+Ω must vanish due to the boundary condition satisfied by ψ and φ, or Ψ and Φ (see Appendix
+II). Additionally, if we make ψ = φ, or Ψ = Φ, in Eq. (14), we obtain the result
+d
+dt⟨⟨Ψ, Ψ⟩⟩ = − [ j ]|b
+a = d
+dt⟨ψ, ψ⟩KFG,
+(15)
+where j = j(x, t) = (iℏ/2m)(ψ∗
+x ψ−ψ∗ψx) would be the probability current density, although we
+know that this quantity, as well as ̺, cannot be interpreted as probability quantities [3, 4]. The
+disappearance of the boundary term in Eq. (15) implies that the pseudo norm remains constant,
+and because j(a, t) = j(b, t), we have that ˆh must be a pseudo-Hermitian operator. In the case
+that Ω = R, the scalar product ⟨⟨Ψ, Φ⟩⟩ is a time-independent constant whenever Ψ and Φ are
+two normalizable solutions, i.e., solutions that have their pseudo norm finite. The square of the
+pseudo norm of these functions could be negative, but their magnitude cannot be infinite if the
+boundary term in Eq. (14) is expected to be zero.
+Next, we use the pseudo inner product given in Eq. (7), which is defined over an indefinite
+inner product space [14]. For a collection of basic properties of this scalar product (but also of
+general results on Hamiltonians of the type given in Eq. (2)), see Ref. [12]. Using integration
+by parts twice, it can be demonstrated that the Hamiltonian differential operator ˆh in Eq. (2)
+satisfies the following relation:
+⟨⟨Ψ, ˆhΦ⟩⟩ = ⟨⟨ˆhadjΨ, Φ⟩⟩ + f[Ψ, Φ],
+(16)
+where the boundary term f[Ψ, Φ] is given by
+f[Ψ, Φ] ≡ ℏ2
+2m
+�
+Ψ†
+x ˆτ3 (ˆτ3 + iˆτ2)Φ − Ψ† ˆτ3 (ˆτ3 + iˆτ2)Φx
+���b
+a .
+(17)
+
+7
+This quantity can also be written in a way that will be especially important, namely,
+f[Ψ, Φ] ≡ ℏ2
+2m
+1
+2
+�
+((ˆτ3 + iˆτ2)Ψx)† (ˆτ3 + iˆτ2)Φ − ((ˆτ3 + iˆτ2)Ψ)† (ˆτ3 + iˆτ2)Φx
+����
+b
+a .
+(18)
+The latter somewhat unexpected expression is true because the singular matrix ˆτ3 +iˆτ2 obeys the
+following relation: (ˆτ3 +iˆτ2)†(ˆτ3 +iˆτ2) = 2ˆτ3 (ˆτ3 +iˆτ2); however, (ˆτ3 +iˆτ2)2 = ˆ0. The differential
+operator ˆhadj in Eq. (16) is the generalized Hermitian conjugate, or the formal generalized adjoint
+of ˆh, namely,
+ˆhadj = ˆη−1 ˆh† ˆη = ˆτ3 ˆh† ˆτ3
+(19)
+(ˆη = ˆτ3 = ˆη−1 is sometimes called the metric operator; in this case, ˆη is a bounded operator
+and satisfies ˆη3 = ˆη) and therefore (just formally, i.e., by using only the scalar product definition
+given in Eq. (7)),
+⟨⟨Ψ, ˆhΦ⟩⟩ = ⟨⟨ˆhadjΨ, Φ⟩⟩.
+(20)
+The latter is essentially the relation that defines the generalized adjoint differential operator ˆhadj
+on an indefinite inner product space. Clearly, the latter definition requires that f[Ψ, Φ] in Eq.
+(16) vanishes.
+The Hamiltonian operator also formally satisfies the following relation:
+ˆh = ˆhadj,
+(21)
+that is, ˆh is formally generalized Hermitian (or formally pseudo-Hermitian), or formally generalized
+self-adjoint (or formally pseudo self-adjoint). However, if the boundary conditions imposed on Ψ
+and Φ at the endpoints of the interval Ω lead to the cancellation of the boundary term in Eq.
+(16), then the differential operator ˆh is indeed generalized Hermitian (or pseudo-Hermitian), and
+as shown in Appendix II, it is also generalized self-adjoint (or pseudo self-adjoint), i.e.,
+⟨⟨Ψ, ˆhΦ⟩⟩ = ⟨⟨ˆhΨ, Φ⟩⟩.
+(22)
+Precisely, we want to obtain a general set of boundary conditions for the generalized Hermitian
+Hamiltonian differential operator. Thus, if we impose Ψ = Φ in the latter relation and in Eq.
+(16) (with the result in Eq. (21)), we obtain the following condition:
+f[Ψ, Ψ] = ℏ
+i [ j ]|b
+a = 0
+( ⇒ j(b, t) = j(a, t) ) ,
+(23)
+where j = j(x, t) is given by
+j = iℏ
+2m
+1
+2
+�
+((ˆτ3 + iˆτ2)Ψx)† (ˆτ3 + iˆτ2)Ψ − ((ˆτ3 + iˆτ2)Ψ)† (ˆτ3 + iˆτ2)Ψx
+�
+(24)
+
+8
+(see Eq. (18)). But also because ˆτ3 (ˆτ3 + iˆτ2) = ˆ12 + ˆσx (the latter if we use the expression
+given by Eq. (17)), and the result in Eq. (5), we obtain
+j = iℏ
+2m ( ψ∗
+x ψ − ψ∗ψx ) ,
+(25)
+as expected (see the comment made just after Eq. (15)). Certainly, all the generalized Hermitian
+boundary conditions must lead to the equality of j at the endpoints of the interval Ω. Further-
+more, we also obtain the result ⟨⟨Ψ, ˆhΨ⟩⟩ = ⟨⟨ˆhΨ, Ψ⟩⟩ = ⟨⟨Ψ, ˆhΨ⟩⟩∗ (the superscript ∗ denotes
+the complex conjugate); therefore, ⟨⟨Ψ, ˆhΨ⟩⟩ ≡ ⟨⟨ˆh⟩⟩Ψ ∈ R, i.e., the generalized mean value of
+the Hamiltonian operator is real valued. Other typical properties of operators that are Hermitian
+in the usual sense hold here as well; for example, the eigenvalues are real (see, for example, Refs.
+[1, 3]).
+Substituting j from Eq. (25) into Eq. (23), we obtain the result (we omit the variable t in
+the expressions that follow)
+λ2m
+ℏ2 f[Ψ, Ψ] = [ ψ λψ∗
+x − ψ∗λψx ]|b
+a
+= [ ψ(b) λψ∗
+x(b) − ψ∗(b) λψx(b) ] − [ ψ(a) λψ∗
+x(a) − ψ∗(a) λψx(a) ] = 0,
+(26)
+where λ ∈ R is a parameter required for dimensional reasons. It is very convenient to rewrite the
+latter two terms using the following identity:
+z1z∗
+2 − z∗
+1z2 = i
+2 [ (z1 + iz2)(z1 + iz2)∗ − (z1 − iz2)(z1 − iz2)∗ ]
+= i
+2
+�
+|z1 + iz2|2 − |z1 − iz2|2 �
+,
+(27)
+where z1 and z2 are complex numbers. Then, the following result is obtained:
+λ2m
+ℏ2 f[Ψ, Ψ] = i
+2
+�
+|ψ(b) + iλψx(b)|2 − |ψ(b) − iλψx(b)|2 �
+− i
+2
+�
+|ψ(a) + iλψx(a)|2 − |ψ(a) − iλψx(a)|2 �
+= i
+2
+�
+|ψ(b) + iλψx(b)|2 + |ψ(a) − iλψx(a)|2 �
+− i
+2
+�
+|ψ(b) − iλψx(b)|2 + |ψ(a) + iλψx(a)|2 �
+= 0,
+(28)
+that is,
+λ2m
+ℏ2 f[Ψ, Ψ] = i
+2
+
+ ψ(b) + iλψx(b)
+ψ(a) − iλψx(a)
+
+
+† 
+ ψ(b) + iλψx(b)
+ψ(a) − iλψx(a)
+
+
+
+9
+− i
+2
+
+ ψ(b) − iλψx(b)
+ψ(a) + iλψx(a)
+
+
+† 
+ ψ(b) − iλψx(b)
+ψ(a) + iλψx(a)
+
+ = 0.
+(29)
+Let us now consider the following general matrix boundary condition:
+
+ ψ(b) + iλψx(b)
+ψ(a) − iλψx(a)
+
+ = ˆM
+
+ ψ(b) − iλψx(b)
+ψ(a) + iλψx(a)
+
+ ,
+(30)
+where ˆM is an arbitrary complex matrix. By substituting Eq. (30) into Eq. (29), we obtain
+i
+2
+
+ ψ(b) − iλψx(b)
+ψ(a) + iλψx(a)
+
+
+† �
+ˆM† ˆM − ˆ12
+�
+
+ ψ(b) − iλψx(b)
+ψ(a) + iλψx(a)
+
+ = 0;
+therefore, ˆM is a unitary matrix (the justification for this result is given in the comment that
+follows Eq. (A14)). Thus, a general set of generalized Hermitian boundary conditions for the 1D
+KFG particle in a box can be written as follows:
+
+ ψ(b) − iλψx(b)
+ψ(a) + iλψx(a)
+
+ = ˆU(2×2)
+
+ ψ(b) + iλψx(b)
+ψ(a) − iλψx(a)
+
+ ,
+(31)
+where ˆU(2×2) = ˆM−1 is also unitary. This family of boundary conditions is similar to the one
+corresponding to the problem of the 1D Schrödinger particle enclosed in a box; for example, see
+Eq. (28) in Ref. [15]. In relation to this, we can also take the nonrelativistic approximation of the
+general boundary condition given in Eq. (31). For that purpose, it is convenient to first write the
+KFG wavefunction ψ = ψ(x, t) as follows: ψ = ψS exp(−i mc2t/ℏ), where ψS = ψS(x, t) is the
+Schrödinger wavefunction. Because in this approximation we have that | iℏ(ψS)t | ≪ mc2 | ψS |,
+we can write ψt = (−i mc2t/ℏ)ψ, and therefore ψ1 =
+�
+1 −
+V
+2mc2
+�
+ψ and ψ2 =
+V
+2mc2ψ (see Eq.
+(4)). Thus, for weak external potentials and to the lowest order in v/c (and for positive energy
+solutions), ψ1 ≈ ψ satisfies the Schrödinger equation in the potential V + mc2 (the latter mc2
+can be eliminated by using the expression ψ1 ≈ ψ = ψS exp(−i mc2t/ℏ)) but also (ψ1)x ≈ ψx
+(see, for example, Refs. [2, 3, 9]). It is then clear that, in the problem of the particle in a box, the
+one-component KFG wavefunction satisfies the same boundary conditions as the one-component
+Schrödinger wavefunction. Incidentally, a similar result to Eq. (31) had already been obtained
+by taking the nonrelativistic limit of the most general family of boundary conditions for the 1D
+Dirac particle enclosed in a box [16]. Additionally, in the analogous problem of a 1D Schrödinger
+particle in the presence of a point interaction at the point x = 0 (or a hole at the origin), the most
+
+10
+general family of boundary conditions is similar to that given in Eq. (31) [17]. Indeed, all these
+results substantiate that the set of boundary conditions dependent on the four real parameters
+given in Eq. (31) is also the most general for a 1D KFG particle in the interval [a, b]. Moreover,
+by making the replacements a → 0+ and b → 0− in Eq. (31), we obtain the respective most
+general set of boundary conditions for the case in which the 1D KFG particle moves along the
+real line with a hole at the origin. Some examples of boundary conditions for this system can be
+seen in Refs. [7, 9] and will be briefly discussed in Section III.
+For all the boundary conditions that are part of the general set of boundary conditions in
+Eq. (31), ˆh is a pseudo-Hermitian operator, but it is also a pseudo self-adjoint operator (see
+Appendix II). Certainly, the result in Eq. (31) is given in terms of the wavefunction ψ, but if the
+relation in Eq. (5) is used, it can also be written in terms of the components of Ψ = [ ψ1 ψ2 ]T,
+i.e., in terms of ψ1 + ψ2, and its spatial derivative (ψ1)x + (ψ2)x, evaluated at the edges x = a
+and x = b. Actually, the general family of boundary conditions given in Eq. (31) must be written
+in terms of (ˆτ3 + iˆτ2)Ψ and (ˆτ3 + iˆτ2)Ψx evaluated at the ends of the box. We work on this
+in the next section. We give below some examples of boundary conditions that are contained
+in Eq.
+(31): ψ(a) = ψ(b) = 0 (ˆU(2×2) = −ˆ12), i.e., ψ can satisfy the Dirichlet boundary
+condition; ψx(a) = ψx(b) = 0 (ˆU(2×2) = +ˆ12), i.e., ψ can satisfy the Neumann boundary
+condition; ψ(a) = ψ(b) and ψx(a) = ψx(b) (ˆU(2×2) = +ˆσx), ψ can satisfy the periodic boundary
+condition; ψ(a) = −ψ(b) and ψx(a) = −ψx(b) (ˆU(2×2) = −ˆσx), ψ can satisfy the antiperiodic
+boundary condition; ψ(a) = ψx(b) = 0 (ˆU(2×2) = ˆσz), i.e., ψ can satisfy a mixed boundary
+condition; ψx(a) = ψ(b) = 0 (ˆU(2×2) = −ˆσz), i.e., ψ can satisfy another mixed boundary
+condition; ψ(a) − λψx(a) = 0 and ψ(b) + λψx(b) = 0 (ˆU(2×2) = iˆ12), ψ can satisfy a kind of
+Robin boundary condition. In fact, the latter boundary condition would be the KFG version of
+the boundary condition commonly used in the so-called (one-dimensional) MIT bag model for
+hadronic structures (see, for example, Ref. [16]). All these boundary conditions are typical of
+wave equations that are of the second order in the spatial derivative.
+Of all the boundary conditions included in the four-parameter family of boundary conditions,
+only those arising from a diagonal unitary matrix describe a particle in an impenetrable box. This
+is because, for these boundary conditions, the probability current density satisfies the relation
+j(b) = j(a) = 0 for all t. Thus, the most general family of confining boundary conditions for
+a 1D KFG particle in a box only has two (real) parameters. The latter result is due to the
+similarity between the general set of boundary conditions given in Eq. (31) and the general sets
+
+11
+of boundary conditions for the 1D Dirac and Schrödinger particles, and because we already know
+that the confining boundary conditions come from a matrix ˆU(2×2) that is diagonal [16].
+III.
+BOUNDARY CONDITIONS FOR THE 1D KFG PARTICLE IN A BOX II
+Here, we show that the most general family of pseudo self-adjoint boundary conditions can also
+be expressed in terms of Ψ and Ψx evaluated at the endpoints of the box. Specifically, in terms
+of (ˆτ3 + iˆτ2)Ψ and (ˆτ3 + iˆτ2)Ψx. Indeed, following a procedure similar to that used above to
+obtain Eq. (26), namely, substituting j from Eq. (24) into Eq. (23), we obtain
+λ2m
+ℏ2 f[Ψ, Ψ] = 1
+2
+�
+((ˆτ3 + iˆτ2)λΨx)† (ˆτ3 + iˆτ2)Ψ − ((ˆτ3 + iˆτ2)Ψ)† (ˆτ3 + iˆτ2)λΨx
+����
+b
+a
+= 1
+2
+�
+((ˆτ3 + iˆτ2)λΨx(b))† (ˆτ3 + iˆτ2)Ψ(b) − ((ˆτ3 + iˆτ2)Ψ(b))† (ˆτ3 + iˆτ2)λΨx(b)
+�
+− 1
+2
+�
+((ˆτ3 + iˆτ2)λΨx(a))† (ˆτ3 + iˆτ2)Ψ(a) − ((ˆτ3 + iˆτ2)Ψ(a))† (ˆτ3 + iˆτ2)λΨx(a)
+�
+= 0,
+(32)
+where again, we insert the real parameter λ for dimensional reasons. Now, we use the following
+matrix identity twice:
+ˆZ†
+2 ˆZ1 − ˆZ†
+1 ˆZ2 = i
+2
+�
+(ˆZ1 + iˆZ2)†(ˆZ1 + iˆZ2) − (ˆZ1 − iˆZ2)†(ˆZ1 − iˆZ2)
+�
+.
+(33)
+Then, we obtain the following result:
+λ2m
+ℏ2 f[Ψ, Ψ] = 1
+2
+i
+2
+�
+((ˆτ3 + iˆτ2)(Ψ + iλΨx)(b))† (ˆτ3 + iˆτ2)(Ψ + iλΨx)(b)
+− ((ˆτ3 + iˆτ2)(Ψ − iλΨx)(b))† (ˆτ3 + iˆτ2)(Ψ − iλΨx)(b)
+�
+−1
+2
+i
+2
+�
+((ˆτ3 + iˆτ2)(Ψ + iλΨx)(a))† (ˆτ3 + iˆτ2)(Ψ + iλΨx)(a)
+− ((ˆτ3 + iˆτ2)(Ψ − iλΨx)(a))† (ˆτ3 + iˆτ2)(Ψ − iλΨx)(a)
+�
+= 0,
+(34)
+that is,
+λ2m
+ℏ2 f[Ψ, Ψ] = 1
+2
+i
+2
+
+ (ˆτ3 + iˆτ2)(Ψ + iλΨx)(b)
+(ˆτ3 + iˆτ2)(Ψ − iλΨx)(a)
+
+
+† 
+ (ˆτ3 + iˆτ2)(Ψ + iλΨx)(b)
+(ˆτ3 + iˆτ2)(Ψ − iλΨx)(a)
+
+
+− 1
+2
+i
+2
+
+ (ˆτ3 + iˆτ2)(Ψ − iλΨx)(b)
+(ˆτ3 + iˆτ2)(Ψ + iλΨx)(a)
+
+
+† 
+ (ˆτ3 + iˆτ2)(Ψ − iλΨx)(b)
+(ˆτ3 + iˆτ2)(Ψ + iλΨx)(a)
+
+ = 0.
+(35)
+
+12
+Now, we propose writing a general matrix boundary condition as follows:
+
+ (ˆτ3 + iˆτ2)(Ψ + iλΨx)(b)
+(ˆτ3 + iˆτ2)(Ψ − iλΨx)(a)
+
+ = ˆA
+
+ (ˆτ3 + iˆτ2)(Ψ − iλΨx)(b)
+(ˆτ3 + iˆτ2)(Ψ + iλΨx)(a)
+
+ ,
+(36)
+where ˆA is an arbitrary 4 × 4 complex matrix. By substituting Eq. (36) into Eq. (35), we obtain
+1
+2
+i
+2
+
+ (ˆτ3 + iˆτ2)(Ψ − iλΨx)(b)
+(ˆτ3 + iˆτ2)(Ψ + iλΨx)(a)
+
+
+† �
+ˆA† ˆA − ˆ14
+�
+
+ (ˆτ3 + iˆτ2)(Ψ − iλΨx)(b)
+(ˆτ3 + iˆτ2)(Ψ + iλΨx)(a)
+
+ = 0,
+then ˆA is a unitary matrix (ˆ14 is the 4 × 4 identity matrix). Note that the components of the
+column vectors in Eq. (36) are themselves 2 × 1 column matrices and are given by
+(ˆτ3 + iˆτ2)(Ψ ± iλΨx)(x) =
+
+ (ψ ± iλψx)(x)
+−(ψ ± iλψx)(x)
+
+ ,
+x = a, b.
+(37)
+Thus, the general boundary condition in Eq. (36) can be written as follows:
+
+
+(ψ + iλψx)(b)
+−(ψ + iλψx)(b)
+(ψ − iλψx)(a)
+−(ψ − iλψx)(a)
+
+
+= ˆA
+
+
+(ψ − iλψx)(b)
+−(ψ − iλψx)(b)
+(ψ + iλψx)(a)
+−(ψ + iλψx)(a)
+
+
+.
+(38)
+On the other hand, this relation can also be written as follows:
+
+
+(ψ + iλψx)(b)
+(ψ − iλψx)(a)
+(ψ + iλψx)(b)
+(ψ − iλψx)(a)
+
+
+= ˆSˆAˆS†
+
+
+(ψ − iλψx)(b)
+(ψ + iλψx)(a)
+(ψ − iλψx)(b)
+(ψ + iλψx)(a)
+
+
+,
+(39)
+where ˆS is given by
+ˆS =
+
+
+1
+0
+0
+0
+0
+0
+1
+0
+0 −1 0
+0
+0
+0
+0 −1
+
+
+= 1
+2
+�
+ˆσz ⊗ ˆ12 + iˆσy ⊗ ˆσx + iˆσx ⊗ ˆσy + ˆ12 ⊗ ˆσz
+�
+,
+(40)
+
+13
+where ⊗ denotes the Zehfuss-Kronecker product of matrices, or the matrix direct product
+ˆF ⊗ ˆG ≡
+
+
+F11 ˆG · · · F1n ˆG
+...
+...
+...
+Fm1 ˆG · · · Fmn ˆG
+
+ ,
+(41)
+which is bilinear and associative and satisfies, among other properties, the mixed-product prop-
+erty: (ˆF ⊗ ˆG)(ˆJ ⊗ ˆK) = (ˆFˆJ ⊗ ˆGˆK) (see, for example, Ref. [18]). The matrix ˆS is unitary, and
+therefore, ˆSˆAˆS† is also a unitary matrix. Now, notice that the left-hand side of the relation in
+Eq. (39) is given by (see Eq. (30))
+
+
+
+ (ψ + iλψx)(b)
+(ψ − iλψx)(a)
+
+
+
+ (ψ + iλψx)(b)
+(ψ − iλψx)(a)
+
+
+
+
+=
+
+
+ˆM
+
+ (ψ − iλψx)(b)
+(ψ + iλψx)(a)
+
+
+ˆM
+
+ (ψ − iλψx)(b)
+(ψ + iλψx)(a)
+
+
+
+
+=
+
+
+ˆM ˆ0
+ˆ0
+ˆM
+
+
+
+
+(ψ − iλψx)(b)
+(ψ + iλψx)(a)
+(ψ − iλψx)(b)
+(ψ + iλψx)(a)
+
+
+,
+(42)
+and substituting the latter relation into Eq. (39), we obtain
+ˆSˆAˆS† =
+
+
+ˆM ˆ0
+ˆ0
+ˆM
+
+ = ˆ12 ⊗ ˆM
+(43)
+(because ˆM is a unitary matrix, the block diagonal matrix in Eq. (43) is also unitary). Then,
+from Eq. (43), we can write the matrix ˆA as follows:
+ˆA = ˆS†
+
+
+ˆM ˆ0
+ˆ0
+ˆM
+
+ ˆS = ˆS†(ˆ12 ⊗ ˆM)ˆS.
+(44)
+Thus, the most general set of generalized self-adjoint boundary conditions for the 1D KFG particle
+in a box can be written as follows (see Eq. (36)):
+
+ (ˆτ3 + iˆτ2)(Ψ − iλΨx)(b)
+(ˆτ3 + iˆτ2)(Ψ + iλΨx)(a)
+
+ = ˆU(4×4)
+
+ (ˆτ3 + iˆτ2)(Ψ + iλΨx)(b)
+(ˆτ3 + iˆτ2)(Ψ − iλΨx)(a)
+
+ ,
+(45)
+where
+ˆU(4×4) = ˆA−1 = ˆA† = ˆS†
+
+
+ˆM†
+ˆ0
+ˆ0
+ˆM†
+
+ ˆS = ˆS†
+
+
+ˆM−1
+ˆ0
+ˆ0
+ˆM−1
+
+ ˆS
+= ˆS†
+
+
+ˆU(2×2)
+ˆ0
+ˆ0
+ˆU(2×2)
+
+ ˆS = ˆS†(ˆ12 ⊗ ˆU(2×2))ˆS
+(46)
+
+14
+(to reach this result, we use Eq. (44) and the fact that ˆU(2×2) = ˆM−1, the latter two results
+and only some properties of the matrix direct product could also be used). Note that the general
+matrix boundary condition in Eq. (45) could also be written as follows:
+(ˆ12 ⊗ (ˆτ3 + iˆτ2))
+
+ (Ψ − iλΨx)(b)
+(Ψ + iλΨx)(a)
+
+ = ˆU(4×4)(ˆ12 ⊗ (ˆτ3 + iˆτ2))
+
+ (Ψ + iλΨx)(b)
+(Ψ − iλΨx)(a)
+
+ ;
+(47)
+however, the matrix ˆ12 ⊗ (ˆτ3 + iˆτ2) does not have an inverse and the column vector on the left
+side of this relation cannot be cleared. Thus, the expression given in Eq. (47) is an elegant way
+to write the general boundary condition, but it is not functional and could lead to errors.
+The boundary conditions that were presented just before the last paragraph of Sect. II can be
+extracted from Eq. (45) if the matrix ˆU(2×2) is known. In effect, the Dirichlet boundary condition
+is (ˆτ3 + iˆτ2)Ψ(a) = (ˆτ3 + iˆτ2)Ψ(b) = 0 (ˆU(4×4) = −ˆ14 = −ˆ12 ⊗ ˆ12); the Neumann boundary
+condition is (ˆτ3 + iˆτ2)Ψx(a) = (ˆτ3 + iˆτ2)Ψx(b) = 0 (ˆU(4×4) = +ˆ14 = +ˆ12 ⊗ ˆ12); the periodic
+boundary condition is (ˆτ3 + iˆτ2)Ψ(a) = (ˆτ3 + iˆτ2)Ψ(b) and (ˆτ3 + iˆτ2)Ψx(a) = (ˆτ3 + iˆτ2)Ψx(b)
+(ˆU(4×4) = ˆσx ⊗ ˆ12); the antiperiodic boundary condition is (ˆτ3 + iˆτ2)Ψ(a) = −(ˆτ3 + iˆτ2)Ψ(b)
+and (ˆτ3 + iˆτ2)Ψx(a) = −(ˆτ3 + iˆτ2)Ψx(b) (ˆU(4×4) = −ˆσx ⊗ ˆ12); a mixed boundary condition is
+(ˆτ3 + iˆτ2)Ψ(a) = (ˆτ3 + iˆτ2)Ψx(b) = 0 (ˆU(4×4) = ˆσz ⊗ ˆ12); another mixed boundary condition is
+(ˆτ3 + iˆτ2)Ψx(a) = (ˆτ3 + iˆτ2)Ψ(b) = 0 (ˆU(4×4) = −ˆσz ⊗ ˆ12); a kind of Robin boundary condition
+(and a kind of MIT bag boundary condition for a 1D KFG particle) is (ˆτ3+iˆτ2)(Ψ(a)−λΨx(a)) =
+0 and (ˆτ3 + iˆτ2)(Ψ(b) + λΨx(b)) = 0 (ˆU(4×4) = iˆ14 = iˆ12 ⊗ ˆ12). Then, to write all these
+boundary conditions in terms of ψ(a) and ψ(b), and ψx(a) and ψx(b), we must use the fact that
+Ψ = [ ψ1 ψ2 ]T and ψ = ψ1 + ψ2 (Eq. (5)). If we wish to obtain explicit relations between the
+components of Ψ and Ψx at x = a and Ψ and Ψx at x = b, we must use the relations given in
+Eqs. (5) and (6). Additionally, it can be shown that when the matrix ˆU(2×2) is diagonal, then
+the matrix ˆU(4×4) is also diagonal; consequently, diagonal matrices ˆU(4×4) in Eq. (45) lead to
+confining boundary conditions (see the last paragraph of Sect. II).
+In general, the boundary conditions imposed on (ˆτ3 +iˆτ2)Ψ and (ˆτ3 +iˆτ2)Ψx at the endpoints
+of the box do not imply that Ψ and Ψx must also satisfy them. For example, let us consider
+the problem of the 1D KFG particle in the step potential (V (x) = V0 Θ(x), where Θ(x) is
+the Heaviside step function).
+This problem was also considered in Refs.
+[7, 9].
+The step
+potential is a (soft) point interaction in the neighborhood of the origin, that is, between the
+points x = a → 0+ and x = b → 0−, and the boundary condition is the periodic boundary
+
+15
+condition, which in this case becomes the continuity condition of (ˆτ3 + iˆτ2)Ψ and (ˆτ3 + iˆτ2)Ψx
+at x = 0, i.e., (ˆτ3 + iˆτ2)Ψ(0−) = (ˆτ3 + iˆτ2)Ψ(0+) and (ˆτ3 + iˆτ2)Ψx(0−) = (ˆτ3 + iˆτ2)Ψx(0+).
+As we know, from this condition, it is obtained that ψ(0−) = ψ(0+) and ψx(0−) = ψx(0+).
+If the relations ψ1 + ψ2 = ψ (Eq. (5)) and ψ1 − ψ2 = (E − V )ψ/mc2 (Eq. (6)) are used
+(in the latter, we also assumed that ψ is an energy eigenstate), one can find relations between
+{Ψ(0+), Ψx(0+)} and {Ψ(0−), Ψx(0−)}. We find that the relation given in Eq. (30) in Ref.
+[7] is none other than the boundary condition (ˆτ3 + iˆτ2)Ψ(0−) = (ˆτ3 + iˆτ2)Ψ(0+), with Eqs.
+(5) and (6) evaluated at x = 0±. Likewise, the relation given in Eq. (31) of the same reference
+is none other than (ˆτ3 + iˆτ2)Ψx(0−) = (ˆτ3 + iˆτ2)Ψx(0+), with the spatial derivatives of Eqs.
+(5) and (6) also evaluated at x = 0±. Finally, adding the latter two boundary conditions, we
+obtain Eq. (32) of Ref. [7]. Clearly, if the height of the step potential is not zero, then Ψ(0+)
+is different from Ψ(0−), and Ψx(0+) is different from Ψx(0−). Similarly, in Ref. [9], it was
+explicitly proven that Ψ(0+) ̸= Ψ(0−) and Ψx(0+) ̸= Ψx(0−) (see Eqs. (19) and (20) in
+that reference), but it was also shown that the boundary condition should be written in the form
+(ˆτ3+iˆτ2)Ψ(0−) = (ˆτ3+iˆτ2)Ψ(0+) and (ˆτ3+iˆτ2)Ψx(0−) = (ˆτ3+iˆτ2)Ψx(0+). Incidentally, in the
+same reference, it was shown that the latter boundary condition can be obtained by integrating
+the 1D FV equation from x = 0− to x = 0+.
+On the other hand, in the problem of the 1D KFG particle inside the box Ω = [a, b], and
+subjected to the potential V , with the Dirichlet boundary condition, (ˆτ3 + iˆτ2)Ψ(a) = (ˆτ3 +
+iˆτ2)Ψ(b) = 0, we know that ψ also satisfies this condition, namely, ψ(a) = ψ(b) = 0. The
+latter boundary condition together with Eqs. (5) and (6) lead us to the boundary condition
+Ψ(a) = Ψ(b) = 0. Indeed, in addition to ψ1(a) + ψ2(a) = ψ1(b) + ψ2(b) = 0, ψ1(a) − ψ2(a) =
+ψ1(b) − ψ2(b) = 0 (because ψt(a, t) = ψt(b, t) = 0 also holds). Finally, Ψ also satisfies the
+Dirichlet boundary condition at the edges of the box (the latter boundary condition was precisely
+the one used in Ref. [8]).
+In short, let us suppose that the one-component wavefunction ψ can vanish at a point on
+the real line, for example, at x = 0 (also V (0+) and V (0−) must be finite numbers there).
+The latter is the Dirichlet boundary condition, namely, ψ(0−) = ψ(0+) = 0 ≡ ψ(0). Certainly,
+this result is obtained from the disappearance of (ˆτ3 + iˆτ2)Ψ at that same point, i.e., from the
+fact that the Hamiltonian operator with the latter boundary condition is a pseudo self-adjoint
+operator; then, the latter condition implies that the entire two-component wavefunction Ψ has
+to disappear at that point (use Eqs. (5) and (6)). In other words, the 1D FV wave equation
+
+16
+is a second-order equation in the spatial derivative that accepts the vanishing of the entire two-
+component wavefunction at a point. On the other hand, let us now suppose that ψx can vanish
+at a point on the real line, for example, at x = 0, but ψ is nonzero there (also Vx(0+) and
+Vx(0−) must be finite numbers there). The latter is the Neumann boundary condition, namely,
+ψx(0−) = ψx(0+) = 0 ≡ ψx(0). Indeed, we also have that (ˆτ3 + iˆτ2)Ψx vanishes at that same
+point. Then, it can be shown that (ψ1)x and (ψ2)x do not have to vanish at the point in question,
+and therefore, Ψx is not zero there either (use Eqs. (5) and (6)).
+IV.
+APPENDIX I
+The 1D KFG wave equation given in Eq. (3) can also be written as follows:
+�
+−ℏ2 ∂2
+∂t2 − i2ℏ V (x) ∂
+∂t + (V (x))2
+�
+ψ =
+�
+−ℏ2c2 ∂2
+∂x2 + (mc2)2
+�
+ψ,
+(A1)
+and therefore,
+ψtt = c2ψxx −
+�mc2
+ℏ
+�2
+ψ + 2V
+iℏ ψt + V 2
+ℏ2 ψ.
+(A2)
+The scalar product for the two-component column state vectors Ψ = [ ψ1 ψ2 ]T and Φ =
+[ φ1 φ2 ]T, where ψ1 + ψ2 = ψ and φ1 + φ2 = φ, is given by
+⟨⟨Ψ, Φ⟩⟩ ≡
+ˆ
+Ω
+dx Ψ†ˆτ3Φ =
+iℏ
+2mc2
+ˆ
+Ω
+dx
+�
+ψ∗
+� ∂
+∂t − V
+iℏ
+�
+φ −
+�� ∂
+∂t − V
+iℏ
+�
+ψ
+�∗
+φ
+�
+=
+iℏ
+2mc2
+ˆ
+Ω
+dx
+�
+ψ∗φt − ψ∗
+t φ − 2V
+iℏ ψ∗φ
+�
+≡ ⟨ψ, φ⟩KFG.
+(A3)
+The latter quantity is preserved in time; in fact, taking its time derivative and using the result in
+Eq. (A2), and a similar relation for φ (ψ and φ are solutions of the 1D KFG wave equation in
+its standard form), one obtains the same relation given in Eq. (14), namely,
+d
+dt⟨⟨Ψ, Φ⟩⟩ = d
+dt⟨ψ, φ⟩KFG = − iℏ
+2m [ ψ∗
+x φ − ψ∗φx ]|b
+a .
+(A4)
+As follows from the results obtained in Appendix II, if ψ and φ both satisfy any boundary condition
+included in the most general set of boundary conditions, the boundary term in Eq. (A4) always
+vanishes.
+
+17
+V.
+APPENDIX II
+The goal of this section is to show that if the functions belonging to the domain of ˆh (considered
+a densely defined operator) obey any of the boundary conditions included in Eq. (31), then the
+functions of the domain of ˆhadj must obey the same boundary condition. This means that for
+the general family of boundary conditions given in Eq. (31), the operator ˆh = ˆhadj is pseudo
+self-adjoint. Our results are obtained using simple arguments that are part of the general theory
+of linear operators in an indefinite inner product space (see, for example, Refs. [19, 20]).
+Let us return to the result given in Eq. (16), namely,
+⟨⟨Ξ, ˆhΦ⟩⟩ = ⟨⟨ˆhadjΞ, Φ⟩⟩ + f[Ξ, Φ],
+(A5)
+where f[Ξ, Φ] is given by (see Eq. (18))
+f[Ξ, Φ] ≡ ℏ2
+2m
+1
+2
+�
+((ˆτ3 + iˆτ2)Ξx)† (ˆτ3 + iˆτ2)Φ − ((ˆτ3 + iˆτ2)Ξ)† (ˆτ3 + iˆτ2)Φx
+����
+b
+a .
+(A6)
+Here, ˆh can act on column vectors Φ = [ φ1 φ2 ]T ∈ D(ˆh), where D(ˆh) is the domain of ˆh, a set
+of column vectors on which we allow the differential operator ˆh to act, which includes boundary
+conditions, and ˆhadj can act on column vectors Ξ = [ ξ1 ξ2 ]T ∈ D(ˆhadj) (in general, D(ˆhadj)
+may not coincide with D(ˆh)). By virtue of the result given in Eq. (5), the respective solutions
+of Eq. (3) are the following:
+φ1 + φ2 = φ
+and
+ξ1 + ξ2 = ξ.
+(A7)
+The boundary term in Eq. (A6) can be written in terms of φ and ξ, namely,
+f[Ξ, Φ] = ℏ2
+2m [ ξ∗
+x φ − ξ∗φx ]|b
+a .
+(A8)
+First, let us suppose that every column vector Φ ∈ D(ˆh) satisfies the boundary conditions
+(ˆτ3 + iˆτ2)Φ(a) = (ˆτ3 + iˆτ2)Φ(b) = 0 and (ˆτ3 + iˆτ2)Φx(a) = (ˆτ3 + iˆτ2)Φx(b) = 0, or, equivalently,
+φ(a) = φ(b) = 0 and φx(a) = φx(b) = 0 (remember the first relation in Eq. (A7)). In this case,
+the boundary term in Eq. (A5) vanishes, and we have the result
+⟨⟨Ξ, ˆhΦ⟩⟩ = ⟨⟨ˆhadjΞ, Φ⟩⟩.
+(A9)
+The latter relation is precisely the one that defines the generalized adjoint differential operator.
+It is clear that its verification did not require the imposition of any boundary condition on the
+
+18
+vectors Ξ ∈ D(ˆhadj). Thus, until now, we have that D(ˆh) ̸= D(ˆhadj) (in fact, we have that
+D(ˆh) ⊂ D(ˆhadj), i.e., D(ˆh) is a restriction of D(ˆhadj)).
+If the operator ˆh is to be a generalized self-adjoint differential operator, the relation given in
+Eq. (21), namely, ˆh = ˆhadj, must be verified, and therefore, D(ˆh) = D(ˆhadj). To achieve this, we
+must allow every vector Φ ∈ D(ˆh) to satisfy more general boundary conditions, that is, we must
+relax the domain of ˆh. Let us suppose that we have a set of boundary conditions to be imposed
+on a vector Φ ∈ D(ˆh); if the cancellation of the boundary term f[Ξ, Φ] by these boundary
+conditions only depends on imposing the same boundary conditions on the vector Ξ ∈ D(ˆhadj),
+then ˆh will be a generalized self-adjoint differential operator.
+First, from Eq. (A8), we write the boundary term in Eq. (A5) as follows:
+λ2m
+ℏ2 f[Ξ, Φ] = [ φ λξ∗
+x − ξ∗λφx ]|b
+a
+= [ φ(b) λξ∗
+x(b) − ξ∗(b) λφx(b) ] − [ φ(a) λξ∗
+x(a) − ξ∗(a) λφx(a) ] = 0.
+(A10)
+It is fairly convenient to rewrite the latter two terms using the following identity:
+z1z∗
+2 − z∗
+3z4 = i
+2 [ (z1 + iz4)(z3 + iz2)∗ − (z1 − iz4)(z3 − iz2)∗ ] ,
+(A11)
+where z1, z2, z3 and z4 are complex numbers. The latter relation is the generalization of that
+given in Eq. (27). In fact, making the replacements z3 → z1 and z4 → z2 in Eq. (A11), the
+relation given in Eq. (27) is obtained. Then, the following result is derived:
+λ2m
+ℏ2 f[Ξ, Φ] = i
+2 [(φ(b) + iλφx(b)) (ξ(b) + iλξx(b))∗ − (φ(b) − iλφx(b)) (ξ(b) − iλξx(b))∗]
+− i
+2 [(φ(a) + iλφx(a)) (ξ(a) + iλξx(a))∗ − (φ(a) − iλφx(a)) (ξ(a) − iλξx(a))∗]
+= i
+2 [(φ(b) + iλφx(b)) (ξ(b) + iλξx(b))∗ + (φ(a) − iλφx(a)) (ξ(a) − iλξx(a))∗]
+− i
+2 [(φ(b) − iλφx(b)) (ξ(b) − iλξx(b))∗ + (φ(a) + iλφx(a)) (ξ(a) + iλξx(a))∗] = 0,
+this means that
+λ2m
+ℏ2 f[Ξ, Φ] = i
+2
+
+ ξ(b) + iλξx(b)
+ξ(a) − iλξx(a)
+
+
+† 
+ φ(b) + iλφx(b)
+φ(a) − iλφx(a)
+
+
+− i
+2
+
+ ξ(b) − iλξx(b)
+ξ(a) + iλξx(a)
+
+
+† 
+ φ(b) − iλφx(b)
+φ(a) + iλφx(a)
+
+ = 0.
+(A12)
+
+19
+Let us now consider a more general set of boundary conditions to be imposed on a vector
+Φ ∈ D(ˆh) (i.e., more general than the boundary conditions that we presented after Eq. (A8)),
+namely,
+
+ φ(b) + iλφx(b)
+φ(a) − iλφx(a)
+
+ = ˆN
+
+ φ(b) − iλφx(b)
+φ(a) + iλφx(a)
+
+ ,
+(A13)
+where ˆN in an arbitrary complex matrix. By substituting the latter relation in Eq. (A12), we
+obtain the following result:
+λ2m
+ℏ2 f[Ξ, Φ]
+= i
+2
+
+
+
+
+
+
+
+
+
+ ξ(b) + iλξx(b)
+ξ(a) − iλξx(a)
+
+
+†
+ˆN −
+
+ ξ(b) − iλξx(b)
+ξ(a) + iλξx(a)
+
+
+†
+
+
+
+ φ(b) − iλφx(b)
+φ(a) + iλφx(a)
+
+
+
+
+
+
+
+= 0,
+and therefore,
+
+ ξ(b) + iλξx(b)
+ξ(a) − iλξx(a)
+
+
+†
+ˆN =
+
+ ξ(b) − iλξx(b)
+ξ(a) + iλξx(a)
+
+
+†
+(A14)
+(This result is because, at this point, we cannot impose any boundary conditions that would
+completely annul the column vectors in Eq. (A13), for example). Every vector Ξ ∈ D(ˆhadj)
+should satisfy the same boundary conditions that Φ ∈ D(ˆh) satisfies, i.e., the boundary conditions
+in Eq. (A13), namely,
+
+ ξ(b) + iλξx(b)
+ξ(a) − iλξx(a)
+
+ = ˆN
+
+ ξ(b) − iλξx(b)
+ξ(a) + iλξx(a)
+
+ .
+(A15)
+Taking the Hermitian conjugate of the matrix relation in Eq. (A14) and substituting this result
+into Eq. (A15), we obtain
+
+ ξ(b) + iλξx(b)
+ξ(a) − iλξx(a)
+
+ = ˆNˆN†
+
+ ξ(b) + iλξx(b)
+ξ(a) − iλξx(a)
+
+ ;
+therefore, ˆN is a unitary matrix. Thus, the most general family of generalized self-adjoint, or
+pseudo self-adjoint boundary conditions, for the 1D KFG particle in a box can be written in the
+form given by Eq. (31), namely,
+
+ ξ(b) − iλξx(b)
+ξ(a) + iλξx(a)
+
+ = ˆU
+
+ ξ(b) + iλξx(b)
+ξ(a) − iλξx(a)
+
+ ,
+(A16)
+
+20
+where ˆU = ˆN−1. The fact that the boundary condition for Φ ∈ D(ˆh) (for example, given in
+terms of φ) is the same boundary condition for Ξ ∈ D(ˆhadj) (given in terms of ξ) ensures that
+D(ˆh) = D(ˆhadj); therefore, ˆh, which was already a pseudo-Hermitian operator, is also a pseudo
+self-adjoint operator. Additionally, the boundary term given in Eq. (14), or in Eq. (A4), vanishes,
+and therefore, the pseudo inner product is conserved.
+VI.
+CONCLUDING REMARKS
+The KFG Hamiltonian operator, or the Hamiltonian present in the 1D FV wave equation, is
+formally pseudo-Hermitian; this is well known. In addition, when this operator describes a 1D
+KFG particle in a box, it is a pseudo-Hermitian operator, but is also a pseudo self-adjoint operator.
+We obtained the most general set of boundary conditions for this problem, which depends on four
+real parameters. These results can be extended to the problem of the 1D KFG particle moving
+on the real line with a penetrable or an impenetrable obstacle at one point, i.e., with a point
+interaction there.
+As we have seen, the general set of boundary conditions can be written in terms of the one-
+component wavefunction for the 1D KFG wave equation ψ and its derivative ψx, evaluated at
+the ends of the box. More interestingly, the general set can also be precisely written in terms
+of the two-component column vectors (ˆτ3 + iˆτ2)Ψ and (ˆτ3 + iˆτ2)Ψx, evaluated at the ends of
+the box. As seen from the examples presented in Section III, Ψ and Ψx, evaluated at x = a
+and x = b, do not necessarily satisfy by themselves the same boundary conditions of the general
+set. However, any specific boundary condition on Ψ and Ψx can be obtained from the respective
+boundary condition that ψ and ψx satisfy at the ends of the box and by using the relations that
+arise between the components of the column vector Ψ = [ ψ1 ψ2 ]T, and ψ, ψt, and the potential
+V (see Eqs. (5) and (6)). We hope that our article will be of interest to those interested in the
+fundamental and also technical aspects of relativistic wave equations. Furthermore, to the best
+of our knowledge, the main results of our article, i.e., those related to general sets of boundary
+conditions in the 1D KFG theory, do not appear to have been considered before.
+
+21
+Conflicts of interest
+The author declares no conflicts of interest.
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