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+Toward Building General Foundation Models for Language, Vision, and
+Vision-Language Understanding Tasks
+Xinsong Zhang 1 Yan Zeng 1 Jipeng Zhang 2 Hang Li 1
+Abstract
+Foundation models or pre-trained models have
+substantially improved the performance of various
+language, vision, and vision-language understand-
+ing tasks. However, existing foundation models
+can only perform the best in one type of tasks,
+namely language, vision, or vision-language. It is
+still an open question whether it is possible to con-
+struct a foundation model performing the best for
+all the understanding tasks, which we call a gen-
+eral foundation model. In this paper, we propose
+a new general foundation model, X-FM (the X-
+Foundation Model). X-FM has one language en-
+coder, one vision encoder, and one fusion encoder,
+as well as a new training method. The training
+method includes two new techniques for learning
+X-FM from text, image, and image-text pair data.
+One is to stop gradients from the vision-language
+training when learning the language encoder. The
+other is to leverage the vision-language training
+to guide the learning of the vision encoder. Exten-
+sive experiments on benchmark datasets show that
+X-FM can significantly outperform existing gen-
+eral foundation models and perform better than or
+comparable to existing foundation models specif-
+ically for language, vision, or vision-language
+understanding.
+1. Introduction
+With the enormous power of foundation models, also known
+as pre-trained models, remarkable performance gains have
+recently been achieved in a variety of understanding tasks in
+natural language processing (NLP), computer vision (CV),
+and other fields (Devlin et al., 2019; Liu et al., 2019; Lewis
+et al., 2020; Raffel et al., 2020; Brown et al., 2020; Doso-
+vitskiy et al., 2021; He et al., 2022; Bao et al., 2021; Lu
+1ByteDance AI Lab 2The Hong Kong University of Science
+and Technology. Correspondence to: Xinsong Zhang <zhangxin-
+[email protected]>.
+Copyright 2023 by the author(s). The code and pre-trained models
+will be released upon publication.
+et al., 2019; Tan & Bansal, 2019a; Chen et al., 2020; Li
+et al., 2020; 2021a; Zeng et al., 2021; 2022) . Foundation
+models are usually equipped with Transformer (Vaswani
+et al., 2017) as the backbone, pre-trained with a tremendous
+amount of unlabeled data, and then fine-tuned with small
+amounts of labeled data in downstream tasks. The strong
+representation ability of the model, the massive amount of
+data, and the effective means of training make the founda-
+tion models powerful for successfully solving the tasks of
+vision, language, and vision-language (Li et al., 2021b;c;
+Singh et al., 2021; Wang et al., 2021b; 2022b; Diao et al.,
+2022; Wang et al., 2022a).
+The state-of-the-art foundation models usually work the
+best for one type of tasks, namely language, vision, and
+vision-language. For example, RoBERTa (Liu et al., 2019),
+BEiTv2 (Peng et al., 2022), and X-VLM (Zeng et al., 2021;
+2022) are language, vision, and vision-language founda-
+tion models respectively, and can achieve state-of-the-art
+performances for the specific type of tasks. It is still very
+challenging, however, to build a general foundation model
+that can perform the best in all types of tasks. Existing
+models, such as FLAVA (Singh et al., 2021), OFA (Wang
+et al., 2022b), DaVinci (Diao et al., 2022) and Uni-Perceiver-
+MoE (Zhu et al., 2022), are trying to achieve the goal. Their
+performances are still not satisfactory, however, when com-
+pared with the best performing foundation models for the
+individual types of tasks, as shown in Table 1. Previous
+work (Bingel & Søgaard, 2017; Wang et al., 2020) also
+shows that it is difficult to train a general foundation model
+in a multi-task learning setting that can effectively learn and
+utilize representations for all types of tasks. The reason is
+that language, vision, and vision-language are very different
+in nature, and a simple way of jointly training a model from
+language, vision, and vision-language data can easily create
+a suboptimal solution.
+To address the challenge, we propose a new general founda-
+tion model, X-FM (X-Foundation Model). X-FM consists of
+three modular encoders for language (text) encoding, vision
+(image) encoding, and fusion encoding, as shown in Fig 1.
+The language encoder, the vision encoder, and the entire
+model can be used in downstream tasks of language, vision,
+and vision-language understanding, respectively. All three
+arXiv:2301.05065v1  [cs.CV]  12 Jan 2023
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+Methods
+Text Tasks
+Vision Tasks
+Multi-modal Tasks (MSCOCO Retriveal & VQA)
+GLUE
+ImageNet
+Zero-Shot
+Fine-Tune
+MNLI
+RTE
+FT/LE
+TR
+IR
+TR
+IR
+VQA
+Foundation models specifically for language, vision, or vision-language understanding
+RoBERTa (Liu et al., 2019)
+87.6
+78.7
+–
+–
+–
+–
+–
+–
+BEiTv2 (Peng et al., 2022)
+–
+–
+85.5/80.1
+–
+–
+–
+–
+–
+X-VLM (Zeng et al., 2021)
+–
+–
+–
+70.8/92.1/96.5
+55.6/82.7/90.0
+80.4/95.5/98.2
+63.1/85.7/91.6
+78.1
+X2-VLM (Zeng et al., 2022)
+–
+–
+–
+–
+–
+80.5/95.5/97.8
+62.7/84.7/90.7
+79.2
+General foundation models
+UNIMO-2 (Li et al., 2021c)
+87.5
+–
+80.8/-
+–
+–
+–
+–
+76.3
+SimVLM (Wang et al., 2021c)
+83.4
+63.9
+-/80.6
+–
+–
+–
+–
+77.9
+FLAVA (Singh et al., 2021)
+80.3
+57.8
+-/75.5
+42.7/76.8/-
+38.4/67.5/-
+61.5/82.1/89.6
+50.1/74.4/83.2
+72.8
+OFA (Wang et al., 2022b)
+84.3
+70.8
+82.2/–
+–
+–
+–
+–
+78.0
+DaVinci (Diao et al., 2022)
+83.1
+64.2
+83.9/78.8
+–
+–
+–
+–
+76.3
+OmniVL (Wang et al., 2022a)
+–
+–
+–
+–
+–
+76.8/93.6/97.3
+58.5/82.6/89.5
+78.3
+Uni-Perceiver-MoE (Zhu et al., 2022)
+81.5
+75.8
+84.5/–
+64.6/–/–
+51.6/–/–
+70.5/–/–
+54.1/–/–
+–
+X-FMbase
+87.7
+83.2
+85.3/81.0
+73.8/93.9/97.2
+59.4/83.6/90.0
+81.8/96.0/98.3
+64.7/86.1/91.6
+79.1
+Table 1: Performance comparisons between foundation models. All results are from base-size models. MSCOCO is a
+cross-modal retrieval task, and IR and TR are image-retrieval and text-retrieval, respectively. MNLI results are average
+accuracies of MNLI-m and MNLI-mm. Accuracy is reported for RTE. For ImageNet1k classification, we report linear
+evaluation (LE) performance and fine-tuning (FT) performance, respectively. We report R@1/R@5/R@10 for all retrieval
+tasks at both zero-shot and fine-tune settings. We report the VQA test-dev result. bold denotes the best number across
+general foundation models. underline denotes the best across all models.
+encoders are stacked Transformer layers. The language en-
+coder and the vision encoder follow the implementations
+of BERT (Devlin et al., 2019) and ViT (Dosovitskiy et al.,
+2021), respectively. The fusion encoder has the same ar-
+chitecture as BERT except that there is a cross-attention
+sub-layer after the self-attention sub-layer in each Trans-
+former layer.
+In learning of X-FM, the language encoder, vision encoder,
+and fusion encoder are jointly trained with text data, im-
+age data, and image-text pair data as input. Given the text
+data, we train the language encoder by masked language
+modeling (MLM). Given the image data, we train the vi-
+sion encoder by masked image modeling (MIM). Given the
+image-text pair data, we train the fusion encoder by image
+text matching (ITM), image-conditioned masked language
+modeling (IMLM), bounding box prediction (BBP), train
+the vision encoder and the language encoder by image-text
+contrastive learning (ITC), and train the vision encoder by
+MIM. (See Fig 1.)
+The essential thinking of our learning method is that lan-
+guage is more abstract than vision, and there is an asymmet-
+ric relationship between language and vision. Therefore, we
+separate the learning of the three encoders. The language
+encoder is trained mainly from text data and is isolated from
+the training of the fusion encoder. The vision encoder is
+simultaneously trained from image data and image-text pair
+data, guided by the vision-language training. The fusion
+encoder is trained from image-text pair data.
+Our learning method includes two new techniques. One
+technique is to stop gradients from the vision-language train-
+ing when learning the language encoder. The gradient flow
+is stopped from the fusion encoder to the language encoder
+in training, while the activation flow from the language en-
+coder to the fusion encoder is as usual. As a result, the
+language encoder is not affected by training of the fusion
+encoder with image-text pair data. Moreover, the training of
+the fusion encoder concentrates on learning the alignments
+between language and vision features.
+The other technique is to leverage the vision-language train-
+ing to guide the learning of the vision encoder with masked
+image modeling (MIM). In MIM, the masked image is com-
+pared with the original image by the differences between the
+predicted representations and target representations at the
+masked and [CLS] positions. The vision encoder creates
+both the predicated and target representations, while there
+is gradient flow from the predicted representations but no
+gradient flow from the target representations. The vision
+encoder can create the target representations because it is
+also trained in the vision-language training.
+We conduct experiments on a variety of twenty-two tasks of
+language, vision, and vision-language understanding. X-FM
+can outperform other general foundation models by a large
+margin and can even achieve better or comparable perfor-
+mance than SOTA foundation models specifically designed
+for language, vision, or vision-language understanding tasks,
+as shown in Table 1.
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+2. Related Work
+Following the success of language model pre-training, vi-
+sion pre-training and vision-language pre-training with
+Transformer as the backbone (Vaswani et al., 2017) have
+also made significant progress recently, pushing the state-of-
+the-art of various understanding tasks of language, vision,
+and vision-language.
+In language understanding, BERT (Devlin et al., 2019) is
+the first model adopting masked language modeling (MLM)
+for pre-training, which achieves remarkable performance
+on a wide range of tasks. Several other models are then
+developed to improve training robustness (Liu et al., 2019),
+sample efficiency (Sun et al., 2019; Joshi et al., 2020; Clark
+et al., 2020), and prediction accuracy of BERT (Lan et al.,
+2020; Zhang et al., 2020; He et al., 2021).
+In vision understanding, ViT (Dosovitskiy et al., 2021; Tou-
+vron et al., 2021) is proposed, utilizing Transformer as the
+backbone. Inspired by MLM, subsequent work proposes
+using masked image modeling (MIM) with the objective of
+recovering masked images. The learning targets vary from
+pixels (He et al., 2022) to image tokens (Bao et al., 2021;
+Peng et al., 2022).
+In vision-language understanding, there are generally two
+approaches. One is “dual encoders,” in which image and text
+are encoded separately, followed by a shallow interaction
+layer. The other is “fusion encoder(s)” in which attention
+or self-attention is used to fuse information from the two
+modalities after encoding. The former approach includes
+CLIP (Radford et al., 2021) and ALIGN (Jia et al., 2021)
+and performs well in vision tasks and cross-modal retrieval
+tasks. However, it cannot perform so well in multi-modal
+fusion tasks such as visual question answering (VQA (Goyal
+et al., 2017)) and visual reasoning (NLVR2 (Suhr et al.,
+2019b)). The latter approach varies depending on the way
+of using image features. Early work feeds pre-extracted
+object features along with texts into Transformer models
+and trains the models to make multi-modal modeling and
+multi-modal alignments with suitable objectives (Lu et al.,
+2019; Tan & Bansal, 2019b; Li et al., 2020; Chen et al.,
+2020; Cho et al., 2021; Zhang et al., 2021). Later work
+uses patch embeddings directly with new architectures such
+as vision Transformer (Li et al., 2021a; 2022) or multiway
+Transformer (Wang et al., 2021a; Bao et al., 2022) and uses
+new objectives such as bounding box prediction (Zeng et al.,
+2021; 2022).
+Recently, the fact that Transformer can model multi-modal
+data within a single architecture has inspired research to
+develop general foundation models that can solve lan-
+guage, vision, and vision-language tasks at the same time.
+UNIMO (Li et al., 2021b;c) jointly learns from image
+and text data vision representations, language representa-
+tions, and vision-language alignments in a shared space.
+FLAVA (Singh et al., 2021), a general foundation model, per-
+forms pre-training with masked uni-modal and multi-modal
+modeling objectives. OFA (Wang et al., 2022c) formulates
+vision-language tasks as sequence-to-sequence (seq2seq)
+problems and pre-trains a seq2seq model in multi-task learn-
+ing. SimVLM (Wang et al., 2021c) pre-trains a seq2seq
+model with a single objective of language generation (prefix
+language modeling). DaVinci (Diao et al., 2022) combines
+prefix language modeling and prefix image modeling to
+learn a general foundation model for a wide range of tasks.
+Uni-Perceiver (Zhu et al., 2021; 2022) builds a unified per-
+ception architecture that processes various modalities and
+tasks with a single Transformer network and shared parame-
+ters.
+Previous studies on general foundation models have shown
+that different capabilities can be established with only one
+model. Still, few studies demonstrate that the best perfor-
+mance can be achieved in all tasks with one model. In this
+paper, we propose a new general foundation model and show
+that it can perform the best for all the understanding tasks
+of language, vision, and vision-language. We compare our
+model extensively with recent general foundation models
+on multiple dimensions, as shown in Appendix A.
+Several super-large foundation models (over 1B parame-
+ters) are proposed recently, most of which are trained on
+super-large in-house datasets (over 400M image-text pairs).
+The authors do not report results at the base (about 280M
+parameters) and large (about 800M parameters) scale on
+public datasets, which we consider in this paper. CoCa (Yu
+et al., 2022) pre-trains an image-text sequence-to-sequence
+model with contrastive loss and captioning loss. BEiT-
+3 (Wang et al., 2022d) uses a multi-way Transformer and a
+unified objective of masked “language” modeling for learn-
+ing from image (Imglish1), text, and image-text pair data.
+Florence (Yuan et al., 2021) first scales the web-scale image-
+text pairs to 900M representations and then adapts to vari-
+ous computer vision tasks. Flamingo (Alayrac et al., 2022)
+makes use of a large language model in vision-language
+pre-training to solve the “in-context learning” problem for
+vision-language tasks. PaLI (Chen et al., 2022) jointly scales
+up the vision encoder and language encoder to cover a vari-
+ety of language, vision, vision-language, and multilingual
+tasks.
+3. Method
+3.1. Model Architecture and Training Process
+We propose a new general foundation model X-FM, having
+a language encoder, a vision encoder, and a fusion encoder,
+shown as Fig 1. The language encoder is a stack of Trans-
+1They view the image as a foreign language.
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+Text Encoder
+Cross-Attention
+Fusion Encoder
+Feed Forward
+Self-Attention
+Vision Encoder
+Feed Forward
+Self-Attention
+two
+brown
+and
+white
+dogs.
+Mask
+Feed Forward
+Self-Attention
+stop grad
+MIM
+MLM
+IMLM, ITM, BBP
+ITC
+x M
+x N
+x L
+Mask
+target
+Figure 1: The architecture and pre-training process of X-FM, a Transformer-based general foundation model. Given
+a text, we learn the language encoder by MLM. Given an image, we learn the vision encoder by MIM. Given an image-text
+pair, we learn the fusion encoder by BBP, ITM, IMLM and ITC, and further learn the vision encoder by MIM. The gradients
+of BBP, ITM, and IMLM are stopped from the fusion encoder to the language encoder. The vision encoder is trained by
+MIM with both the image-text pair data and the image data. M, N and P denote numbers of encoder layers.
+former layers like that of BERT (Devlin et al., 2019), while
+the vision encoder is a stack of Transformer layers like that
+of ViT (Dosovitskiy et al., 2021). The language encoder uses
+a post-layer-norm, while the vision encoder uses a pre-layer-
+norm. The fusion encoder is similar to that of ALBEF (Li
+et al., 2021a) and X-VLM (Zeng et al., 2021), in which
+each layer has an attention sub-layer after a self-attention
+sub-layer. In the self-attention sub-layers, the queries are
+from language and the keys & values are from vision.
+We propose a new method for learning X-FM, also shown in
+Fig 1. Text, image, and image-text pair data are used as input
+to train X-FM. The language encoder is trained by masked
+language modeling (MLM) and image text contrastive learn-
+ing (ITC). The vision encoder is trained by masked image
+modeling (MIM) and ITC. The fusion encoder is trained
+by image text matching (ITM), image-conditioned masked
+language modeling (IMLM), and bounding box prediction
+(BBP). There are two new techniques developed for the
+training.
+Stop Gradient. We stop gradients from the vision-language
+training when learning the language encoder. Specifically,
+when the fusion encoder is trained with image-text pair
+data by ITM, IMLM, and BBP, there are forward flows
+(activations) from the language encoder to the fusion en-
+coder, but there are no backward flows (gradients) from the
+fusion encoder to the language encoder. In this way, the
+language encoder is only trained with text data by MLM
+and with image-text pair data by ITC. The former helps the
+language encoder to learn text representations, and the latter
+helps the language encoder and the vision encoder to make
+alignments between their respective text representations and
+image representations. Meanwhile, the training of the fusion
+encoder is performed separately with the focus of learning
+from image-text pair data.
+Masked Image Modeling. The training of vision encoder
+by MIM is carried out as follows. The image data is first
+masked and then predicted by the vision encoder. The dif-
+ferences between predicted representations and ‘target’ rep-
+resentations at masked positions and [CLS] position are
+then measured with MSE (mean squared error) loss. The
+target representations are obtained from the same image
+data (without masking) by the vision encoder. There are
+no gradients from the target representations in the learning
+of the vision encoder. The vision encoder can create target
+representations because it is also trained with image-text
+pair data. In this way, the vision encoder is trained by both
+the cross-modal objectives (ITC, ITM, BBP, IMLM) with
+image-text pair data and the uni-modal objective (MIM)
+with image data. The representations obtained from the
+vision-language training are highly semantic, which is nec-
+essary for MIM as demonstrated in previous work (Bao
+et al., 2021; Peng et al., 2022; Wei et al., 2022a;b).
+There are three advantages by exploiting the new MIM
+technique. First, it becomes possible to leverage image data
+for learning of the vision encoder, which is relatively easy
+to obtain. Second, it is convenient to conduct MIM with the
+signals from the vision-language training. Note that most
+previous work for MIM makes use of an external image
+tokenizer such as VQ-VAE (Bao et al., 2021; Singh et al.,
+2021), CLIP (Wei et al., 2022b), and VQ-KL (Peng et al.,
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+2022). Third, the learning of the vision encoder and that
+of the fusion encoder are mutually enhanced. Once the
+vision encoder is trained, it is also utilized to train the fusion
+encoder.
+3.2. Pre-training Objectives
+We explain six objectives in learning of X-FM. Here, T
+represents the distribution of text data, I represents the
+distribution of image data, and D represents the distribution
+of image-text pair data.
+Masked Language Modeling (MLM) We perform MLM
+on text data to learn the language encoder of X-FM. Specifi-
+cally we recover the masked tokens in a text by minimizing
+the cross entropy loss below.
+Lmlm = ET ∼T H(⃗y( ¯T), ˆ⃗p( ¯T))
+(1)
+where T denotes a text, ¯T denotes the masked text of T, ˆ⃗p
+denotes the predicted probability vectors of masked tokens
+of ¯T, ⃗y denotes the one-hot vectors representing the original
+tokens of ¯T, and H denotes cross-entropy.
+Image-Text Contrastive Learning (ITC). We use an
+image-text contrastive loss as in CLIP (Radford et al., 2021)
+to learn the alignments between images and texts in ITC.
+Given a batch of images and texts, we calculate the cosine
+similarities between all image-text pairs. For each image,
+there is one text matched and the rest is unmatched. For each
+text, there is one image matched and the rest is unmatched.
+The contrastive loss is defined as follows.
+Litc = 1
+2E(I,T )∼D
+�
+H(⃗yi2t(I), ⃗pi2t(I))
++ H(⃗yt2i(T), ⃗pt2i(T))
+�
+(2)
+where (I, T) denotes an image-text pair, ⃗pi2t(I) denotes the
+in-batch image-to-text similarities, ⃗pt2i(T) denotes the in-
+batch text-to-image similarities, ⃗yi2t(I) denotes the one-hot
+vectors representing the image-to-text matching relations,
+⃗yt2i(T) denotes the one-hot vectors representing the text-to-
+image matching relations, and H denotes cross-entropy.
+Image-Text Matching (ITM). We also learn the align-
+ments between images and texts in ITM, using a loss in-
+dicating whether an image-text pair is matched. For each
+image in a batch there is a matched (positive) text, and we
+sample an unmatched (negative) text in the batch. For each
+text there is a matched (positive) image, and we sample
+an unmatched image in the batch. The loss is defined as
+follows.
+Litm = E(I,T )∼D
+�
+H(pmatch(I, T))
++H(pmatch(˜I, T))
+(3)
++H(pmatch(I, ˜T))
+�
+where (I, T) denotes a positive image-text pair, (˜I, T) and
+(I, ˜T) denote negative image-text pairs, pmatch(I, T) de-
+notes a predicted matching probability of (I, T), and H
+denotes logistic loss.
+Image-conditioned
+Masked
+Language
+Modeling
+(IMLM) We conduct IMLM on image-text pair data to
+learn the fusion encoder.
+Specifically, we recover the
+masked tokens of the text given for an image-text pair by
+minimizing the cross entropy loss below.
+Limlm = E(I,T )∼DH(⃗y( ¯T), ˆ⃗p(I, ¯T))
+(4)
+where (I, T) denotes an image-text pair, ¯T denotes the
+masked text of T, ˆ⃗p(I, ¯T) denotes the predicted probability
+vectors of the masked tokens of ¯T based on I, ⃗y denotes the
+one-hot vectors representing the original tokens of ¯T, and
+H denotes cross-entropy.
+Bounding Box Prediction (BBP) We adopt the BBP in X-
+VLM (Zeng et al., 2021; 2022), which locates the visual
+concept in the image by a bounding box given the text. With
+BBP we learn the alignments between the images and texts
+in multi-granularity. In BBP, two losses are simultaneously
+minimized to measure the differences between the predicted
+bounding box and the ground-truth bounding box. One
+is generalized intersection over union GIoU (Rezatofighi
+et al., 2019) and the other is ℓ1 distance.
+Lbbp = E(I,T )∼D{GIoU(⃗b,ˆ⃗b) + ∥⃗b − ˆ⃗b∥1}
+(5)
+where⃗b = (cx, cy, w, h) denotes the ground truth bounding
+box, ˆ⃗b = ( ˆcx, ˆcy, ˆw, ˆh) denotes the predicted bounding box.
+A bounding box is represented by two coordinates, width,
+and height.
+Masked Image Modeling (MIM) We perform MIM on im-
+age data and image-text pair data to learn the vision encoder.
+Specifically, we recover the masked image patches in an
+image by minimizing the loss below.
+Lmim = E(I,T )∼D||⃗v(¯I) − ˆ⃗v(¯I)||2 + EI∼I||⃗v(¯I) − ˆ⃗v(¯I)||2
+(6)
+where (I, T) and I denote an image-text pair and a single
+image respectively, ¯I denotes the masked image I, ˆ⃗v(¯I) de-
+notes the predicted representations at the masked positions
+and [CLS] of ¯I, and ⃗v(¯I) denotes the target representa-
+tions at the masked positions and [CLS] of ¯I. ||˙||2 is the
+MSE loss. We employ block masking following previous
+work (Bao et al., 2021; Peng et al., 2022). Note that (I, T)
+and I are independently sampled from D and I, and the
+sample sizes are not necessarily equal.
+Finally, the pre-training objective of X-FM is defined as the
+sum of the losses described above.
+L = Lmlm + Litc + Litm + Limlm + Lbbp + Lmim (7)
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+Base-Size Models
+Large-Size Models
+RoBERTa
+BEiTv2
+X2-VLM
+UNIMO-2
+FLAVA
+SimVLM
+OFA
+DaVinci
+Uni-Per.
+OmniVL
+X-FM
+RoBERTa
+BEiTv2
+X2-VLM
+SimVLM
+OFA
+Uni-Per.
+X-FM
+Task
+Eval.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+MNLI
+FT
+87.6
+–
+–
+87.5
+80.3
+83.4
+84.3
+82.3
+81.5
+–
+87.7
+90.2
+–
+–
+–
+84.3
+85.7
+90.4
+CoLA
+FT
+63.6
+–
+–
+62.1
+50.7
+46.7
+52.3
+52.1
+52.2
+–
+65.3
+68.0
+–
+–
+–
+52.3
+57.4
+69.9
+MRPC
+FT
+90.2
+–
+–
+–
+84.2
+79.8
+88.7
+83.1
+–
+–
+91.7
+90.9
+–
+–
+–
+88.7
+–
+92.4
+QQP
+FT
+91.9
+–
+–
+–
+88.7
+90.4
+91.3
+88.2
+–
+–
+91.8
+92.2
+–
+–
+–
+91.3
+–
+92.2
+SST-2
+FT
+94.8
+–
+–
+94.7
+90.9
+90.9
+92.7
+90.5
+90.9
+–
+95.0
+96.4
+–
+–
+–
+92.7
+93.4
+96.7
+QNLI
+FT
+92.8
+–
+–
+–
+87.3
+88.6
+91.1
+87.2
+88.2
+–
+92.9
+94.7
+–
+–
+–
+91.1
+91.9
+94.8
+RTE
+FT
+78.70
+–
+–
+–
+57.8
+63.9
+70.8
+60.7
+75.8
+–
+83.8
+86.6
+–
+–
+–
+70.8
+78.4
+87.4
+STS-B
+FT
+91.2
+–
+–
+91.2
+85.7
+87.2
+–
+86.3
+–
+–
+90.8
+92.4
+–
+–
+–
+–
+–
+92.1
+Language Avg.
+86.4
+–
+–
+–
+78.2
+78.9
+–
+78.8
+–
+–
+87.4
+88.9
+–
+–
+–
+–
+–
+89.5
+ImageNet
+FT
+–
+85.5
+–
+80.8
+–
+–
+82.2
+83.9
+84.5
+–
+85.3
+–
+87.3
+–
+–
+–
+86.4
+86.3
+ImageNet
+LE
+–
+80.1
+–
+–
+75.5
+80.6
+71.4†
+75.9
+–
+–
+81.0
+–
+66.8†
+–
+82.3
+74.7†
+–
+81.0
+Food101
+LE
+–
+88.2†
+–
+–
+88.5
+–
+75.2†
+89.3
+–
+87.4
+88.7
+–
+52.2†
+–
+–
+81.6†
+–
+88.9
+CIFAR10
+LE
+–
+95.3†
+–
+–
+92.9
+–
+86.1†
+93.0
+–
+96.2
+97.2
+–
+63.5†
+–
+–
+91.9†
+–
+97.2
+CIFAR100
+LE
+–
+81.5†
+–
+–
+77.7
+–
+66.7†
+79.0
+–
+83.2
+86.7
+–
+39.7†
+–
+–
+75.6†
+–
+85.1
+Pets
+LE
+–
+93.1†
+–
+–
+84.8
+–
+81.0†
+85.5
+–
+87.1
+90.8
+–
+38.9†
+–
+–
+86.8†
+–
+90.0
+DTD
+LE
+–
+78.4†
+–
+–
+77.3
+–
+70.3†
+77.1
+–
+76.2
+78.4
+–
+44.4†
+–
+–
+74.4†
+–
+79.0
+Flowers102
+LE
+–
+95.7†
+–
+–
+96.4
+–
+86.3†
+96.1
+–
+89.8
+97.1
+–
+66.6†
+–
+–
+92.6†
+–
+95.8
+Vision Avg.
+–
+88.7
+–
+–
+86.3
+–
+79.2
+86.7
+–
+86.7
+89.8
+–
+50.9
+–
+–
+83.8
+–
+89.3
+VQAv2
+FT
+–
+–
+79.2
+76.3
+72.5
+77.9
+78.0
+73.9
+–
+78.3
+79.1
+–
+–
+80.5
+79.3
+80.3
+–
+79.5
+NLVR2
+FT
+–
+–
+86.1
+–
+–
+81.8
+–
+77.9
+–
+–
+86.7
+–
+–
+87.6
+84.8
+–
+–
+87.8
+Flickr30K TR R@1
+ZS
+–
+–
+85.1†
+88.5
+67.7
+–
+–
+–
+82.1
+–
+90.1
+–
+–
+86.8†
+–
+–
+83.6
+89.7
+Flickr30K IR R@1
+ZS
+–
+–
+77.3†
+72.7
+65.2
+–
+–
+–
+72.4
+–
+79.1
+–
+–
+80.5†
+–
+–
+75.9
+79.1
+Flickr30K TR R@1
+FT
+–
+–
+97.4
+92.0
+–
+–
+–
+–
+93.6
+94.9
+97.4
+–
+–
+99.1
+–
+–
+94.1
+97.9
+Flickr30K IR R@1
+FT
+–
+–
+90.0
+80.1
+–
+–
+–
+–
+79.8
+83.4
+88.6
+–
+–
+91.1
+–
+–
+83.7
+89.4
+COCO TR R@1
+ZS
+–
+–
+68.4†
+–
+42.7
+–
+–
+–
+64.6
+–
+73.8
+–
+–
+69.7†
+–
+–
+67.9
+74.4
+COCO IR R@1
+ZS
+–
+–
+55.2†
+–
+38.4
+–
+–
+–
+51.6
+–
+59.4
+–
+–
+58.3†
+–
+–
+55.3
+59.4
+COCO TR R@1
+FT
+–
+–
+80.5
+–
+–
+–
+–
+–
+70.5
+76.8
+81.8
+–
+–
+82.3
+–
+–
+74.7
+82.1
+COCO IR R@1
+FT
+–
+–
+62.7
+–
+–
+–
+–
+–
+52.6
+58.5
+64.7
+–
+–
+65.2
+–
+–
+57.1
+65.4
+Vision-Language Avg.
+–
+–
+78.2
+–
+–
+–
+–
+–
+–
+–
+80.1
+–
+–
+80.1
+–
+–
+–
+80.5
+Table 2: Experimental results on vision, language and vision-language tasks. MNLI results are average of MNLI-m
+and MNLI-mm. MRPC results are average accuracies and F1 scores. Matthews correlation coefficient (MCC) is reported for
+CoLA, and Pearson correlation coefficient (PCC) is reported for STS-B. We report accuracies for all the vision and multi-
+modal tasks. FT is short for fine-tuning, LE for linear evaluation, ZS for zero-shot, TR for text retrieval, and IR for image
+retrieval. Results for RoBERTa are from its corresponding paper (Liu et al., 2019), and they use the mid-training (Phang
+et al., 2018) on MNLI for RTE, MRPC, and STS-B while other models (e.g., BERT, SimVLM, DaVinci, X-FM) do not
+use this trick. Language Avg. is the average score of all the language tasks, while Vision Avg. is the average score of six
+line evaluation tasks except ImageNet. Vision-Language Avg. is the average score of all vision-language tasks. † are our
+reproduced results with the officially released models. Uni-Per. stands for Uni-Perceiver-MoE (Zhu et al., 2022).
+4. Experiments
+4.1. Pre-training Datasets
+We conduct our experiments on several widely used pub-
+lic datasets, which consist of two in-domain datasets,
+COCO (Lin et al., 2014) and Visual Genome (VG) (Kr-
+ishna et al., 2017), and two out-of-domain datasets, SBU
+Captions (Ordonez et al., 2011) and Conceptual Captions
+(CC) (Sharma et al., 2018). Following X-VLM (Zeng et al.,
+2021; 2022), we also include annotations of objects and
+regions from RefCOCO (Yu et al., 2016), Objects365 (Shao
+et al., 2019) and OpenImages (Kuznetsova et al., 2018).
+Since we assume also using uni-modal data, we include
+RoBERTa corpus (Liu et al., 2019), C4 datasets (Raffel
+et al., 2020) and Imagenet21K (Ridnik et al., 2021). All
+pre-training datasets are listed in Table 3.
+4.2. Implementation Details
+Pre-training
+Our model is of base size and large size, and
+the parameters are listed in Table 5. The vision encoder is
+initialized with BEiTv2 (Peng et al., 2022). The language
+encoder is initialized with RoBERTa (Liu et al., 2019). The
+fusion encoder is trained from scratch. X-FM is pre-trained
+at image resolution of 224 × 224 with patch size of 16 × 16.
+Dataset
+# Images
+# Texts
+# Objects
+# Regions
+COCO
+0.11M
+0.55M
+0.45M
+-
+VG
+0.10M
+-
+2.0M
+3.7M
+SBU
+0.86M
+0.86M
+-
+-
+CC-3M
+2.9M
+2.9M
+-
+-
+Objects365
+0.58M
+-
+2.0M
+-
+OpenImages
+1.7M
+-
+4.2M
+-
+C4
+-
+800GB
+-
+-
+RoBERTa Corpus
+-
+160GB
+-
+-
+ImageNet-21k
+14M
+-
+-
+-
+Table 3: Statistics of the pre-training datasets.
+We pre-train X-FMbase for 200K steps with a batch size of
+3072 image-text pairs, 3072 images, and 8192 sentences on
+32 A100 and pre-train X-FMlarge with the same batch for
+160K steps on 64 A100, which takes about six days. The
+learning rate for both models is warmed-up to 1e−4 in the
+first 2500 steps and decayed following a linear schedule. We
+set the maximum number of text tokens to 30 for image-text
+pairs, while that of pure text corpus is set to 128. We apply
+mixed precision for pre-training.
+Fine-tuning
+We choose widely used downstream tasks
+whose details are shown in Appendix B. We report overall
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+Model
+# Params
+MSCOCO (5K test set)
+Flickr30K (1K test set)
+MSCOCO (5K test set)
+Flickr30K (1K test set)
+TR-Fine-Tune
+IR-Fine-Tune
+TR-Fine-Tune
+IR-Fine-Tune
+TR-Zero-Shot
+IR-Zero-Shot
+TR-Zero-Shot
+IR-Zero-Shot
+R@1/R@5/R@10
+R@1/R@5/R@10
+R@1/R@5/R@10
+R@1/R@5/R@10
+R@1/R@5/R@10
+R@1/R@5/R@10
+R@1/R@5/R@10
+R@1/R@5/R@10
+ALBEF
+210M
+73.1/91.4/96.0
+56.8/81.5/89.2
+94.3/99.4/99.8
+82.8/96.7/98.4
+–
+–
+90.5/98.8/99.7
+76.8/93.7/96.7
+VLMobase
+175M
+74.8/93.1/96.9
+57.2/82.6/89.8
+92.3/99.4/99.9
+79.3/95.7/97.8
+–
+–
+–
+–
+VL-BEiT
+175M
+79.5/–/–
+61.5/–/–
+95.8/–/–
+83.9/–/–
+–
+–
+–
+–
+OmniVL
+288M
+76.8/93.6/97.3
+58.5/82.6/89.5
+94.9/9.6/99.9
+83.4/97.0/98.6
+–
+–
+–
+–
+X-VLM
+216M
+80.4/95.5/98.2
+63.1/85.7/91.6
+96.8/99.8/100
+86.1/97.4/98.7
+70.8/92.1/96.5
+55.6/82.7/90.0
+85.3/97.8/99.6
+71.9/93.3/96.4
+X2-VLMbase
+255M
+80.5/95.5/97.8
+62.7/84.7/90.7
+97.4/99.9/100
+90.0/98.6/99.3
+68.4†/92.5†/96.8†
+55.2†/82.2†/89.3†
+85.1†/99.2†/100.0†
+77.3†/95.3†/97.6†
+X-FMbase
+284M
+81.8/96.0/98.3
+64.7/86.1/91.6
+97.4/100/100
+88.6/97.9/98.9
+73.8/93.9/97.2
+59.4/83.6/90.0
+90.1/99.2/99.9
+79.1/95.2/97.3
+VLMolarge
+562M
+78.2/94.4/97.4
+60.6/84.4/91.0
+95.3/99.9/100
+84.5/97.3/98.6
+–
+–
+–
+–
+X2-VLMlarge
+593M
+82.3/96.2/98.3
+65.2/86.4/91.9
+99.1/100/100
+91.1/98.6/99.4
+69.7†/93.0†/97.2†
+58.3†/83.8†/90.5†
+86.8†/98.9†/99.9†
+80.5†/96.4†/98.3†
+X-FMlarge
+807M
+82.1/96.2/98.2
+65.4/86.6/91.9
+97.9/100/100
+89.4/98.2/99.1
+74.4/94.1/97.3
+59.4/84.4/90.7
+89.7/99.1/100
+79.1/95.4/97.9
+Super-Large Models or Super-Large Datasets
+CLIP
+490M
+–
+–
+88.7/98.0/99.2
+76.7/93.6/96.4
+58.4/81.5/88.1
+37.8/62.4/72.2
+88.0/98.7/99.4
+68.7/90.6/95.2
+ALIGN
+490M
+77.0/93.5/96.9
+59.9/83.3/89.8
+95.3/99.8/100
+84.9/97.4/98.6
+58.6/83.0/89.7
+45.6/69.8/78.6
+88.6/98.7/99.7
+75.7/93.8/96.8
+Florence
+893M
+81.8/95.2/–
+63.2/85.7/–
+97.2/99.9/–
+87.9/98.1/–
+64.7/85.9/–
+47.2/71.4/–
+90.9/99.1/–
+76.7/93.6/–
+CoCa
+2.1B
+–
+–
+–
+–
+66.3/86.2/91.8
+51.2/74.2/82.0
+92.5/99.5/99.9
+80.4/95.7/97.7
+BEiT-3
+1.9B
+84.8/96.5/98.3
+67.2/87.7/92.8
+98.0/100/100
+90.3/98.7/99.5
+–
+–
+94.9/99.9/100.0
+81.5/95.6/97.8
+X2-VLMlarge
+593M
+84.4/96.5/98.5
+67.7/87.5/92.5
+98.8/100/100
+91.8/98.6/99.5
+–
+–
+–
+–
+Table 4: Results of text-retrieval (TR) and image-retrieval (IR) on COCO and Flickr30K. † denotes our reproduced results
+with the officially released models. Giant models with over 1B parameters (e.g., BEiT-3) and models are pre-trained with
+over 400M data (e.g., CLIP and X2-VLMlarge) are in grey since they are not directly comparable with other models.
+Model
+Param
+Hidden
+Layers
+Vision
+Text
+Fusion
+X-FMbase
+284
+768
+12
+12
+12
+X-FMlarge
+807
+1024
+24
+24
+12
+Table 5: Size variants of X-FM. All modules consist of
+transformer layers. Param indicates the parameter number
+of transformer layers.
+performance on eight language tasks from GLUE (Wang
+et al., 2019), eight vision tasks following OmniVL (Wang
+et al., 2022a), four multi-modal tasks, which are text-
+image retrieval on MSCOCO and Flickr, visual question
+answering (VQA (Goyal et al., 2017)) and visual reason-
+ing (NLVR2 (Suhr et al., 2019b)). For image-text retrieval
+task, we report both zero-shot results and fine-tuned results.
+For the ImageNet classification task, we report both linear
+evaluation results and fine-tuning results. The other vision
+tasks are evaluated in the linear evaluation setting. All the
+other tasks are evaluated in the fine-tuning setting. Because
+the image resolution differs between pre-training and fine-
+tuning, the position parameters are adapted using linear
+interpolation. For all downstream tasks, we apply random
+resize crops and horizontal flips augmentation for the im-
+ages during training. More details of network architectures
+and hyper-parameters setups are given in Appendix C.
+4.3. Comparison with SOTA Foundation Models
+We extensively compare the performance of X-FM with
+state-of-the-art foundation models on vision, language, and
+multi-modal tasks. We first compare our model with general
+foundation models, including UNIMO-v2 (Li et al., 2021c),
+FLAVA (Singh et al., 2021), SimVLM (Wang et al., 2021c),
+OFA (Wang et al., 2022b), DaVinci (Diao et al., 2022), Om-
+niVL (Wang et al., 2022a), and Uni-Perceiver-MoE (Zhu
+et al., 2022). We also include comparisons with SOTA
+foundation models specifically designed for language, vi-
+sion, or vision-language tasks, RoBERTa (Liu et al., 2019),
+BEiTv2 (Peng et al., 2022), and X2-VLM (Zeng et al., 2022).
+There are several observations in Table 2. First, X-FMbase
+(column 11) outperforms all the previous general foundation
+models (column 4-10) across almost all tasks by a large mar-
+gin, becoming a new and stronger general foundation model.
+Compared to the previous general foundation models, X-
+FMbase improves at least 3.2% and the most even to 9.7%
+on the average of all the reported numbers. Second, we com-
+pare X-FM with state-of-the-art foundation models specif-
+ically designed for language, vision, and vision-language
+tasks, RoBERTa, BEiTv2 and X2-VLM. We observe that
+X-FM is also better than or comparable with the foundation
+models at both base and large scale (column 1,2,3 vs 11 and
+12,13,14 vs 18).
+4.4. Comparison with SOTA Vision-Language Models
+In addition to general foundation models, we also compare
+X-FM with state-of-the-art vision-language models. The re-
+sults are shown in Table 4 and Table 7. X-FM demonstrates
+its superiority on MSCOCO retrieval and NLVR2, while
+achieving competitive performance on Flickr retrieval and
+VQA. Note that X-FMbase outperforms CLIP, ALIGN and
+Florence on image-text retrieval tasks with fewer parame-
+ters and much less training data. Compared to the recently
+released SOTA vision-language model, X2-VLM, X-FM is
+much better on image-text retrieval tasks at the zero-shot
+setting.
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+X-FMbase
+RoBERTa†
+S-MLM
+S-ITM
+wostop
+BEiTv2†
+woMIM
+wBEiTv2 Tokenizer
+X2-VLM†
+Multi-task
+ALL
+Task
+Eval.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+MNLI
+FT
+87.7
+87.4
+87.3
+87.7
+–
+–
+–
+–
+87.4
+87.6
+CoLA
+FT
+63.2
+61.6
+63.6
+64.2
+–
+–
+–
+–
+62.2
+65.2
+MRPC
+FT
+90.7
+92.2
+91.1
+90.7
+–
+–
+–
+–
+92.0
+92.5
+QQP
+FT
+91.5
+91.6
+91.6
+91.6
+–
+–
+–
+–
+91.6
+91.6
+SST-2
+FT
+95.0
+95.1
+94.2
+94.6
+–
+–
+–
+–
+94.4
+95.3
+QNLI
+FT
+93.1
+93.0
+93.2
+92.5
+–
+–
+–
+–
+92.8
+92.9
+RTE
+FT
+80.9
+79.1
+81.6
+81.2
+–
+–
+–
+–
+79.8
+81.9
+STS-B
+FT
+90.9
+90.7
+90.7
+90.4
+–
+–
+–
+–
+90.1
+90.8
+Language Avg.
+86.6
+86.4
+86.7
+86.6
+–
+–
+–
+–
+86.3
+87.2
+ImageNet
+FT
+–
+–
+–
+–
+85.5
+84.8
+85.0
+–
+85.0
+85.3
+ImageNet
+LE
+–
+–
+–
+–
+80.5
+79.1
+79.4
+–
+79.3
+81.1
+Food101
+LE
+–
+–
+–
+–
+88.2
+86.9
+87.2
+–
+86.9
+88.7
+CIFAR10
+LE
+–
+–
+–
+–
+95.3
+96.6
+96.5
+–
+96.6
+97.5
+CIFAR100
+LE
+–
+–
+–
+–
+81.5
+83.3
+83.9
+–
+84.1
+86.9
+Pets
+LE
+–
+–
+–
+–
+93.1
+88.1
+88.5
+–
+88.2
+90.7
+DTD
+LE
+–
+–
+–
+–
+78.4
+77.7
+76.9
+–
+78.0
+78.7
+Flowers102
+LE
+–
+–
+–
+–
+95.7
+94.1
+94.5
+–
+94.2
+97.1
+Vision Avg.
+–
+–
+–
+–
+87.3
+86.3
+86.5
+–
+86.5
+88.2
+VQAv2
+FT
+–
+78.8
+78.5
+78.7
+–
+78.3
+78.2
+78.0
+78.2
+78.6
+NLVR2
+FT
+–
+86.3
+86.0
+86.4
+–
+85.9
+85.5
+86.2
+86.1
+86.7
+Flickr30K TR R@1
+ZS
+–
+88.3
+87.2
+87.1
+–
+87.1
+87.2
+87.7
+85.0
+89.3
+Flickr30K IR R@1
+ZS
+–
+76.6
+74.9
+75.8
+–
+76.1
+75.3
+75.1
+75.6
+77.4
+Flickr30K TR R@1
+FT
+–
+97.5
+97.0
+97.2
+–
+96.4
+96.7
+97.0
+97.0
+97.7
+Flickr30K IR R@1
+FT
+–
+87.4
+86.9
+87.3
+–
+86.2
+86.6
+86.2
+86.4
+87.4
+COCO TR R@1
+ZS
+–
+72.0
+72.1
+70.5
+–
+73.0
+72.1
+73.2
+69.9
+72.8
+COCO IR R@1
+ZS
+–
+58.4
+57.1
+57.7
+–
+58.2
+57.7
+57.7
+56.5
+59.0
+COCO TR R@1
+FT
+–
+81.2
+80.2
+80.9
+–
+80.6
+80.1
+80.3
+80.0
+81.2
+COCO IR R@1
+FT
+–
+64.2
+63.4
+63.6
+–
+63.7
+63.0
+63.1
+63.0
+64.0
+Vision-Language Avg.
+–
+79.1
+78.3
+78.5
+–
+78.6
+78.2
+78.5
+77.8
+79.4
+Table 6: Ablation studies on vision, language, and vision-language tasks. We use the same settings as Table 2. “ALL” for
+X-FMbase is trained with the same data under the same settings for pre-training and fine-tuning compared to all the variants.
+Language Avg. is the average of all language tasks, while Vision Avg. is the average of all vision tasks. Vision-Language
+Avg. is the average of all vision-language tasks. Note that performance of “ALL” is slightly different from X-FMbase in
+Table 2, because we use less training steps (160k) for ablation to save the computational resources.
+4.5. Ablation Study
+To verify the contributions of different modules in our frame-
+work, we ablate them and evaluate the performance of X-
+FM on all downstream tasks. The results are shown in
+Table 6. We first explain several abbreviations in the table.
+S-MLM means that we only separate the language represen-
+tations in the learning IMLM task, while S-ITM means that
+language representations for computing ITM and BBP are
+separated. wostop indicates without stopping the gradients
+of all language representations. woMIM means that we do
+not learn by MIM, while wBEiTv2 tokenizer means that
+we learn by MIM with the image tokenizer used in BEiTv2.
+Multi-task is a variation that uses straightforward multi-task
+learning to optimize the three encoders in X-FM. To make
+a fair comparison, we also train RoBERTa, BEiTv2 and
+X2-VLM with the same data noted as RoBERTa†, BEiTv2†
+and X2-VLM†. Note that we also increase the fusion layers
+in X2-VLM† to make the parameter sizes comparable to
+our models. RoBERTa†, BEiTv2† and X2-VLM† all have
+slightly better results on average than the official ones. From
+the results, we have the following observations.
+First, both designs (stop gradient and masked image mod-
+eling) bring improvements, and the combination can make
+further improvements on all three downstream tasks (col-
+umn 10 vs. others). Second, without separated language
+representations, models always perform worse on language
+understanding tasks (column 10 vs. 2,3,4). Besides, the sep-
+arate language representations in the IMLM task on image-
+text data are helpful for multi-modal tasks (column 2 vs. 4).
+As we point out in section 1, the fusion encoder can learn
+better cross-modal feature alignments by IMLM task from
+image-text pairs instead of utilizing text tokens. Although S-
+ITM shows slight side effects (column 4 vs. 3), stopping the
+gradients of language representation in the fusion encoder
+is necessary to simultaneously achieve strong language un-
+derstanding and vision-language understanding capability.
+Third, the MIM task is useful for vision-language and vi-
+sion learning (column 10 vs. 6). Meanwhile, the targets
+in our MIM task are better than the BEiTv2 tokenizer (col-
+umn 10 vs. 7). Four, X-FM is much better than a naive
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+Method
+# Params
+VQA
+NLVR2
+test-dev
+test-std
+dev
+test-P
+ALBEF
+210M
+74.5
+74.7
+80.2
+80.5
+VLMobase
+175M
+76.6
+76.9
+82.8
+83.3
+METER
+341M
+77.7
+77.6
+82.3
+83.1
+VL-BEiT
+175M
+77.5
+77.8
+81.9
+82.7
+BLIPbase
+240M
+78.2
+78.2
+82.5
+83.1
+X-VLM
+216M
+78.1
+78.1
+84.2
+84.2
+OFAbase
+182M
+78.0
+78.1
+-
+-
+OmniVL
+288M
+78.3
+78.4
+-
+-
+X2-VLMbase
+255M
+79.2
+79.3
+85.9
+86.1
+X-FMbase
+284M
+79.1
+79.2
+86.3
+86.5
+VLMolarge
+562M
+79.9
+80.0
+85.6
+86.9
+OFAlarge
+472M
+80.3
+80.5
+-
+-
+X2-VLMlarge
+593M
+80.5
+80.5
+87.2
+87.6
+X-FMlarge
+807M
+79.5
+79.6
+86.2
+87.8
+Super-Large Models or Super-Large Datasets
+SimVLMbase
+273M
+77.9
+78.1
+81.7
+X2-VLMbase
+255M
+80.4
+80.2
+86.2
+87.0
+SimVLMlarge
+783M
+79.3
+79.6
+84.1
+84.8
+X2-VLMlarge
+593M
+81.9
+81.8
+88.7
+89.4
+Florence
+893M
+80.2
+80.3
+–
+–
+CoCa
+2.1B
+82.3
+82.3
+86.1
+87.0
+BEiT-3
+1.9B
+84.2
+84.0
+91.5
+92.6
+Table 7: Results on VQA and visual reasoning. Giant mod-
+els with over 1B parameters (e.g., CoCa and BEiT-3) or
+models are pre-trained with over 400M data (e.g., SimVLM
+and X2-VLMlarge) are in grey because they are not directly
+comparable with other models.
+multi-task learning strategy for a foundation model (column
+10 vs. 8). Compared with the straightforward multi-task
+strategy, X-FMbase improves an average of 0.9%, 1.7%
+and 1.6% on language, vision, and vision-language tasks,
+respectively. Five, X-FM is also slightly better than foun-
+dation models specifically designed for language, vision,
+and vision-language tasks with the same training corpus
+(column 10 vs. 1,5,8).
+5. Conclusion and Limitation
+5.1. Conclusion
+In this work, we address the problem of how to build a
+general foundation model that can perform the best for all
+the understanding tasks of language, vision, and vision-
+language. We propose a general foundation model with two
+new and effective training techniques, X-FM, to learn rich
+language, vision and vision-language representations at the
+same time. Experimental results explicitly imply that X-FM
+outperforms other general foundation models by a large
+margin. Moreover, X-FM can even be better or comparable
+compared with the SOTA foundation models specifically
+designed for language, vision, or vision-language under-
+standing tasks.
+5.2. Limitation
+Like most existing work on foundation models, the entire
+project consumed over 5 A100 GPU years on a computing
+cluster with high electricity costs, although we only tested
+base and large models. There is still potential for efficiency
+improvement through sparse attention (Zaheer et al., 2020)
+or the lottery ticket hypothesis (Frankle & Carbin, 2018).
+We will explore the techniques to improve the training effi-
+ciency and reduce the carbon footprint so that we can adhere
+to the proposals on “green” deep learning (Schwartz et al.,
+2020; Xu et al., 2021).
+Due to considerations of fair comparisons and computa-
+tional resources, we did not try super-large models which
+use at least 1.9B or more parameters like BEITv3 (Wang
+et al., 2022d), CoCa (Yu et al., 2022) and PaLI (Chen et al.,
+2022). However, scalability is also an important factor for
+foundation models. We leave the investigations to future
+work.
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+model with multi-grained tokenization. arXiv preprint
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+and Dai, J. Uni-perceiver-moe: Learning sparse gen-
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+arXiv:2206.04674, 2022.
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+tasks. arXiv preprint arXiv:2112.01522, 2021.
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+A. Comparison of Recent Foundation Models
+Table 8 shows an extensive comparison of recent foundation
+models and X-FM on multiple axes. Previous work either (i)
+perform best on uni-modal tasks (Liu et al., 2019; Peng et al.,
+2022) or vision-language tasks (Zeng et al., 2021; 2022); (2)
+target a specific uni-modal domain along with part of vision-
+and-language tasks (Wang et al., 2021a; Radford et al., 2021;
+Jia et al., 2021; Wang et al., 2021c; Yu et al., 2022; Wang
+et al., 2022b; Diao et al., 2022); or (3) target all domains
+but cannot perform best on all the tasks (Li et al., 2021c;
+Singh et al., 2021; Zhu et al., 2022). Our model, X-FM, is
+a general foundation model that can perform the best for
+all the understanding tasks of language, vision, and vision
+language.
+B. Details of Downstream Tasks
+Language Understanding.
+We conduct experiments on GLUE benchmark including
+MNLI (Williams et al., 2018), CoLA (Warstadt et al., 2019),
+MRPC (Dolan & Brockett, 2005), QQP (Iyer et al., 2017),
+SST-2 (Socher et al., 2013), QNLI (Rajpurkar et al., 2016),
+RTE (Dagan et al., 2005; Haim et al., 2006; Giampiccolo
+et al., 2007; Bentivogli et al., 2009), and STS-B (Agirre
+et al., 2007). We follow the practice of BERT (Devlin et al.,
+2019; Liu et al., 2019) and feed the input into the language
+encoder, and the hidden state of the [CLS] is fed into a
+new multi-class linear classifier or regression head.
+Vision Understanding.
+We conduct vision experiments on both fine-tuning and lin-
+ear evaluation (linear eval). The linear evaluation follows
+a common practice (Caron et al., 2021; He et al., 2020;
+Singh et al., 2021) in self-supervised learning to evaluate
+the representation quality, where the pre-trained backbone
+model is frozen, and an MLP head is appended on top of it.
+We choose 7 popular datasets following OmnVL (Wang
+et al., 2022a):
+ImageNet (Russakovsky et al., 2015),
+Food101 (Bossard et al., 2014), CIFAR10 (Krizhevsky et al.,
+2009), CIFAR100 (Krizhevsky et al., 2009), DTD (Cimpoi
+et al., 2014), Pets (Parkhi et al., 2012) and Flowers102 (Nils-
+back & Zisserman, 2008).
+Vision-Language Understanding.
+Image-Text Retrieval We evaluate X-FM on both
+MSCOCO and Flickr30K datasets. We adopt the widely
+used Karpathy split (Karpathy & Li, 2015) for both datasets.
+Following the previous work (Li et al., 2021a; Zeng et al.,
+2021; 2022), we first encode images and texts separately
+and calculate s(I, T) to obtain the top-k candidates, and
+then use the fusion encoder to re-rank the candidates.
+Visual Question Answering The task requires the model
+to predict an answer given an image and a question. We
+evaluate X-FM on the VQA v2.0 dataset (Goyal et al., 2017).
+Following the previous work (Zeng et al., 2021), we use
+a Transformer decoder to generate answers based on the
+outputs of the fusion module. The decoder network shares
+the same network architecture with the fusion encoder. Note
+that we use an image resolution of 768*768 for the final
+result of X-FMbase, and use an image resolution of 480*480
+for X-FMlarge and X-FMbase in ablation studies for efficient
+fine-tuning.
+Visual Reasoning We evaluate X-FM on a widely used
+benchmark NLVR2 (Suhr et al., 2019a). The task allows the
+model to determine whether a text describes the relations
+between two images. Following previous work (Wang et al.,
+2021a; Bao et al., 2022), we formulate the triplet input into
+two image-text pairs, each containing the text description
+and an image. We then concatenate the final output [CLS]
+features of the fusion module of the two pairs to predict the
+label.
+C. Details of hyper parameters
+Pre-training
+X-FMbase is implemented with a 12-layer
+language encoder, a 12-layer vision encoder, and a 12-layer
+fusion encoder, 768 dimensions for hidden states, 3072 for
+intermediate size, and 128 for maximum input length. X-
+FMlarge is implemented with a 24-layer language encoder, a
+24-layer vision encoder, and a 12-layer fusion encoder, 1024
+dimensions for hidden states, 4096 for intermediate size, and
+128 for maximum input length. We initialize the language
+encoder with RoBERTa and the vision encoder with BEiTv2.
+The weight decay is set to 0.01 with β1 = 0.9, β2 = 0.98.
+The learning rate is 1e-4 with a warm-up period for the first
+2500 steps and then linearly decayed to 0. In each batch,
+there are 3072 image-text pairs, 3072 images, and 8192 text-
+only sentences. We use center-crop to resize each image
+to the size of 224×224. The default settings are shown in
+Table 9.
+Fine-tuning
+The learning rate is ∈ {1e-5, 2e-5, 5e-5} and
+our model is optimized by AdamW. Because the image
+resolution differs between pre-training and fine-tuning, the
+position parameters are adapted using linear interpolation.
+For all downstream tasks, we apply random resize crops
+and horizontal flips augmentation during training. The de-
+fault settings for text classification, image classification and
+vision-language understanding are shown in Tables 10, 11,
+12 and 13, respectively. Note that the resolution for VQA is
+different as described in Section B.
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+Methods
+Multimodal data
+Pretraining Objectives
+Fusion Arch.
+Target Modalities
+public
+dataset(s)
+size
+Contr.
+ITM
+BBP
+(M/P)LM
+Unimodal
+ST
+CT
+MT
+V
+CV&L
+MV&L
+L
+RoBERTa (Liu et al., 2019)
+–
+–
+–
+–
+–
+–
+–
+MLM
+–
+–
+–
+–
+–
+–
+�
+BEiTv2 (Peng et al., 2022)
+–
+–
+–
+–
+–
+–
+–
+MIM
+–
+–
+–
+�
+–
+–
+–
+X-VLM (Zeng et al., 2021; 2022)
+�
+Combination
+5M
+�
+�
+�
+MLM
+–
+–
+�
+–
+–
+�
+�
+–
+VLMo (Wang et al., 2021a)
+�
+Combination
+5M
+�
+�
+–
+MLM
+MLM+MIM
+–
+–
+�
+–
+�
+�
+–
+CLIP (Radford et al., 2021)
+�
+WebImageText
+400M
+�
+–
+–
+–
+–
+–
+–
+–
+�
+�
+–
+–
+ALIGN (Jia et al., 2021)
+�
+JFT
+1.8B
+�
+–
+–
+–
+–
+–
+–
+–
+�
+�
+–
+–
+SimVLM (Wang et al., 2021c)
+�
+JFT
+1.8B
+–
+–
+–
+PrefixLM
+PrefixLM
+�
+–
+–
+∗
+–
+�
+�
+CoCa (Yu et al., 2022)
+�
+JFT
+4.8B
+�
+–
+–
+LM
+–
+�
+–
+–
+�
+�
+�
+–
+UNIMO-2 (Li et al., 2021c)
+�
+Combination
+5M
+–
+�
+–
+MLM
+VCL
+�
+–
+–
+�
+�
+�
+�
+OFA (Wang et al., 2022b)
+�
+Combination
+15M
+–
+–
+–
+LM
+LM
+�
+–
+–
+∗
+–
+�
+�
+DaVinci (Diao et al., 2022)
+�
+Combination
+46M
+–
+–
+–
+PrefixLM + PrefixIM
+PrefixLM
+�
+–
+–
+�
+–
+�
+�
+FLAVA (Singh et al., 2021)
+�
+Combination
+70M
+�
+�
+–
+MLM
+MLM+MIM
+�
+–
+–
+�
+�
+�
+�
+Uni-Perceiver-MoE (Zhu et al., 2022)
+�
+Combination
+116M
+–
+�
+–
+LM+MLM
+LM+MLM+Classify.
+�
+–
+–
+�
+�
+�
+�
+X-FM
+�
+Combination
+5M
+�
+�
+�
+MLM+MIM
+MLM+MIM
+–
+�
+–
+�
+�
+�
+�
+Super-Large Models
+Flamingo (Alayrac et al., 2022)
+�
+Combination
+2.2B
+–
+–
+–
+LM
+–
+�
+–
+–
+–
+�
+�
+–
+BEiT-v3 (Wang et al., 2022d)
+�
+Combination
+21M
+–
+–
+–
+MLM
+MLM+MIM
+–
+–
+�
+∗
+�
+�
+–
+PaLI (Chen et al., 2022)
+�
+WebImageText
+41B
+–
+–
+–
+LM
+–
+�
+–
+–
+�
+�
+�
+�
+Table 8: Comparison of recent foundation models in different modalities. Contr. indicates contrastive learning. ITM is
+short for image-text matching. BBP represents boundary box prediction. (M/P)LM means image-conditioned (masked/prefix)
+language modeling. V, CV&L, MV&L and L stand for vision tasks, cross-modal retrieval tasks, multi-modal fusion tasks
+and language tasks respectively. ST, CT and MT are abbreviations for single Transformer, cross-attention Transformer and
+multiway Transformer. VCL stands for visual contrastive learning. ∗ means the modality is partially targeted (SimVLM
+and OFA include ImageNet.). Giant models with over 1B parameters (e.g. BEiT-3) are in grey since they are not directly
+comparable with other models.
+config
+value
+optimizer
+AdamW
+learning rate
+1e-4
+weight decay
+0.01
+optimizer momentum
+β1, β2=0.9, 0.999
+language batch size
+8192
+vision batch size
+3072
+vision-language batch size
+3072
+learning rate schedule
+linear decay
+warmup steps
+2500
+training steps
+200k
+augmentation
+RandomResizedCrop
+image res
+224*224
+patch size
+16
+text length for MLM
+128
+text length for IMLM
+30
+Table 9: Pre-training setting.
+config
+value
+optimizer
+AdamW
+learning rate
+{1e-5, 2e-5, 5e-5}
+weight decay
+0.0
+optimizer momentum
+β1, β2=0.9, 0.999
+batch size
+{16, 32, 64}
+learning rate schedule
+linear decay
+warmup ratio
+0.0
+training epochs
+{5, 10, 20}
+Table 10: Text classification: GLUE setting.
+config
+value
+optimizer
+AdamW
+learning rate
+[2e-5, 4e-5]
+weight decay
+0.01
+optimizer momentum
+β1, β2=0.9, 0.999
+batch size
+[256, 2048]
+learning rate schedule
+linear decay
+warmup rate
+0.1
+training epochs
+100
+augmentation
+RandomResizedCrop
+image res
+224*224
+patch size
+16
+Table 11: Image classification: Linear probing setting.
+
+Toward Building General Foundation Models for Language, Vision, and Vision-Language Understanding Tasks
+config
+value
+optimizer
+AdamW
+learning rate
+4e-5
+minimal learning rate
+1e-7
+weight decay
+0.01
+optimizer momentum
+β1, β2=0.9, 0.999
+batch size
+1024
+learning rate schedule
+linear decay
+warmup rate
+0.1
+training epochs
+100
+augmentation
+RandomResizedCrop
+image res
+224*224
+patch size
+16
+label smoothing
+0.1
+mixup prob.
+1.0
+cutmix prob.
+1.0
+Table 12: ImageNet classification: Fine-tuning setting.
+config
+value
+optimizer
+AdamW
+learning rate
+{1e-5, 2e-5, 5e-5}
+weight decay
+0.01
+optimizer momentum
+β1, β2=0.9, 0.999
+batch size
+{64, 192, 512}
+learning rate schedule
+linear decay
+warmup rate
+0.1
+training epochs
+{10, 15, 20}
+augmentation
+RandomResizedCrop
+image res
+384*384
+patch size
+16
+Table 13: Vision-Language understanding: fine-tuning set-
+ting.
+

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+arXiv:2301.00990v1  [math.NA]  3 Jan 2023
+The energy method for high-order invariants in shallow water wave equations
+Qifeng Zhanga, Tong Yana, Guang-hua Gaob
+aDepartment of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018, China
+bDepartment of Mathematics, Nanjing University of Posts and Telecommunications, Nanjing, 210096, China
+Abstract
+Third order dispersive evolution equations are widely adopted to model one-dimensional long waves and have
+extensive applications in fluid mechanics, plasma physics and nonlinear optics. Among them are the KdV equation,
+the Camassa–Holm equation and the Degasperis–Procesi equation. They share many common features such as
+complete integrability, Lax pairs and bi-Hamiltonian structure. In this paper we revisit high-order invariants
+for these three types of shallow water wave equations by the energy method in combination of a skew-adjoint
+operator (1 − ∂xx)−1. Several applications to seek high-order invariants of the Benjamin-Bona-Mahony equation,
+the regularized long wave equation and the Rosenau equation are also presented.
+Keywords: Energy method; High-order invariant; Shallow water wave equation
+1. Introduction
+A family of third order dispersive evolution equations of the form
+ut − α2uxxt + γuxxx + c0ux = (c1u2 + c2u2
+x + c3uuxx)x,
+x ∈ R, t > 0
+(1.1)
+frequently appeared in the simulation of the shallow water waves, see e.g., [1], where α, γ and ci (i = 0, 1, 2, 3) are
+real constants; u denotes a horizontal velocity field with the independent spatial variable x and temporal variable t.
+A typical such equation (1.1) with α2 = c0 = c2 = c3 = 0, c1 = 2, γ = −2 is the KdV equation
+ut − 4uux − 2uxxx = 0,
+x ∈ R, t > 0,
+(1.2)
+which describes the unidirectional propagation of waves at the free surface of shallow water under the influence
+of gravity. The first four invariants of (1.2) are respectively as (see e.g., [2], although there is a minor typo in the
+coefficient of the fourth invariant, it does not affect the reading of this classic review)
+M1 =
+�
+R
+udx,
+M2 =
+�
+R
+u2dx,
+M3 =
+�
+R
+�
+u2
+x − 2
+3u3�
+dx,
+M4 =
+�
+R
+�
+u2
+xx − 10
+3 uu2
+x + 5
+9u4�
+dx.
+Taking α2 = c3 = 1, γ = c0 = 0, c1 = − 3
+2, c2 =
+1
+2, we have another example called the Camassa–Holm
+equation [3]
+ut − uxxt + 3uux = 2uxuxx + uuxxx,
+x ∈ R, t > 0,
+(1.3)
+which models the unidirectional propagation of shallow water waves over a flat bottom. The first three invariants
+are listed as follows
+E1 =
+�
+R
+(u − uxx)dx,
+E2 = 1
+2
+�
+R
+(u2 + u2
+x)dx,
+E3 = 1
+2
+�
+R
+u(u2 + u2
+x)dx.
+The third example by assigning α2 = c2 = c3 = 1, γ = c0 = 0, c1 = −2 is called the Degasperis–Procesi
+equation
+ut − uxxt + 4uux = 3uxuxx + uuxxx,
+x ∈ R, t > 0,
+(1.4)
+∗E-mail address: [email protected] (Q. Zhang), [email protected] (Tong Yan), [email protected] (G. Gao)
+Preprint submitted to Elsevier
+January 4, 2023
+
+which can be regarded as a model for nonlinear shallow water dynamics [4]. The frequently discussed invariants
+are
+H1 =
+�
+R
+(u − uxx)dx,
+H2 =
+�
+R
+(u − uxx)vdx,
+H3 =
+�
+R
+u3dx,
+where 4v − vxx = u.
+Up to now, there have been thousands of papers focusing on the theoretical and numerical studies on these three
+equations. It is worth mentioning that the invariant-preserving property is a key index of the success for numerical
+methods. However, high-order invariants are usually difficult to preserve numerically. Liu et al. also pointed out “it
+appears a rather difficult task to preserve all three conservation laws” in [5]. In this work, higher-order invariants of
+these equations will be re-derived in view of the energy method, which may be possible to provide some thoughts
+for invariant-preserving numerical methods. Actually, the energy method originated from conservation laws in
+physics was first proposed in 1928 by Courant, Friedrichs and Lewy [6]. From then on, it has been widely applied
+to the mathematical and numerical analysis of nonlinear evolution equations. We trust the readers with [7] instead
+of a long list of references to relevant works.
+The rest of the paper is arranged as follows.
+In Section 2, combining the energy method and a skew-
+adjoint operator, we show the high-order invariants for the KdV equation, the Camassa–Holm equation and the
+Degasperis–Procesi equation, respectively. Then we list several applications for seeking some high-order invariants
+of other types of the shallow water wave equations in Section 3.
+2. Main results
+In what follows, we directly show that Mi (i = 1, 2, 3, 4), Ei (i = 1, 2, 3) and Hi (i = 1, 2, 3) are invariants of
+(1.2), (1.3) and (1.4) subjected to the periodic boundary conditions based on the energy method, respectively.
+2.1. Invariants of the KdV equation
+Proof: (I) Multiplying by 1, u and (u2 + uxx), respectively, with (1.2), we have Mi (i = 1, 2, 3). In what follows, we
+show the fourth invariant M4 of the KdV equation by the energy method.
+Multiplying both sides of (1.2) by 2uxxxx + 10
+3 u2
+x + 20
+3 uuxx + 20
+9 u3 and integrating the result, we have
+0 =
+�
+R
+�
+2uxxxx + 10
+3 u2
+x + 20
+3 uuxx + 20
+9 u3�
+· utdx
+−
+�
+R
+�
+2uxxxx + 10
+3 u2
+x + 20
+3 uuxx + 20
+9 u3�
+· (4uux + 2uxxx)dx
+=
+�
+R
+�
+2uxxuxxt − 10
+3 (utu2
+x + 2uuxuxt) + 20
+9 u3ut
+�
+dx
+−
+�
+R
+�
+2uxxxx + 10
+3 u2
+x + 20
+3 uuxx + 20
+9 u3�
+· (4uux + 2uxxx)dx
+= d
+dt M4 − 8
+�
+R
+uuxuxxxxdx − 40
+3
+�
+R
+uu3
+xdx − 80
+3
+�
+R
+u2uxuxxdx − 80
+9
+�
+R
+u4uxdx
+− 4
+�
+R
+uxxxuxxxxdx − 20
+3
+�
+R
+uxxxu2
+xdx − 40
+3
+�
+R
+uuxxuxxxdx − 40
+9
+�
+R
+u3uxxxdx.
+(2.1)
+It remains to check that the sum of all the integral terms in the above equation is zero. Calculating each term in
+(2.1) using the integration by parts, we have
+− 8
+�
+R
+uuxuxxxxdx = −20
+�
+R
+uxu2
+xxdx,
+(2.2)
+− 80
+3
+�
+R
+u2uxuxxdx = 80
+3
+�
+R
+uu3
+xdx,
+(2.3)
+− 80
+9
+�
+R
+u4uxdx = 0,
+(2.4)
+− 4
+�
+R
+uxxxuxxxxdx = 0,
+(2.5)
+2
+
+− 20
+3
+�
+R
+uxxxu2
+xdx = 40
+3
+�
+R
+uxu2
+xxdx,
+(2.6)
+− 40
+3
+�
+R
+uuxxuxxxdx = 20
+3
+�
+R
+uxu2
+xxdx,
+(2.7)
+− 40
+9
+�
+R
+u3uxxxdx = −40
+3
+�
+R
+uu3
+xdx.
+(2.8)
+Substituting (2.2)–(2.8) into (2.1), we have d
+dt M4 = 0, which completes the proof.
+Remark 1. Suppose the general form of the KdV equation is
+ut − auux − buxxx = 0,
+and the corresponding high-order invariant
+M(t) =
+�
+R
+(u2
+xx − Auu2
+x + Bu4)dx.
+Using the same method above, we could derive
+� 5a = 3Ab,
+12Bb = Aa,
+which can be rewritten as
+a
+b = 3A
+5 = 12B
+A .
+Therefore, it follows
+A2 = 20B.
+For instance, when a = −6, b = −1, we have A = 10, B = 5, which deduces to the KdV equation as
+ut + 6uux + uxxx = 0,
+with a fourth-order invariant
+M(t) =
+�
+R
+(u2
+xx − 10uu2
+x + 5u4)dx.
+2.2. Invariants of the Camassa–Holm equation
+Proof: Multiplying by 1 and u on both sides of (1.3), respectively, and then integrating the results, which implies
+E1 and E2 through the integration by parts. Below, we prove E3 by the energy method. Firstly, noticing that (1.3)
+can be written with a skew-adjoint operator (1 − ∂xx)−1 as
+ut + uux + ∂x(1 − ∂xx)−1�
+u2 + 1
+2u2
+x
+�
+= 0.
+Let g = (1 − ∂xx)−1�
+u2 + 1
+2u2
+x
+�
+. Then we see from the above equation that (1.3) is equivalent to
+
+ut + uux + gx = 0,
+(2.9)
+g − gxx = u2 + 1
+2u2
+x.
+(2.10)
+Multiplying (2.9) by 3u2 + u2
+x − 2(uux)x and integrating the result on both sides, we have
+0 =
+�
+R
+(ut + uux + gx) · (3u2 + u2
+x − 2(uux)x)dx
+=
+�
+R
+ut · (3u2 + u2
+x − 2(uux)x)dx +
+�
+R
+(uux + gx) · (3u2 + u2
+x − 2(uux)x)dx
+≜ A + B.
+(2.11)
+3
+
+Calculating each term derives that
+A =
+�
+R
+ut · (3u2 + u2
+x − 2(uux)x)dx
+=
+�
+R
+ut · (3u2 + u2
+x)dx +
+�
+R
+2uux · uxtdx
+=
+�
+R
+ut · 3u2dx +
+�
+R
+ut · u2
+xdx +
+�
+R
+u · (u2
+x)tdx
+=
+�
+R
+(u3)tdx +
+�
+R
+(u · u2
+x)tdx
+= d
+dt
+�
+R
+(u3 + uu2
+x)dx
+(2.12)
+and
+B =
+�
+R
+(uux + gx) · (3u2 + u2
+x − 2(uux)x)dx
+=
+�
+R
+u · u3
+xdx +
+�
+R
+gx · (3u2 + u2
+x)dx −
+�
+R
+gx · 2(uux)xdx
+=
+�
+R
+u · u3
+xdx +
+�
+R
+gx · (3u2 + u2
+x)dx + 2
+�
+R
+gxx · uuxdx
+=
+�
+R
+u · u3
+xdx +
+�
+R
+gx · (3u2 + u2
+x)dx + 2
+�
+R
+(g − u2 − 1
+2u2
+x) · uuxdx
+=
+�
+R
+gx · (3u2 + u2
+x)dx + 2
+�
+R
+g · uuxdx
+=
+�
+R
+gx · (3u2 + u2
+x)dx −
+�
+R
+gx · u2dx
+=
+�
+R
+gx · (2u2 + u2
+x)dx
+= 2
+�
+R
+gx · (g − gxx)dx = 0.
+(2.13)
+Substituting (2.12) and (2.13) into (2.11), we have
+d
+dt
+�
+R
+(u3 + uu2
+x)dx = 0,
+which implies E3.
+2.3. Invariants of the Degasperis–Procesi equation
+Proof: Integrating on both sides of (1.4), it easily obtains H1. Then we show invariants H2 and H3 of (1.4),
+respectively. Firstly let g = (1 − ∂xx)−1� 3
+2u2�
+, then (1.4) is equivalent to
+
+ut + uux + gx = 0,
+(2.14)
+g − gxx = 3
+2u2.
+(2.15)
+Multiplying by 2u − 6v on both sides of (2.14) and then integrating the result, we have
+0 =
+�
+R
+(ut + uux + gx) · (2u − 6v)dx
+=
+�
+R
+ut · (2u − 6v)dx +
+�
+R
+uux · (2u − 6v)dx +
+�
+R
+gx · (2u − 6v)dx
+≜ C + D.
+(2.16)
+4
+
+The each term in the above identity is estimated as
+C =
+�
+R
+ut · (2u − 6v)dx = 2
+�
+R
+ut · udx − 6
+�
+R
+ut · vdx = 2
+�
+R
+ut · udx − 6
+�
+R
+(4vt − vxxt) · vdx
+= 2
+�
+R
+ut · udx − 24
+�
+R
+vt · vdx − 6
+�
+R
+vxt · vxdx = d
+dt
+�
+R
+(u2 − 12v2 − 3v2
+x)dx
+= d
+dt
+�
+R
+�
+u2 − 3(4v − vxx) · v
+�
+dx = d
+dt
+�
+R
+(u2 − 3uv)dx = d
+dt
+�
+R
+u · (u − 3v)dx
+= d
+dt
+�
+R
+u · (v − vxx)dx = d
+dt
+�
+R
+(u − uxx) · vdx
+(2.17)
+and
+D =
+�
+R
+uux · (2u − 6v)dx +
+�
+R
+gx · (2u − 6v)dx
+= −6
+�
+R
+uux · vdx +
+�
+R
+gx · (2u − 6v)dx
+= 3
+�
+R
+u2 · vxdx +
+�
+R
+gx · (2u − 6v)dx
+= 2
+�
+R
+(g − gxx) · vxdx +
+�
+R
+gx · (2u − 6v)dx
+= 2
+�
+R
+g · vxdx − 2
+�
+R
+gxx · vxdx +
+�
+R
+gx · (2v − 2vxx)dx
+= 2
+�
+R
+g · vxdx + 2
+�
+R
+gx · vdx − 2
+�
+R
+gxx · vxdx − 2
+�
+R
+gx · vxxdx
+= 2
+�
+R
+(gv)xdx − 2
+�
+R
+(gx · vx)xdx = 0.
+(2.18)
+Substituting (2.17) and (2.18) into (2.16), we have
+d
+dt
+�
+R
+(u − uxx) · vdx = 0,
+which implies H2.
+Finally, we show H3. Multiplying (2.14) on both sides by u2 and integrating the result, it yields by noting (2.15)
+0 =
+�
+R
+(ut + uux + gx) · u2dx
+=
+�
+R
+ut · u2dx +
+�
+R
+u3 · uxdx +
+�
+R
+gx · u2dx
+=
+�
+R
+�1
+3u3�
+tdx + 2
+3
+�
+gx · (g − gxx)dx
+= 1
+3
+d
+dt
+�
+R
+u3dx,
+which implies the invariant H3.
+3. Applications to other periodic nonlinear dispersive waves
+3.1. Benjamin-Bona-Mahony equation
+Consider the Benjamin-Bona-Mahony equation [8] of the form
+ut − uxxt + ux + εuux = 0,
+x ∈ R.
+(3.1)
+5
+
+It can be written as
+ut + ∂x(1 − ∂xx)−1�
+u + ε
+2u2�
+= 0,
+x ∈ R.
+Let g = (1 − ∂xx)−1�
+u + ε
+2u2�
+, then the equation (3.1) turns out to be
+
+ut + gx = 0,
+(3.2)
+g − gxx = u + ε
+2u2.
+(3.3)
+Multiplying both sides of (3.2) by u2 and integrating the result, and then using (3.3), we have
+0 =
+�
+R
+(ut + gx) · u2dx =
+�
+R
+ut · u2dx +
+�
+R
+gx · u2dx
+=
+�
+R
+ut · u2dx + 2
+ε
+�
+R
+gx · (g − gxx − u)dx =
+�
+R
+ut · u2dx − 2
+ε
+�
+R
+gx · udx
+=
+�
+R
+ut · u2dx + 2
+ε
+�
+R
+ut · udx = d
+dt
+�
+R
+�1
+3u3 + 1
+εu2�
+dx,
+which indicates
+�
+R
+1
+3
+�
+u3 + 1
+εu2�
+dx
+is a three-order invariant for (3.1).
+3.2. Regularized long wave equation
+Consider the regularized long wave equation [9] of the form
+ut − µuxxt + ux + upux = 0,
+(3.4)
+where µ > 0 is a positive constant. When p = 2, it is called modified regularized long wave equation; when p ⩾ 3,
+it is called generalized regularized long wave equation. Similar to the foregoing argument, (3.4) can be written as
+an equivalent form of
+
+ut + gx = 0,
+(3.5)
+g − µgxx = u +
+1
+p + 1up+1.
+(3.6)
+Multiplying both sides of (3.5) by up+1, integrating the result, and then using (3.6), we have
+0 =
+�
+R
+(ut + gx) · up+1dx =
+�
+R
+ut · up+1dx +
+�
+R
+gx · up+1dx
+=
+�
+R
+ut · up+1dx + (p + 1)
+�
+R
+gx · (g − µgxx − u)dx
+=
+�
+R
+ut · up+1dx − (p + 1)
+�
+R
+gx · udx
+=
+�
+R
+ut · up+1dx + (p + 1)
+�
+R
+ut · udx
+= d
+dt
+�
+R
+�
+1
+p + 2up+2 + p + 1
+2
+u2�
+dx,
+which indicates
+�
+R
+�
+1
+p + 2up+2 + p + 1
+2
+u2�
+dx
+is a high-order invariant for (3.4). This corrects an invariant I3 in Example 4 appeared in [10] (pp. 492).
+6
+
+3.3. Rosenau equation
+Consider the Rosenau equation [11]
+ut + uxxxxt + ux + uux = 0,
+(3.7)
+which is equivalent to
+
+ut + gx = 0,
+(3.8)
+g + gxxxx = u + 1
+2u2.
+(3.9)
+Multiplying both sides of (3.8) by u2 and noticing (3.9), similar to the argument in the above, we have a third-order
+invariant for (3.7) of the form
+�
+R
+�1
+3u3 + u2�
+dx.
+Acknowledgement
+We appreciate Prof. Zhi-zhong Sun for many useful discussions. This work is dedicated to Prof. Zhi-zhong Sun
+on the occasion of his 60th birthday. The work is supported by Natural Science Foundation of Zhejiang Province
+(Grant No. LZ23A010007).
+References
+References
+[1] J. Escher, Y. Liu, Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation. J. Funct. Anal., 241 (2006)
+457–485.
+[2] T. Tao, Low-regularity global solutions to nonlinear dispersive equations. Surveys in analysis and operator theory (Canberra, 2001),
+19–48, Proc. Centre Math. Appl. Austral. Nat. Univ., 40, Austral. Nat. Univ., Canberra, (2002).
+[3] R. Camassa, D. D. Holm, An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71 (1993) 1661–1664.
+[4] A. Degasperis, M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.). Symmetry and Perturbation Theory, World
+Scientific, Singapore, (1999) 23–37.
+[5] H. Liu, Y. Xing, An invariant preserving discontinuous Galerkin method for the Camassa-Holm equation. SIAM J. Sci. Comput., 38
+(2016) A1919–A1934.
+[6] R. Courant, K.O. Friedrichs, H. Lewy, ¨Uber die partiellen Differenzenglei-chungen der mathematischen physik, Math. Ann., 100 (1928)
+32–74.
+[7] Z. Sun. Finite Difference Methods for Nonlinear Evolution Equations, Science Press, Beijing, (2018).
+[8] L.A. Medeiros, G.P. Menzala, Existence and uniqueness for periodic solutions of the Benjamin-Bona-Mahony equation. SIAM J. Math.
+Anal., 8(5) (1977) 792–799.
+[9] C.E. Seyler, D.L. Fenstermacher, A symmetric regularized-long-wave equation. The Physics of Fluids, 27(4) (1984) 4–7.
+[10] A. Ghiloufi, K. Omrani, New conservative difference schemes with fourth-order accuracy for some model equation for nonlinear
+dispersive waves. Numer. Methods Partial Differential Equation, 34 (2018) 451–500.
+[11] M.A. Park, On the Rosenau equation. Math. Appl. Comput., 9 (1990) 145–152.
+7
+

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='00990v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='NA]  3 Jan 2023  The energy method for high-order invariants in shallow water wave equations  Qifeng Zhanga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Tong Yana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Guang-hua Gaob  aDepartment of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Zhejiang Sci-Tech University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Hangzhou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' 310018,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' China  bDepartment of Mathematics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Nanjing University of Posts and Telecommunications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Nanjing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' 210096,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' China  Abstract  Third order dispersive evolution equations are widely adopted to model one-dimensional long waves and have  extensive applications in fluid mechanics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' plasma physics and nonlinear optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Among them are the KdV equation,  the Camassa–Holm equation and the Degasperis–Procesi equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' They share many common features such as  complete integrability, Lax pairs and bi-Hamiltonian structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' In this paper we revisit high-order invariants  for these three types of shallow water wave equations by the energy method in combination of a skew-adjoint  operator (1 − ∂xx)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Several applications to seek high-order invariants of the Benjamin-Bona-Mahony equation,  the regularized long wave equation and the Rosenau equation are also presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Keywords: Energy method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' High-order invariant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Shallow water wave equation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Introduction  A family of third order dispersive evolution equations of the form  ut − α2uxxt + γuxxx + c0ux = (c1u2 + c2u2  x + c3uuxx)x,  x ∈ R, t > 0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1)  frequently appeared in the simulation of the shallow water waves, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=', [1], where α, γ and ci (i = 0, 1, 2, 3) are  real constants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' u denotes a horizontal velocity field with the independent spatial variable x and temporal variable t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  A typical such equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1) with α2 = c0 = c2 = c3 = 0, c1 = 2, γ = −2 is the KdV equation  ut − 4uux − 2uxxx = 0,  x ∈ R, t > 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2)  which describes the unidirectional propagation of waves at the free surface of shallow water under the influence  of gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' The first four invariants of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2) are respectively as (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=', [2], although there is a minor typo in the  coefficient of the fourth invariant, it does not affect the reading of this classic review)  M1 =  �  R  udx,  M2 =  �  R  u2dx,  M3 =  �  R  �  u2  x − 2  3u3�  dx,  M4 =  �  R  �  u2  xx − 10  3 uu2  x + 5  9u4�  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Taking α2 = c3 = 1, γ = c0 = 0, c1 = − 3  2, c2 =  1  2, we have another example called the Camassa–Holm  equation [3]  ut − uxxt + 3uux = 2uxuxx + uuxxx,  x ∈ R, t > 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3)  which models the unidirectional propagation of shallow water waves over a flat bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' The first three invariants  are listed as follows  E1 =  �  R  (u − uxx)dx,  E2 = 1  2  �  R  (u2 + u2  x)dx,  E3 = 1  2  �  R  u(u2 + u2  x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  The third example by assigning α2 = c2 = c3 = 1, γ = c0 = 0, c1 = −2 is called the Degasperis–Procesi  equation  ut − uxxt + 4uux = 3uxuxx + uuxxx,  x ∈ R, t > 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4)  ∗E-mail address: zhangqifeng0504@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='com (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Zhang), tyan0320@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='zstu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='cn (Tong Yan), gaogh@njupt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='cn (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Gao)  Preprint submitted to Elsevier  January 4, 2023  which can be regarded as a model for nonlinear shallow water dynamics [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' The frequently discussed invariants  are  H1 =  �  R  (u − uxx)dx,  H2 =  �  R  (u − uxx)vdx,  H3 =  �  R  u3dx,  where 4v − vxx = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Up to now, there have been thousands of papers focusing on the theoretical and numerical studies on these three  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' It is worth mentioning that the invariant-preserving property is a key index of the success for numerical  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' However, high-order invariants are usually difficult to preserve numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' also pointed out “it  appears a rather difficult task to preserve all three conservation laws” in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' In this work, higher-order invariants of  these equations will be re-derived in view of the energy method, which may be possible to provide some thoughts  for invariant-preserving numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Actually, the energy method originated from conservation laws in  physics was first proposed in 1928 by Courant, Friedrichs and Lewy [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' From then on, it has been widely applied  to the mathematical and numerical analysis of nonlinear evolution equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' We trust the readers with [7] instead  of a long list of references to relevant works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  The rest of the paper is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  In Section 2, combining the energy method and a skew-  adjoint operator, we show the high-order invariants for the KdV equation, the Camassa–Holm equation and the  Degasperis–Procesi equation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Then we list several applications for seeking some high-order invariants  of other types of the shallow water wave equations in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Main results  In what follows, we directly show that Mi (i = 1, 2, 3, 4), Ei (i = 1, 2, 3) and Hi (i = 1, 2, 3) are invariants of  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4) subjected to the periodic boundary conditions based on the energy method, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Invariants of the KdV equation  Proof: (I) Multiplying by 1, u and (u2 + uxx), respectively, with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2), we have Mi (i = 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' In what follows, we  show the fourth invariant M4 of the KdV equation by the energy method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Multiplying both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2) by 2uxxxx + 10  3 u2  x + 20  3 uuxx + 20  9 u3 and integrating the result,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' we have  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='0 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2uxxxx + 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3 u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='x + 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3 uuxx + 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9 u3�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='utdx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2uxxxx + 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3 u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='x + 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3 uuxx + 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9 u3�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='(4uux + 2uxxx)dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2uxxuxxt − 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3 (utu2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='x + 2uuxuxt) + 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9 u3ut  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2uxxxx + 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3 u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='x + 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3 uuxx + 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9 u3�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='(4uux + 2uxxx)dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='= d  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='dt M4 − 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='uuxuxxxxdx − 40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='uu3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='xdx − 80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='u2uxuxxdx − 80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='u4uxdx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='− 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='uxxxuxxxxdx − 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='uxxxu2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='xdx − 40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='uuxxuxxxdx − 40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='u3uxxxdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1)  It remains to check that the sum of all the integral terms in the above equation is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Calculating each term in  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1) using the integration by parts, we have  − 8  �  R  uuxuxxxxdx = −20  �  R  uxu2  xxdx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2)  − 80  3  �  R  u2uxuxxdx = 80  3  �  R  uu3  xdx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3)  − 80  9  �  R  u4uxdx = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4)  − 4  �  R  uxxxuxxxxdx = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='5)  2  − 20  3  �  R  uxxxu2  xdx = 40  3  �  R  uxu2  xxdx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='6)  − 40  3  �  R  uuxxuxxxdx = 20  3  �  R  uxu2  xxdx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='7)  − 40  9  �  R  u3uxxxdx = −40  3  �  R  uu3  xdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='8)  Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='8) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1), we have d  dt M4 = 0, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Suppose the general form of the KdV equation is  ut − auux − buxxx = 0,  and the corresponding high-order invariant  M(t) =  �  R  (u2  xx − Auu2  x + Bu4)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Using the same method above, we could derive  � 5a = 3Ab,  12Bb = Aa,  which can be rewritten as  a  b = 3A  5 = 12B  A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Therefore, it follows  A2 = 20B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  For instance, when a = −6, b = −1, we have A = 10, B = 5, which deduces to the KdV equation as  ut + 6uux + uxxx = 0,  with a fourth-order invariant  M(t) =  �  R  (u2  xx − 10uu2  x + 5u4)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Invariants of the Camassa–Holm equation  Proof: Multiplying by 1 and u on both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3), respectively, and then integrating the results, which implies  E1 and E2 through the integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Below, we prove E3 by the energy method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Firstly, noticing that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3)  can be written with a skew-adjoint operator (1 − ∂xx)−1 as  ut + uux + ∂x(1 − ∂xx)−1�  u2 + 1  2u2  x  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Let g = (1 − ∂xx)−1�  u2 + 1  2u2  x  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Then we see from the above equation that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3) is equivalent to  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  ut + uux + gx = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9)  g − gxx = u2 + 1  2u2  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='10)  Multiplying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9) by 3u2 + u2  x − 2(uux)x and integrating the result on both sides, we have  0 =  �  R  (ut + uux + gx) · (3u2 + u2  x − 2(uux)x)dx  =  �  R  ut · (3u2 + u2  x − 2(uux)x)dx +  �  R  (uux + gx) · (3u2 + u2  x − 2(uux)x)dx  ≜ A + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='11)  3  Calculating each term derives that  A =  �  R  ut · (3u2 + u2  x − 2(uux)x)dx  =  �  R  ut · (3u2 + u2  x)dx +  �  R  2uux · uxtdx  =  �  R  ut · 3u2dx +  �  R  ut · u2  xdx +  �  R  u · (u2  x)tdx  =  �  R  (u3)tdx +  �  R  (u · u2  x)tdx  = d  dt  �  R  (u3 + uu2  x)dx  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='12)  and  B =  �  R  (uux + gx) · (3u2 + u2  x − 2(uux)x)dx  =  �  R  u · u3  xdx +  �  R  gx · (3u2 + u2  x)dx −  �  R  gx · 2(uux)xdx  =  �  R  u · u3  xdx +  �  R  gx · (3u2 + u2  x)dx + 2  �  R  gxx · uuxdx  =  �  R  u · u3  xdx +  �  R  gx · (3u2 + u2  x)dx + 2  �  R  (g − u2 − 1  2u2  x) · uuxdx  =  �  R  gx · (3u2 + u2  x)dx + 2  �  R  g · uuxdx  =  �  R  gx · (3u2 + u2  x)dx −  �  R  gx · u2dx  =  �  R  gx · (2u2 + u2  x)dx  = 2  �  R  gx · (g − gxx)dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='13)  Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='12) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='13) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='11), we have  d  dt  �  R  (u3 + uu2  x)dx = 0,  which implies E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Invariants of the Degasperis–Procesi equation  Proof: Integrating on both sides of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4), it easily obtains H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Then we show invariants H2 and H3 of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Firstly let g = (1 − ∂xx)−1� 3  2u2�  , then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4) is equivalent to  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  ut + uux + gx = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='14)  g − gxx = 3  2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='15)  Multiplying by 2u − 6v on both sides of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='14) and then integrating the result, we have  0 =  �  R  (ut + uux + gx) · (2u − 6v)dx  =  �  R  ut · (2u − 6v)dx +  �  R  uux · (2u − 6v)dx +  �  R  gx · (2u − 6v)dx  ≜ C + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='16)  4  The each term in the above identity is estimated as  C =  �  R  ut · (2u − 6v)dx = 2  �  R  ut · udx − 6  �  R  ut · vdx = 2  �  R  ut · udx − 6  �  R  (4vt − vxxt) · vdx  = 2  �  R  ut · udx − 24  �  R  vt · vdx − 6  �  R  vxt · vxdx = d  dt  �  R  (u2 − 12v2 − 3v2  x)dx  = d  dt  �  R  �  u2 − 3(4v − vxx) · v  �  dx = d  dt  �  R  (u2 − 3uv)dx = d  dt  �  R  u · (u − 3v)dx  = d  dt  �  R  u · (v − vxx)dx = d  dt  �  R  (u − uxx) · vdx  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='17)  and  D =  �  R  uux · (2u − 6v)dx +  �  R  gx · (2u − 6v)dx  = −6  �  R  uux · vdx +  �  R  gx · (2u − 6v)dx  = 3  �  R  u2 · vxdx +  �  R  gx · (2u − 6v)dx  = 2  �  R  (g − gxx) · vxdx +  �  R  gx · (2u − 6v)dx  = 2  �  R  g · vxdx − 2  �  R  gxx · vxdx +  �  R  gx · (2v − 2vxx)dx  = 2  �  R  g · vxdx + 2  �  R  gx · vdx − 2  �  R  gxx · vxdx − 2  �  R  gx · vxxdx  = 2  �  R  (gv)xdx − 2  �  R  (gx · vx)xdx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='18)  Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='18) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='16), we have  d  dt  �  R  (u − uxx) · vdx = 0,  which implies H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Finally, we show H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Multiplying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='14) on both sides by u2 and integrating the result, it yields by noting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='15)  0 =  �  R  (ut + uux + gx) · u2dx  =  �  R  ut · u2dx +  �  R  u3 · uxdx +  �  R  gx · u2dx  =  �  R  �1  3u3�  tdx + 2  3  �  gx · (g − gxx)dx  = 1  3  d  dt  �  R  u3dx,  which implies the invariant H3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Applications to other periodic nonlinear dispersive waves  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Benjamin-Bona-Mahony equation  Consider the Benjamin-Bona-Mahony equation [8] of the form  ut − uxxt + ux + εuux = 0,  x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1)  5  It can be written as  ut + ∂x(1 − ∂xx)−1�  u + ε  2u2�  = 0,  x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Let g = (1 − ∂xx)−1�  u + ε  2u2�  , then the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1) turns out to be  \uf8f1\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f3  ut + gx = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2)  g − gxx = u + ε  2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3)  Multiplying both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2) by u2 and integrating the result, and then using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3), we have  0 =  �  R  (ut + gx) · u2dx =  �  R  ut · u2dx +  �  R  gx · u2dx  =  �  R  ut · u2dx + 2  ε  �  R  gx · (g − gxx − u)dx =  �  R  ut · u2dx − 2  ε  �  R  gx · udx  =  �  R  ut · u2dx + 2  ε  �  R  ut · udx = d  dt  �  R  �1  3u3 + 1  εu2�  dx,  which indicates  �  R  1  3  �  u3 + 1  εu2�  dx  is a three-order invariant for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Regularized long wave equation  Consider the regularized long wave equation [9] of the form  ut − µuxxt + ux + upux = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4)  where µ > 0 is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' When p = 2, it is called modified regularized long wave equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' when p ⩾ 3,  it is called generalized regularized long wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Similar to the foregoing argument, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4) can be written as  an equivalent form of  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  ut + gx = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='5)  g − µgxx = u +  1  p + 1up+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='6)  Multiplying both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='5) by up+1, integrating the result, and then using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='6), we have  0 =  �  R  (ut + gx) · up+1dx =  �  R  ut · up+1dx +  �  R  gx · up+1dx  =  �  R  ut · up+1dx + (p + 1)  �  R  gx · (g − µgxx − u)dx  =  �  R  ut · up+1dx − (p + 1)  �  R  gx · udx  =  �  R  ut · up+1dx + (p + 1)  �  R  ut · udx  = d  dt  �  R  �  1  p + 2up+2 + p + 1  2  u2�  dx,  which indicates  �  R  �  1  p + 2up+2 + p + 1  2  u2�  dx  is a high-order invariant for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' This corrects an invariant I3 in Example 4 appeared in [10] (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' 492).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Rosenau equation  Consider the Rosenau equation [11]  ut + uxxxxt + ux + uux = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='7)  which is equivalent to  \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3  ut + gx = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='8)  g + gxxxx = u + 1  2u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9)  Multiplying both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='8) by u2 and noticing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='9), similar to the argument in the above, we have a third-order  invariant for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='7) of the form  �  R  �1  3u3 + u2�  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content='  Acknowledgement  We appreciate Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Zhi-zhong Sun for many useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' This work is dedicated to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Zhi-zhong Sun  on the occasion of his 60th birthday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' The work is supported by Natural Science Foundation of Zhejiang Province  (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' LZ23A010007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/49AzT4oBgHgl3EQfEPqD/content/2301.00990v1.pdf'}
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+Quantum sensing of electric field distributions of liquid electrolytes with NV-centers
+in nanodiamonds
+M. Hollendonner,1, 2 S. Sharma,2 D. B. R. Dasari,3 A. Finkler,4 S. V. Kusminskiy,2, 5 and R. Nagy1, ∗
+1Friedrich-Alexander-University Erlangen-Nuremberg, 91058 Erlangen, Germany
+2Max Planck Institute for the Science of Light, 91058 Erlangen, Germany
+33rd Institute of Physics, IQST, and Research Center SCoPE,
+University of Stuttgart, 70569 Stuttgart, Germany
+4Department of Chemical and Biological Physics,
+Weizmann Institute of Science, Rehovot 7610001, Israel
+5Institute for Theoretical Solid State Physics, RWTH Aachen University, 52074 Aachen, Germany
+(Dated: January 12, 2023)
+To use batteries as large-scale energy storage systems it is necessary to measure and understand
+their degradation in-situ and in-operando. As a battery’s degradation is often the result of molecular
+processes inside the electrolyte, a sensing platform which allows to measure the ions with a high
+spatial resolution is needed. Primary candidates for such a platform are NV-centers in diamonds.
+We propose to use a single NV-center to deduce the electric field distribution generated by the
+ions inside the electrolyte through microwave pulse sequences. We show that the electric field can
+be reconstructed with great accuracy by using a protocol which includes different variations of the
+Free Induction Decay to obtain the mean electric field components and a modified Hahn-echo pulse
+sequence to measure the electric field’s standard deviation σE. From a semi-analytical ansatz we find
+that for a lithium ion battery there is a direct relationship between σE and the ionic concentration.
+Our results show that it is therefore possible to use NV-centers as sensors to measure both the
+electric field distribution and the local ionic concentration inside electrolytes.
+I.
+INTRODUCTION
+Rechargeable batteries play an important role for our
+society and are a key ingredient for the transition to-
+wards renewable energy sources [1–3]. As the production
+of batteries is accompanied with a considerable use of re-
+sources, recyclable [4] batteries with a long lifetime are
+needed. The latter is limited by degradation mechanisms,
+such as the formation of solid-electrolyte interfaces [5] or
+lithium-plating [6] which can reduce the battery’s capac-
+ity with increasing cell age [7]. As these processes happen
+on a molecular level within nanometer scales [5], a sensor
+which is capable of monitoring the ionic concentration
+in-situ and in-operando with high spatial and temporal
+resolutions is needed. Even though MRI allows to recon-
+struct transport properties [8, 9] of a battery, tools which
+allow to perform measurements inside the electrolyte are
+still absent [5].
+It has been demonstrated that nitrogen-vacancy (NV)
+centers in diamond (see Fig. 1(b)) are high-resolution
+quantum sensors, which can detect oscillating or fluctu-
+ating [10–13] magnetic fields with nano- [14, 15] and even
+subpico-Tesla [16] sensitivities. Besides this, NV-centers
+have great ability for the detection of electric fields. They
+can not only detect DC [17, 18] or AC [19] electric fields
+with remarkable precision, but are additionally capable
+of detecting single fundamental charges [20] even within
+the diamond lattice [21]. This electric field sensitivity was
+used by Ref. [22] to show that, based on theoretical con-
+∗ [email protected]
+siderations, bulk NV-centers can work as electrochemical
+sensors if they are in contact with an electrolyte solution.
+Here we show that nanodiamonds equipped with sin-
+gle NV-centers can act as in-situ electric field sensors
+inside liquid electrolytes (Fig. 1(a)). By exploiting how
+transverse and axial electric fields act on the NV-center’s
+ground state spin states, we find variations of the free-
+induction decay (FID) pulse sequence, which allow to
+measure the mean electric field components.
+Further,
+we show that it is possible to use variants of the Hahn-
+echo pulse sequence to additionally obtain the electric
+field’s standard deviation σE.
+From a semi-analytical
+ansatz we demonstrate exemplarily for a lithium ion bat-
+tery (LIB) that there is a direct relationship between the
+electric field’s standard deviation and the local ionic con-
+centration. A nanodiamond with a single NV-center can
+therefore work as a sensor which allows to simultaneously
+reconstruct the electric field distribution and to measure
+the ionic concentration with nm spatial resolution.
+II.
+ELECTRIC FIELD DISTRIBUTION IN
+LIQUID ELECTROLYTES
+Before introducing measurements of the electric field
+distribution by the NV-center, we would like to develop
+an analytic expression of the electric field induced inside
+the nanodiamond by the positive and negative ions of the
+electrolyte.
+The potential Φ at position r inside the nanodiamond
+due to a single charge q at position b, is described by
+arXiv:2301.04427v1  [quant-ph]  11 Jan 2023
+
+2
+FIG. 1. (a) Experimental setting. A nanodiamond which is dissolved in the liquid electrolyte of the battery is surrounded by
+positive (orange) and negative (blue) ions. Two perpendicular aligned gold wires allow to generate polarized microwave drives.
+(b) To work as a quantum sensor, the nanodiamond contains a vacancy (V) next to a nitrogen atom (red). (c) Standard deviation
+of Ez, calculated from 500 repeated sets of randomly placed ions of concentration c around the nanodiamond (rND = 100 nm)
+and inside a sphere of radius R. The relative permittivities are ϵND = 5.8 [22] and ϵe = 17.5 [23]. Solid lines are fits following
+Eq. (3) with A as a fit parameter. (d) Fit parameters A obtained from (c), compared to the theory value.
+Poisson’s equation
+∇2Φ (r) = −ρ (r)
+ϵ
+.
+(1)
+Here ϵ = ϵ0ϵi with i = e, ND, are the permittivities of, re-
+spectively, the electrolyte and the nanodiamond in terms
+of the vacuum permittivity ϵ0 and ρ is the charge density
+induced by q. The solution inside the nanodiamond, ΦND
+(see Methods for the detailed derivation), allows to ob-
+tain the electric field at the center of the nanodiamond,
+which is
+END =
+q
+4πϵ0
+3
+2ϵe + ϵND
+b
+b3 .
+(2)
+By considering the positions of ions of a molar concentra-
+tion c to be normally distributed within a sphere of radius
+R around a nanodiamond (radius rND), the standard de-
+viation of the electric field distribution at the center of
+the nanodiamond is
+σEz = A
+�
+c
+� 1
+rND
+− 1
+R
+�
+A =
+|q|
+ϵ0 (2ϵe + ϵND)
+�
+3NA
+4π .
+(3)
+To validate Eq. (3), we simulated the standard deviation
+of 500 sets of uniformly and randomly placed ions for dif-
+ferent molar ionic concentrations (see Fig. 1(c)). As it is
+the most widely used electrolyte of LIBs [24], we chose
+LiPF−
+6 with ϵe = 17.5 [23]. The total electric field was
+calculated as the linear sum of Eq. (2) for all randomly
+placed ions around a 200 nm spherical nanodiamond [25].
+As it can be seen from Fig. 1(d), the expected A value is
+in fair agreement with the simulations. From Eq. (3) it
+can be calculated that for R = 500 nm, the fluctuations
+will increase only by 3%, compared to σE (R = 400 nm).
+As σE therefore saturates for R ≳ 500 nm, this implies
+that electric field fluctuations only affect the nanodia-
+mond within sub-micrometer range and the system is
+limited by the confocal volume of the experimental setup,
+which typically is ∼ 1 µm3 [26, 27].
+III.
+SENSING OF STATIC ELECTRIC FIELDS
+INSIDE ELECTROLYTES
+An electric field E can in cylindrical coordinates be
+expressed by its axial component Ez, its transverse pro-
+jection E⊥ =
+�
+E2x + E2y and an angle φE, which defines
+the projections onto the x and y axis as Ex = E⊥ cos φE
+and Ey = E⊥ sin φE. The total Hamiltonian which de-
+scribes the NV-center in presence of electric and axial
+magnetic fields will in the following be denoted as ˆH0.
+By taking into account that the NV-center can be driven
+by two perpendicular microwave wires (see Fig. 1(a)) with
+amplitude Ω, frequency ωd and a phase φ between each
+other, the total ground state Hamiltonian in a frame ro-
+tating with ωd is ˆH = ˆH0 + ˆHd (see Methods), where
+ˆH0 = (∆ + ξz) ˆS2
+z + βz ˆSz − ξ⊥
+2
+�
+ˆS2
++eiφE + h.c.
+�
+ˆHd = Ω
+√
+2
+�
+ϵ−σ0,−1 + ϵ+σ†
+0,+1 + h.c.
+�
+.
+(4)
+Here ∆ = D − ωd is the detuning between the zero-
+field splitting, D = 2.87 GHz [28], and the microwave
+drive frequency.
+Si, i = x, y, z, are the spin-1 op-
+erators, which can be used to define ladder operators
+S± = Sx ± iSy. σ0,±1 = |0⟩ ⟨±1| are operators which
+describe transitions between |0⟩ and, respectively, |±1⟩.
+Frequency contributions generated by electric and axial
+magnetic fields are considered through ξz = d∥Ez and
+ξ⊥ = d⊥E⊥ (d∥ = 0.35 Hz cm/V, d⊥ = 17 Hz cm/V [29])
+and βz = γeBz (γe = 28 GHz/T [30]).
+The phase factors ϵ± =
+�
+1 − ie∓iφ�
+/2 which enter into
+Eq. (4), allow to describe the transitions which are caused
+by circularly (φ = ±π/2) or linearly (φ = 0) polarized
+microwave drives [31]. The time-evolution operators of
+ˆHd, ˆR (t) = e−i ˆ
+Hdt (see Methods), show that one can
+induce Rabi oscillations between |0⟩ and |1⟩ for right cir-
+cularly polarized drives and |0⟩ ↔ |−1⟩ for left circular
+polarizations.
+Linearly polarized drives allow to drive
+transitions between |0⟩ and both |±1⟩.
+
+MW
+Pos.Electrode3
+FIG. 2. (a) FID-variations to extract ξ⊥, φE and ξz through subsequent pulse sequences. Here Tπ (Tπ/2) is the duration of
+the microwave pulse such that a π-pulse (π/2-pulse) is performed. Subscripts ± denote circularly polarized drives which cause
+oscillations between |0⟩ and either |1⟩ or |−1⟩. Subscript 0 denotes linear polarization of the drive and the free evolution is
+described through ˆF. (b) FIDξ⊥ for different magnetic fields up to βz = 2.7 MHz, corresponding to Bz = 1 G. For βz = 0 the
+signal has the highest contrast with the lowest frequency of oscillation. (c) Fourier transform of FIDξ⊥,ξz with Ω = 10 MHz
+and Ex,y,z = 10 V/µm. Only for T ∗
+2 > 10 µs the peaks at ξ⊥ ± ξz = 2.4 ± 0.04 MHz and 2ξ⊥ can be resolved.
+In absence of microwave drives, the |±1⟩ states are
+symmetrically mixed by ξ⊥ and axial electric fields ef-
+fectively shift |0⟩ from |±1⟩, which can be seen from
+ˆF (τ) = e−i ˆ
+H0τ (see Methods). As axial and transverse
+electric fields thus act differently on the |ms = 0, ±1⟩
+states of the NV-center, one can derive variations of the
+Free Induction Decay (FID), which allow to extract these
+electric field components.
+A.
+Measurement of electric field components
+The FID consists of two microwave pulses separated
+by a free evolution period τ. Electric field contributions
+ξ⊥, φE and ξz can be sensed through FID-variations,
+as shown in Fig. 2(a). The NV-center can be initialized
+into its |0⟩ state via excitation with green laser light, fol-
+lowed by intersystem-crossing [32]. This state can then
+be driven to −i |1⟩ through a right-polarized π-pulse, de-
+noted as ˆR (Tπ)+, and will be influenced by both axial
+magnetic as well as transverse electric fields. The latter
+induce mixing with |−1⟩. By using a microwave π-pulse
+with the same polarization as the initial one, the trans-
+ferred population from |1⟩ to |−1⟩ can be obtained from
+the FID-signal
+FIDξ⊥ (τ) = | ⟨0| ˆR (Tπ)+ ˆF (τ) ˆR (Tπ)+ |0⟩ |2
+= cos2
+�
+τ
+�
+β2z + ξ2
+⊥
+�
++
+β2
+z
+β2z + ξ2
+⊥
+sin2
+�
+τ
+�
+β2z + ξ2
+⊥
+�
+,
+(5)
+which is a measure of the population which has been
+transferred from |1⟩ to |−1⟩. In Fig. 2(b) one can see this
+FID-signal as a function of the free evolution time τ for
+βz values up to 2.8 MHz, which corresponds to Bz = 1 G.
+Besides having a decreased contrast for βz ̸= 0, the
+frequency
+�
+β2z + ξ2
+⊥ of the FID-oscillations depends on
+both axial magnetic and transverse electric fields. It is
+therefore strongly recommended to perform the measure-
+ments in a magnetically shielded environment, for exam-
+ple by a µ-metal as in Ref. [33]. In the following it will
+be assumed that all measurement are performed without
+any magnetic field being present.
+The transverse electric field components are uniquely
+defined through φE, as ξx
+=
+ξ⊥ cos φE and ξy
+=
+ξ⊥ sin φE. A superposition state −eiπ/4 (|1⟩ + |−1⟩) /
+√
+2
+generated through a linearly polarized π-pulse (consid-
+ered via ˆR (Tπ)0, see Methods) will additionally to ξ⊥
+also be affected by φE as this phase differs in its sign
+for |1⟩ and |−1⟩ (see Methods). If either |1⟩ or |−1⟩ is
+projected to |0⟩ through the final microwave pulse, one
+obtains an FID-signal, which both depends on ξ⊥ and
+φE,
+FIDφE,ξ⊥ (τ) = 1
+2 (1 − sin (2τξ⊥) sin φE) .
+(6)
+One can obtain φE as the relative fraction between the
+value of the FID-signal at τ = 0 and its first maxima at
+2τξ⊥ = π/2,
+FIDφE,ξ⊥
+�
+τ = π
+2
+1
+2ξ⊥
+�
+FIDφE,ξ⊥ (τ = 0)
+= 1 − sin φE .
+(7)
+By using FIDξ⊥ and FIDξ⊥,φE, it is therefore possible to
+not only determine the electric field’s transverse compo-
+nent, but also to obtain the projection onto the x and y
+axes, which are determined through φE.
+Axial electric field contributions ξz cause a Stark
+shift between |0⟩ and |±1⟩.
+A superposition state
+(|0⟩ − i |−1⟩) /
+√
+2 generated by a circularly polarized
+π/2-pulse (see Fig. 2(a)) will therefore be affected both
+by ξz and ξ⊥. If the final microwave π/2-pulse has the
+same polarization as the initial one, an FID-signal is ob-
+tained which depends both on ξ⊥ and ξz,
+FIDξz,ξ⊥ (τ) = 1
+4
+�
+1 − 2 cos (τξ⊥) cos (τξz) + cos2 (τξ⊥)
+�
+,
+(8)
+if the NV-center was driven with ωd = D. The Fourier
+
+4
+transform of Eq. (8) (see Methods),
+�
+FID (ω > 0) = π
+4
+�1
+2δ (2ξ⊥ − ω)
+− δ (ξ⊥ + ξz − ω) − δ (ξ⊥ − ξz − ω)
+�
+, (9)
+shows, that ξz can be measured if it is possible to spec-
+trally resolve ξ⊥ ± ξz.
+To study this, we numerically
+[34, 35] simulated FIDξz,ξ⊥ and included dephasing at
+rates 1/T ∗
+2 through a Lindblad operator
+�
+1/T ∗
+2 Sz for
+T ∗
+2 in the range up to 15 µs (see Fig. 2(c)). One can re-
+solve ξ⊥±ξz for nanodiamonds with T ∗
+2 > 10 µs, which is
+higher than the value of typical nanodiamonds [36]. For
+a nanodiamond with T ∗
+2 ≈ 15 µs it would be possible to
+distinguish between ξ⊥ and ξz and therefore to determine
+the projection of the electric field onto the symmetry axis
+of the NV-center.
+IV.
+INFLUENCE OF FLUCTUATING
+ELECTRIC FIELDS
+It can be assumed that the ions surrounding the nan-
+odiamond will not stay static for the timescales in which
+measurements are performed but will be subject to, for
+instance, drift and diffusion.
+These fluctuations will
+affect the electric field inside the nanodiamond.
+Due
+to the limited T ∗
+2 of nanodiamonds, the FID pulse se-
+quences as introduced before will be mainly suitable for
+the measurement of the average electric fields (see Meth-
+ods).
+The coherence time of a nanodiamond can be
+significantly prolonged if instead of an FID, a Hahn-
+Echo pulse sequence is used [25].
+As it is shown in
+Fig. 3(a), we propose a modified version of the Hahn-
+Echo, where after the first free evolution interval, a π-
+pulse with right-circular polarization is performed, be-
+fore the spin is allowed to precess freely during a sec-
+ond free evolution interval τ. Before being read out, a
+right-circularly polarized π-pulse is applied, which leads
+to a signal Hahn (τ) = (1 − cos (2τξ⊥))2 /4. Simulations
+of this Hahn-Echo variation show that the averages (see
+Methods for an example) can be fit by
+⟨Hahn (τ)⟩ = 1
+4
+�
+1 − cos (2τξ⊥) e−τ/T2�2
+.
+(10)
+Here T2 is the sum of the intrinsic spin coherence time
+T2,int. = 100 µs [25] and a contribution due to the fluc-
+tuating electric fields,
+1
+T2
+=
+1
+T2,int.
++
+1
+T2,E
+.
+(11)
+In Fig. 3(b), one can see T2 as a function of the electric
+field’s standard deviation σE, where solid lines are T2,E =
+αEm/σ2
+E in terms of a fit parameters α. The total spin
+coherence time is therefore strongly affected by σE and
+the mean electric field value Em. If the mean transverse
+electric field has been sensed by the FID sequence as
+shown in Eq. (5), it is therefore possible to derive the
+electric field’s standard deviation, which together with
+ξ⊥, φE and ξz defines the electric field distribution. As
+there is a direct relationship between σE and the local
+ionic concentration (see Fig. 1(c)), the proposed Hahn-
+echo pulse sequence additionally allows to use the NV-
+center inside the nanodiamond as a local concentration
+sensor.
+FIG. 3.
+(a) Hahn-echo pulse sequence used to simulate
+Eq. (10).
+(b) Total T2 for numerically [34, 35] simulated
+Hahn-Echoes with T2,int = 100 µs, with the electric field com-
+ponents sampled from a normal distribution with mean Em
+and standard deviation σE.
+For the simulations a drive of
+Ω = 10 MHz was used. Solid lines are fits of αEm/σ2
+E. Every
+trajectory was obtained from 1000 individual simulations. Er-
+ror bars of one standard deviation are smaller than the data
+points.
+V.
+CONCLUSION AND OUTLOOK
+In conclusion we have shown here a full reconstruc-
+tion of the mean electric field generated in a liquid elec-
+trolyte, through the spin control of a quantum sensor
+immersed in the electrolyte. We have found exact ex-
+pressions correlating the electric field components with
+the free-induction decay of the sensor spin, and the de-
+pendence of the variance on the spin-echo measurements.
+Together we were able to deduce the electric field distri-
+bution and also measure the local ionic concentration, a
+key parameter in characterizing the performance of the
+liquid electrolyte for battery applications. We envisage
+that with improved modeling of the electric field distribu-
+tion in liquid electrolytes and using better quantum con-
+trol methods, for example using correlation spectroscopy
+[37], we could enhance the sensitivity of the sensor to the
+local electric-field environment, allowing for an in-situ
+monitoring of the battery using the liquid electrolyte.
+ACKNOWLEDGMENTS
+R. N. would like to acknowledge financial support by
+the Federal Ministry of Education and Research (BMBF)
+
+5
+project QMNDQCNet and DFG (Project No. 507241320
+and 46256793).
+S. V. K. and D. D. would like to
+acknowledge the funding support from BMBF (Grant
+No. 16KIS1590K). A. F. is the incumbent of the Elaine
+Blond Career Development Chair and acknowledges sup-
+port from Israel Science Foundation (ISF grants 963/19
+and 418/20) as well as the Abramson Family Center for
+Young Scientists and the Willner Family Leadership In-
+stitute for the Weizmann Institute of Science.
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+
+1
+Quantum sensing of electric field distributions of liquid electrolytes with NV-centers
+in nanodiamonds - Supplementary Information
+I.
+ELECTRIC FIELD AT CENTER OF NANODIAMOND
+In the following we would like to deduce the electric field of a single point charge q at a distance b from the origin
+of the nanodiamond with radius rND by following Ref. [S1]. Poisson’s equation describes the electrostatic potential Φ,
+∇2Φ (r) = −ρ (r)
+ϵ
+,
+(S1)
+where ϵ = ϵ0ϵi, i = e, ND is the permittivity of, respectively, the electrolyte and the nanodiamond in terms of the
+vacuum permittivity ϵ0. By exploiting azimuthal symmetry of the problem, the above expression reduces to Laplace’s
+equation for r ̸= b, which in spherical coordinates with |r| = r and θ the angle spanned by r and b is
+∇2Φ (r, θ) = 1
+r2
+∂
+∂r
+�
+r2 ∂Φ
+∂r
+�
++
+1
+r2 sin θ
+∂
+∂θ
+�
+sin θ∂Φ
+∂θ
+�
+= 0 .
+(S2)
+The general solution of this partial differential equation can be expressed in terms of the associated Legendre poly-
+nomials Pl of order l and in terms of two constants Al and Cl as [S1, S2]
+Φ (r, θ) =
+∞
+�
+l=0
+�
+Alrl + Cl
+1
+rl+1
+�
+Pl (cos θ) .
+(S3)
+As the potential inside the nanodiamond must be finite at r = 0, Cl needs to vanish and one therefore has
+ΦND (r, θ) =
+∞
+�
+l=0
+AlrlPl (cos θ) .
+(S4)
+By using that 1/|r − b| = �∞
+l=0
+�
+rl
+</rl+1
+>
+�
+Pl (cos θ) [S1, S2] with r≷ being the greater (smaller) of |r| and |b|, one can
+derive the potential in the electrolyte without discontinuity, i.e. without nanodiamond, to be
+˜Φe (r, θ) =
+q
+4πϵ0ϵe
+∞
+�
+l=0
+rl
+<
+rl+1
+>
+Pl (cos θ) .
+(S5)
+The general solution would then be given as a superposition of this expression with Eq. (S3), i.e. Φe = ˜Φe + Φ, which
+reads
+Φe (r, θ) =
+∞
+�
+l=0
+�
+Cl
+1
+rl+1 +
+q
+4πϵ0ϵe
+rl
+<
+rl+1
+>
+�
+Pl (cos θ) ,
+(S6)
+where it was used that in this case Al = 0 to ensure a vanishing potential at infinite distances to the origin, i.e.
+Φe → 0 for r → ∞. The constants Al and Cl, which enter into, respectively, Eq. (S4) and Eq. (S6), can be determined
+by requiring continuity at the interface between electrolyte and nanodiamond,
+�
+ϵeEe − ϵNDEND�
+· nND = 0
+(S7)
+�
+Ee − END�
+× nND ,
+(S8)
+where nND = r/r is the unit vector normal to the surface of the nanodiamond.
+These boundary conditions are
+satisfied, if
+Al =
+q
+4πϵ0ϵe
+1
+bl+1
+ϵe (2l + 1)
+ϵNDl + ϵe (l + 1)
+(S9)
+Cl =
+q
+4πϵ0ϵe
+lr2l+1
+ND
+bl+1
+ϵe − ϵND
+ϵNDl + ϵe (l + 1) .
+(S10)
+
+2
+The electrostatic potential inside the nanodiamond therefore is
+ΦND (r, θ) =
+q
+4πϵ0ϵe
+∞
+�
+l=0
+1
+bl+1
+ϵe (2l + 1)
+ϵNDl + ϵe (l + 1)rlPl (cos θ)
+(S11)
+and the electric field at the center, i.e. for r = 0, can be calculated as
+E (r = 0, θ) =
+q
+4πϵ0
+3
+2ϵe + ϵND
+b
+b3 ,
+(S12)
+if it is used that in cartesian coordinates one has ez = cos θer − sin θeθ with ez the azimuthally symmetric unit vector
+and er and eθ the radial and altitudinal unit vectors.
+A.
+Electric field variance
+The probability of an ion to be located at b witin a sphere of radius R around the nanodiamond is
+p (b) =
+� 3
+4π
+1
+R3−r3
+ND ,
+rND ≤ b ≤ R
+0,
+otherwise.
+(S13)
+It can be easily verified that this distribution is normalized, i.e.
+�
+R3 d3b p (b) = 1. Direct calculation reveals ⟨Ez⟩ = 0
+and therefore
+σ2
+Ez,ion = ⟨E2
+z⟩
+=
+9q2
+(4πϵ0)2
+1
+(2ϵe + ϵND)2
+1
+R3 − r3
+ND
+� 1
+rND
+− 1
+R
+�
+.
+(S14)
+Under the assumption that the electric fields generated by the single ions are uncorrelated, the total fluctuations
+are given by multiplying the above expression with the number of ions inside the sphere. The standard deviation
+σ2
+Ez = cNAV σ2
+Ez,ion of the electric field components with NA Avogadro’s number, c the molar ionic concentration
+and V the volume in which the ions reside therefore is
+σEz =
+|q|
+ϵ0 (2ϵe + ϵND)
+�
+3NA
+4π
+�
+c
+� 1
+rNV
+− 1
+R
+�
+.
+(S15)
+From this it can be seen that the expected electric field fluctuations increase with the molar concentration, i.e.
+σEz ∝ √c.
+II.
+HAMILTONIAN IN ROTATING FRAME
+As derived by Doherty et al. in Ref. [S3], the Hamiltonian of the NV-center in presence of axial magnetic fields Bz
+and electric field components Ei with i = x, y, z and ℏ = 1 is
+ˆHNV =
+�
+D + d∥Ez
+� ˆS2
+z + γeBz ˆSz
++ d⊥
+�
+Ex
+�
+ˆS2
+y − ˆS2
+x
+�
++ Ey
+�
+ˆSx ˆSy + ˆSy ˆSx
+��
+,
+(S16)
+with γe = 2.8 MHz/G the NV’s gyromagnetic ratio [S4] and d∥ = 0.35 Hz · cm/V and d⊥ = 17 Hz · cm/V the axial
+and transverse dipole moments [S5]. By rewriting this Hamiltonian in terms of its frequency contributions βz = γeBz,
+ξz = d∥Ez and ξ⊥ = d⊥
+�
+E2x + E2y and by introducing the electric field polarization φE, which defines the transverse
+electric field projections via ξx = ξ⊥ cos φE and ξy = ξ⊥ sin φE, Eq. (S16) can be rewritten as
+ˆHNV = (D + ξz) ˆS2
+z + βz ˆSz − ξ⊥
+2
+�
+eiφE ˆS2
++ + h.c.
+�
+,
+(S17)
+where ˆS± = ˆSx ± i ˆSy are spin-1 ladder-operators and h.c. means the hermitian conjugate.
+
+3
+The NV-center can be driven by perpendicular (compared to the NV’s symmetry axis) microwave magnetic fields
+of amplitude Ω = γeBd and frequency ωd.
+To exert polarized drives onto the NV-center, two wires which are
+perpendicular to each other (see Fig. 1(a) main text) are operated with a phase φ between each other. This drive can
+be described by an Hamiltonian [S6]
+ˆHd (t) = Ω
+�
+ˆSx cos (ωdt) + ˆSy cos (ωdt + φ)
+�
+.
+(S18)
+Defining phase-factors ϵ± (φ) =
+�
+1 − ie∓iφ�
+/2, similarly to Ref.
+[S6], allows to compactly account for different
+polarizations as ϵ+ = 1 only if φ = −π/2 (i.e. right-circular polarization) and ϵ− = 1 for left-circular polarized
+microwave fields (φ = +π/2). By transforming ˆHNV + ˆHd (t) into a frame oscillating with ωd through the unitary
+U = eiωdS2
+z, one can derive the Hamiltonian under the rotating-wave approximation, which is
+ˆH = ˆH0 + ˆHd
+ˆH0 = (∆ + ξz) ˆS2
+z + βz ˆSz − ξ⊥
+2
+�
+eiφE ˆS2
++ + h.c.
+�
+ˆHd = Ω
+√
+2 (ϵ− |0⟩ ⟨−1| + ϵ+ |1⟩ ⟨0| + h.c.) .
+(S19)
+A.
+Derivation of time-evolution operators
+To allow for the efficient calculation of pulse-sequences, time evolution operators of the free evolution ˆF (τ) and the
+drive ˆR (T) will be derived in the following.
+1.
+Free Evolution
+A possible set of eigenstates of ˆH0 is {|0⟩ , |+⟩ , |−⟩} with
+|+⟩ = cos θ
+2eiφE/2 |1⟩ + sin θ
+2e−iφE/2 |−1⟩
+|−⟩ = sin θ
+2eiφE/2 |1⟩ − cos θ
+2e−iφE/2 |−1⟩ ,
+(S20)
+where tan θ = −ξ⊥/βz, with corresponding eigenenergies ω0 = 0 and ω± = ∆ + ξz ±
+�
+β2z + ξ2
+⊥. The time evolution
+operator of ˆH0 is ˆF (τ) = �
+i={0,±} e−iωiτ |i⟩ ⟨i|, where the sum is performed over all eigenstates of ˆH0. In the basis
+of {|0⟩ , |±1⟩} this is
+ˆF (τ) = |0⟩ ⟨0| + e−iτ(∆+ξz)�
+iξ⊥
+x sin (τx)
+�
+eiφE |1⟩ ⟨−1| + h.c.
+�
++
+�
+cos (τx) − iβz
+x sin (τx)
+�
+|1��� ⟨1|
++
+�
+cos (τx) + iβz
+x sin (τx)
+�
+|−1⟩ ⟨−1|
+�
+.
+(S21)
+Here the frequency of oscillation has been defined as x =
+�
+β2z + ξ2
+⊥.
+2.
+Microwave Drive
+To derive operators which describe the action of the microwave pulses, it will be assumed that these pulses
+exceed all other frequency scales in magnitude, i.e.
+Ω ≫ ∆, βz, ξz, ξ⊥, such that
+ˆH ≈
+Ω
+√
+2
+ˆ�
+Hd with ˆ�
+Hd =
+
+4
+(ϵ− |0⟩ ⟨−1| + ϵ+ |1⟩ ⟨0| + h.c.). By noting that ˆ�
+H
+3
+d = ˆ�
+Hd, the time evolution
+ˆR (t) = e−it ˆ
+Hd =
+∞
+�
+k=0
+�
+−itΩ
+√
+2
+�n
+n!
+� ˆ�
+Hd
+�n
+,
+(S22)
+can be calculated as
+ˆR (t) = |1⟩ ⟨1|
+�
+1 − |ϵ+|2�
++ |−1⟩ ⟨−1|
+�
+1 − |ϵ−|2�
+− ϵ+ϵ− |1⟩ ⟨−1| − ϵ∗
++ϵ∗
+− |−1⟩ ⟨1|
++ cos
+� tΩ
+√
+2
+� �
+|0⟩ ⟨0| + |ϵ+|2 |1⟩ ⟨1| + |ϵ−|2 |−1⟩ ⟨−1| + ϵ+ϵ− |1⟩ ⟨−1| + ϵ∗
++ϵ∗
+− |−1⟩ ⟨1|
+�
+− i sin
+� tΩ
+√
+2
+�
+(ϵ− |0⟩ ⟨−1| + ϵ+ |1⟩ ⟨0| + h.c.) .
+(S23)
+Depending on the polarization, one can induce Rabi oscillations between |0⟩ and either |−1⟩ for φ = π/2 (denoted as
+ˆR+) or |+1⟩ (φ = −π/2, ˆR−),
+ˆR (t)± = |∓1⟩ ⟨∓1| + cos
+� Ωt
+√
+2
+� �
+|0⟩ ⟨0| + |±1⟩ ⟨±1|
+�
+− i sin
+� Ωt
+√
+2
+� �
+|0⟩ ⟨±1| + h.c.
+�
+.
+(S24)
+The system can be driven to both |±1⟩, if a linearly polarized drive is used,
+R (t)0 = 1
+2 (|1⟩ ⟨1| + |−1⟩ ⟨−1| + i |1⟩ ⟨−1| − i |−1⟩ ⟨1|)
++ cos
+� tΩ
+√
+2
+� �
+|0⟩ ⟨0|
++ 1
+2 (|1⟩ ⟨1| + |−1⟩ ⟨−1| − i |1⟩ ⟨−1| + i |−1⟩ ⟨1|)
+�
+− 1 + i
+2
+sin
+� tΩ
+√
+2
+�
+(|0⟩ ⟨−1| + |1⟩ ⟨0| + h.c.) .
+(S25)
+The last expression can similarly be compactly written by noting that (1 ± i) /2 = e±iπ/4/
+√
+2. These operators can
+then be used to describe the action of (polarized) π- and π/2-pulses onto the |ms = 0, ±1⟩-states of the NV-center.
+III.
+FOURIER TRANSFORMATION OF FID-SIGNAL
+Some arbitrary signals f and ˜f in time- and frequency-domain are connected to each other as
+˜f (ω) = FT [f (τ)] =
+� +∞
+−∞
+dτ f (τ) e−iωτ
+FT−1 �
+˜f (ω)
+�
+= 1
+2π
+� +∞
+−∞
+dω ˜f (ω) eiωτ .
+(S26)
+To simplify the calculation of the Fourier transformed FID-signal, one can rewrite FIDξ⊥,φE (Eq. (6) main text) as
+FIDξz,ξ⊥ (τ) = 1
+4
+�3
+2 + 1
+2 cos (2τξ⊥) − cos (τ [ξ⊥ + ξz])
+− cos (τ [ξ⊥ − ξz])
+�
+.
+(S27)
+From Eq. (S26), one sees that FT [cos (τx)] = π [δ (x − ω) + δ (x + ω)] and therefore
+�
+FID (ω) = π
+4
+�3
+2δ (ω) + 1
+2 [δ (2ξ⊥ − ω) + δ (2ξ⊥ + ω)]
+− [δ (ξ⊥ + ξz − ω) + δ (ξ⊥ + ξz + ω)]
+− [δ (ξ⊥ − ξz − ω) + δ (ξ⊥ − ξz + ω)]
+�
+.
+(S28)
+
+5
+IV.
+SIMULATED PULSE SEQUENCES FOR NORMALLY DISTRIBUTED ELECTRIC FIELDS
+FIG. S1. Simulated expected FID-values of FIDξ⊥ (Eq. (5) main text), calculated from 500 individual FID-simulations with
+drive amplitude of Ω = 10 MHz, intrinsic T ∗
+2,int. and electric field components sampled from a normal distribution with mean
+Em and standard deviation σE. Dephasing is considered through a Lindblad-Operator
+�
+1/T ∗
+2,int.Sz. For both mean electric
+field values of (a) 1.0 V/µm and (b) 4.0 V/µm, it is not possible to resolve ξ⊥.
+To understand how fluctuating electric fields alter the FID-signal, we numerically [S7, S8] simulated FIDξ⊥ (Eq. (5)
+main text) for normally distributed electric fields. Hereby, at every timestep at which the time-evolution is calcuated,
+the electric field components are passed from a beforehand sampled normal distribution with mean Em and standard
+deviation σE. It can be seen from Fig. S1 that the average FIDξ⊥ signal decays rapidly to its steady-state value of
+1/2, which is due to the short T ∗
+2 time of 1 µs. For this reason it is proposed to use the Hahn-Echo pulse sequence for
+measurements of strongly fluctuating electric fields.
+0
+50
+100
+150
+200
+τ in µs
+0.0
+0.2
+0.4
+0.6
+0.8
+⟨Hahn (τ)⟩
+Sim.
+Fit
+FIG. S2.
+Example of the average Hahn-echo signal, which was obtained numerically from 1000 individual simulations of
+the pulse sequence shown in Fig. 3(a) (main text) with a mean electric field value of Em = 1.0 V/µm, standard deviation
+σE = 0.75 V/µm, drive amplitude Ω = 10 MHz and intrisic T2,int. = 100 µs together with the fit following Eq. (10) (main text).
+The total T2 value obtained from this fit is T2 = (39.87 ± 0.86) µs.
+As described in the main text, the numerically obtained Hahn-echo trajectories (see Fig. S2 for an example) are well
+fitted by ⟨Hahn (τ)⟩ = 1
+4
+�
+1 − cos (2τξ⊥) e−τ/T2�2. Here both the intrinsic T2,int. = 100 µs and T2,E due to fluctuating
+elecric fields contribute to the total T2 via
+1
+T2
+=
+1
+T2,int.
++
+1
+T2,E
+.
+(S29)
+The latter can be fitted in terms of Em and σE via
+T2,E = αEm
+σ2
+E
+.
+(S30)
+The values of the fit parameter α can be found in Fig. S3.
+
+6
+1
+2
+Em in V/µm
+30
+35
+40
+α
+FIG. S3. Fit parameter α, obtained by numerically fitting Eq. (S29) and Eq. (S30) with T2,int. = 100 µs to the data from Fig. 3
+(main text).
+[S1] R. Messina, “Image charges in spherical geometry: Application to colloidal systems,” The Journal of Chemical
+Physics, vol. 117, no. 24, pp. 11062-11074, Dec. 2002.
+[S2] J. D. Jackson, “Klassische Elektrodynamik,” De Gruyter, Dec. 2006.
+[S3] M. W. Doherty, F. Dolde, H. Fedder, F. Jelezko, J. Wrachtrup, N. B. Manson, and L. C. L. Hollenberg, “Theory
+of the ground-state spin of the NV center in diamond,” Physical Review B, vol. 85, no. 20, p. 205203, May
+2012.
+[S4] E. Abe and K. Sasaki, “Tutorial: Magnetic resonance with nitrogen-vacancy centers in diamond - microwave
+engineering, materials science, and magnetometry,” Journal of Applied Physics, vol. 123, no. 16, p. 161101,
+Apr. 2018.
+[S5] E. Van Oort and M. Glasbeek, “Electric-field induced modulation of spin echoes of N-V centers in diamond,”
+Chemical Physics Letters, vol. 168, no. 6, pp. 529-532, May 1990.
+[S6] P. London, P. Balasubramanian, B. Naydenov, L. P. McGuiness, and F. Jelezko, “Strong driving of a single spin
+using arbitrarily polarized fields,” Physical Review A, vol. 90, no. 1, p. 012302, July 2014.
+[S7] J. Johansson, P. Nation, and F. Nori, “QuTiP: An open-source Python framework for the dynamics of open
+quantum systems,” Computer Physics Communications, vol. 183, no. 8, pp. 1760-1772, Aug. 2012.
+[S8] J. Johansson, P. Nation, and F. Nori, “QuTiP 2: A Python framework for the dynamics of open quantum
+systems,” Computer Physics Communications, vol. 184, no. 4, pp. 1234-1240, Apr. 2013.
+

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+3D ZEROS IN ELECTROMAGNETIC FIELDS
+Alex J. Vernon1†, Mark R. Dennis2‡, and Francisco J. Rodr´ıguez-Fortu˜no1∗
+1Department of Physics and London Centre for Nanotechnology, King’s College London,
+Strand, London WC2R 2LS, UK
+2School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK
+†[email protected]
+‡[email protected]
+∗francisco.rodriguez [email protected]
+Abstract. We present a study of 3D electromagnetic field zeros, uncovering their re-
+markable characteristic features and propose a classifying framework. These are a spe-
+cial case of general dark spots in optical fields, which sculpt light’s spatial structure into
+matter-moving, information-rich vortices, escape the diffraction limit for single-molecule
+imaging, and can trap particles for nanoscale manipulation. Conventional dark spots
+are two-dimensional in two aspects: localised in a plane and having a non-zero out-of-
+plane field component. We focus on non-paraxial fields, where three-dimensional dark
+spots can exist non-stably at fully localised points, making distinct imprints in the flux
+of energy and momentum, and in the light’s polarisation texture. With this work, we
+hope to enhance current dark spot applications, or inspire new ones impossible with
+lower-dimensional zeros.
+1. Introduction
+An optical vortex is the name commonly given to a zero in a complex scalar field, such
+as a component of the electric E or magnetic H field. Vortices in these components occur
+naturally in general 3D monochromatic interference [1], where they are infinitely thin con-
+tinuous strands either extending infinitely through space, or coiled into knotted, un-knotted
+or linked closed loops [2]–[5]. On a vortex strand, the phase of the complex scalar field (with
+zero real and imaginary parts) is undefined creating circulation in the phase of the rest of the
+field. This phase increases in a clockwise or anti-clockwise sense by an integer multiple of 2π
+along any closed loop containing one vortex line. Vortex lines in optics have direct analogues
+in acoustics and water waves, and as a type of topological defect, are related to vortices in
+(super)fluids [6] and in Bose-Einstein condensates [7], and even cosmic strings [8]. Strong
+research interest in optical vortices over the past 30 years, combined with the availability
+of instruments and the flexibility in generating [9]–[12] and structuring [13] vortex-carrying
+beams, has positioned optics to act as a sandbox for exploring topological phenomena that
+appear more broadly across physics.
+When considering the full 3D vector characteristics of an optical field, vortex lines in
+individual field components like Ex, Ey, and Ez are basis-dependent and not so physically
+meaningful. By picturing these different scalar vortex threads permeating the vector field,
+we can appreciate how unlikely it is that the optical field is zero at a point (i.e. E = 0, all
+1
+arXiv:2301.03540v1  [physics.optics]  9 Jan 2023
+
+2
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+three components simultaneously zero) in typical 3D interference (the vortex line in each of
+the three field components would meet at such a zero point, requiring the manipulation of
+three extra parameters beyond the spatial x, y, z). Despite the rarity of zeros in the wild, a
+lower-dimensional version can be readily manufactured in optical beams, and is remarkably
+well-studied. Paraxial doughnut beams have an axial zero in the transverse field surrounded
+by a bright ring, and are used in modern spectroscopy techniques [14], [15] because of the
+zero’s immunity to the diffraction limit. The transverse field effectively consists of one or
+two scalar components with the vortex line along the beam axis, causing the real part of
+the local wavevector to curl around the axis and imbue the beam with intrinsic orbital an-
+gular momentum. The longitudinal field, meanwhile, is non-zero (albeit very small due to
+paraxiality) in the centre of the beam which, therefore, is better imagined not as an exact
+axial zero, but as a dim line of linear polarisation (an L line) polarised parallel to the beam
+direction. This, and its confinement in only two dimensions, stretching along the third, is
+why we refer to the almost-dark centre of the doughnut beam as a two-dimensional zero.
+Its topological index is straightforward to define by counting how many multiples of 2π the
+phases of the transverse components climb through over an enclosing circuit. The intrinsic
+orbital angular momentum carried by doughnut beams is the key property of the spatial
+structure of light that can rotate matter [16], [17] and store information [18]–[20].
+Surprisingly, the fully localised, three-dimensional optical field zero, E = 0, has been left
+largely unexplored. This is probably due to its unstable nature—a perturbation will destroy
+the zero point (i.e. cause the vortices in the three components no longer to coincide). Never-
+theless, such a point is theoretically possible and can be artificially synthesised [21], but very
+little is understood about how it is imprinted into the surrounding field, and there is no clas-
+sifying topological index like the topological charge of a 2D vortex. The 3D electromagnetic
+field zero is the focus of this work. A zero in the E-field alone has codimension 6, requiring
+that the six total degrees of freedom of two real, three-dimensional vectors (the real and
+imaginary parts of the three components E) are suppressed simultaneously. This means 3D
+zeros exist stably in a six-dimensional parameter space, and is why optical field zeros are
+not natural in random interference patterns spanning only three spatial dimensions, being
+hidden by instability. Instead, 3D zeros must be revealed by tuning an additional three
+parameters (this is discussed in [22] for a zero in two electric field components). Some of
+these parameters could be the polarisation components of a plane wave, for example, and in
+fact, 3D zeros can be very easily manufactured and controlled in pure plane wave interfer-
+ence or near fields with a simple technique [21], and their higher dimensional confinement
+could provide a greater degree of precision in dark spot spectroscopy. Due to their electric
+field dependence, the zero in E is coupled to a collection of singularities, each with its own
+topological signature, in various physical quantities associated with the light field includ-
+ing the complex Poynting vector, canonical momentum, spin momentum and spin angular
+momentum. Learning how energy flow and momentum circulate around a 3D vortex could
+inspire applications which would be otherwise unfeasible using typical lower dimensional
+zeros. Alternatively, the magnetic field H may vanish at a point, or more extremely, both
+E and H might simultaneously vanish, giving a true electromagnetic null with codimension
+12. Here, we report the key features of a 3D field electric or magnetic field zero, including
+the way that polarisation singularities are forced to intersect and the flux of the complex
+Poynting vector and canonical and spin momentum. With these findings, for the first time,
+we propose a framework to classify the physically realisable varieties of 3D field zero.
+
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+3
+2. Results
+To contextualise our study, we begin with some brief intuition on the special features
+which we might expect to find near to a 3D zero.
+If either E or H is zero at a point r0, then of that field, say E, the flux of energy, canon-
+ical momentum, spin angular momentum (and other quantities) are zero too. Since these
+fluxes are vector quantities, their direction is singular at r0 and an imprint is made in the
+surrounding space where they are well-defined. In three spatial dimensions, even if these
+fluxes are divergence-less, there is more than one possible (topologically unique) imprint
+which can be left by and characterise the zero in E. The electric field spin is particularly
+interesting, because its zeros (in non-paraxial fields) are co-dimension 2 objects—meaning
+they are one-dimensional continuous lines, defining the threads of pure linear electric polar-
+isation. This continuity should require at least one zero-spin line, an L line, to pass through
+r0. A similar argument can be made for lines of pure circular electric polarisation, except
+that C lines are defined by a complex quadratic equation, E · E = 0, equivalent to a real
+quartic equation, |E · E|2 = 0, which has either zero, two or four real roots. It turns out,
+as we will show, that a given number of C lines and L lines must always intersect in a 3D
+electric field zero. Before reporting these and other findings in detail from mathematical
+argument and analytical simulations in section 2.3 and beyond, the next two subsections 2.1
+and 2.2 provide an overview of polarisation singularities and set out our way of classifying
+3D field zeros using dyadics associated with the field.
+2.1. Overview of Polarisation Singularities in Paraxial and Non-Paraxial Fields.
+L lines and C lines are called polarisation singularities and are the vector version of scalar
+vortex lines in wave fields, existing in light [23]–[25], acoustic and water waves [26] (both
+acoustic and water waves have a vector nature [27], [28]) where some property of the general
+polarisation ellipse is not defined. In 3D fields, polarisation singularities are often described
+as the underlying skeleton which embeds highly complex topologies into the field’s polar-
+isation texture [29], [30].
+Polarisation singularities have been studied in full 3D and in
+paraxial fields [31], where in paraxial fields (considering only the two transverse field com-
+ponents), polarisation is circular at points and linear along lines. Propagating the paraxial
+field (maintaining the transverse polarisation) draws out the C points and L lines in the
+transverse plane into C lines and L surfaces in three dimensions.
+A polarisation ellipse has orthogonal semi-major and semi-minor axes, telling us which
+way the ellipse is oriented. But because a polarisation circle has no semi-major or semi-
+minor axes, at a C point, the orientation of the circle is undefined causing neighbouring
+polarisation ellipses (almost circular) to rotate when tracked along a C point-enclosing loop.
+The ellipse major axis is described throughout space with a line field, in that the axis is
+oriented some way in space but does not point one way or another—an ellipse looks identi-
+cal to its 180 degree rotated self. This means that along the enclosing circuit, the rotating
+ellipses turn continuously through an integer multiple of π radians, rather than 2π, which
+is why C points are assigned a half-integer index. When the field is fully three dimensional
+and the polarisation ellipse is free to tilt in any Cartesian direction, circular polarisation still
+occurs along one-dimensional threads (C lines which are no longer straight as in the paraxial
+case) but the surrounding polarisation ellipses also twist, so that their major axes sweep out
+M¨obius strips [32]–[34]. Analogues of C lines exist in polychromatic fields, shaping the rest
+of the field into other remarkable topological structures [35].
+
+4
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+L lines/L surfaces in paraxial fields (ignoring longitudinal fields) separate regions of left
+and right handed polarisation ellipses.
+In non-paraxial fields, L lines are strictly one-
+dimensional lines (not surfaces) and complement C lines in shaping the surrounding po-
+larisation structure.
+This reduction of dimension to the L entity occurs because, to be
+linearly polarised, the real and imaginary parts of the field (say E = p + iq) need to be
+(anti)parallel (not necessarily equal). If E is paraxial and linearly polarised, then in the
+transverse plane, the ratio of the x components of p and q must equal the ratio of their y
+components—a single condition, dissolving only one degree of freedom of one vector relative
+to the other. If E is non-paraxial, then an extra condition accounting for the ratio of the
+z components of p and q must be satisfied for linear polarisation [23]. Between paraxial
+and full 3D fields, the linear polarisation object’s codimension, which is the dimension of
+the electric spin angular momentum field SE minus the dimension of the L line/L surface
+which lies in SE, increases from one to two. The spin angular momentum of the field is zero
+when linearly polarised, meaning the direction of the normal to the field oscillations cannot
+be defined. Drawing a circuit around an L line, the spin vector rotates through 2π radians
+in a clockwise or anti-clockwise sense and defines the L line’s topological index.
+The characteristics of scalar vortices and C lines and L lines are visualised in Fig. 1.
+2.2. Indexing Point-like Singularities. Polarisation singularities occur equally often
+among the general polarisation ellipses in E and H fields, and need not coincide with each
+other. Phase singularities, C lines and L lines are all indexed by looking at the circulation or
+rotation of a scalar or vector quantity around a loop enclosing the singularity of interest [36].
+All three of these singularities are threads in 3D fields, but the winding number concept can
+be generalised to higher dimensional singularities and calculated for point-like, 3D vector
+singularities via the topological degree. Instead of integrating a quantity associated with
+a line singularity around a 1D closed circuit, for isolated singular points in 3D, we should
+integrate an appropriate quantity over a closed surface enclosing the point singularity. For
+a vector V(rS) on a surface S (rS ∈ S) in 3D real space, for example, the topological de-
+gree of rS �→ V (the mapping from the real space surface rS to V) is a calculation of the
+integer number of times that every possible direction of V is realised (on a sphere) on all
+the points rS on the surface S. As with other kinds of topological singularities in physical
+fields, the easiest realised topological degrees (winding numbers) are ±1. Mathematically, a
+0, ±1 topological degree is the integral of the determinant of the dyadic D(V) of V over S
+divided by A, the area of S,
+(1)
+deg(V) = 1
+A
+�
+S
+det(D(V))dS.
+The dyadic D(V), also called the Jacobian matrix of V, contains the first order spatial
+derivatives of each component of V.
+The sign of the determinant of D(V) equals the
+product of the signs of its eigenvalues. For 3D vectors where D(V) is a 3 × 3 matrix, it
+is possible for drastically different behaviour of V to be hidden under the same topological
+degree. For example, if V(r = 0) = 0 (meaning the direction of V is singular at the origin)
+and we assume that a linear map from an origin-enclosing surface to V has a topological
+degree of −1, then D(V) at r = 0 could have signed eigenvalues (in any order) of + + −
+or − − −. Physically, the origin could either be a saddle point or a sink for V with no
+distinction in topological degree because both + + − and − − − eigenvalues multiply to a
+negative sign. Rather than calculating the topological degree, to try to classify the flux of
+
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+5
+C-l
+in
+e
+L
+-l
+in
+e
+S
+c
+a
+l
+a
+r
+V
+o
+r
+t
+e
+x
+±
+2
+l
+π
+on-plane view
+on-plane view
+Figure 1. Visualisation of scalar and polarisation singularities in a non-
+paraxial electromagnetic field. Scalar vortices (black line) exist in complex
+scalar fields, such as the components of E, where the scalar field is zero
+and its phase is undefined, forming 1D threads in the interference of three
+or more plane waves. Around a scalar vortex line, the phase of the field
+increases by an integer l multiple of 2π in a clockwise or anticlockwise
+sense. Singular lines exist in the complex vector characteristic of E and
+H fields, called polarisation singularities, which include C lines (lines of
+circular polarisation) and L lines (lines of linear polarisation). In a circuit
+around a point on a C line (blue line), in the plane of the polarisation
+circle at that point, nearby polarisation ellipses rotate through an integer
+multiple of π radians. Around an L line (green line), the normal to nearby
+polarisation ellipses rotates by an integer multiple of 2π radians.
+energy and canonical momentum through a 3D optical field zero, we use the signs of the
+eigenvalues of their first order dyadics evaluated in the position of the field zero.
+We use the ideas discussed here to report our findings in the following sub-sections,
+beginning with the six possible ways that C lines and L lines can intersect in a 3D zero.
+2.3. Polarisation Singularities at a 3D Electric Field Zero. We will focus on a 3D
+electric field zero in a position r0, that is E(r0) = 0, and study the nearby strands of circular
+
+6
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+and linear electric polarisation. Identical arguments to those given here could be made for
+magnetic field zeros (H(r0) = 0) and magnetic polarisation singularities, or for simultaneous
+electric and magnetic field zeros (E(r0) = H(r0) = 0) and polarisation singularities of either
+E or H. Any smooth function of r is nearly linear over small distances, which means all
+fundamental behaviour of the electric field in the immediate vicinity of the zero is captured by
+its Jacobian, JE = D(E), a complex 3×3 matrix containing all first-order spatial derivatives
+of Ex, Ey and Ez, evaluated at r0,
+(2)
+JE = D(E) =
+�
+�
+�
+∂Ex
+∂x
+∂Ex
+∂y
+∂Ex
+∂z
+∂Ey
+∂x
+∂Ey
+∂y
+∂Ey
+∂z
+∂Ez
+∂x
+∂Ez
+∂y
+∂Ez
+∂z
+�
+�
+� = (∇ ⊗ E)T .
+The Jacobian of the magnetic field at r0, JH, can be defined similarly. In free space, JE
+and JH are always traceless because E and H are divergence-free. Maxwell’s equations also
+require that if E(r0) = 0, then JH must be symmetric at r0 and vice versa for H(r0) = 0.
+We make a first-order approximation of the electric field vector near r0 with,
+(3)
+˜E = JEv,
+where v = r−r0.
+Nearby C lines emerge in our approximated field wherever ˜E · ˜E = 0, which we may
+calculate using (3) and separate into real and imaginary parts,
+(4)
+˜E · ˜E = (JEv) · (JEv)
+= vT Mv + ivT Nv,
+where M = Re{JT
+EJE} and N = Im{JT
+EJE}. The two terms in equation (4) are quadric
+surfaces connecting constant valued real and imaginary parts of ˜E · ˜E, and the real and
+imaginary surfaces described by setting (4) equal to zero cross in real space where ˜E is
+circularly polarised. The real 3 × 3 matrices M and N are symmetric and always have real
+eigenvalues. Normally, these eigenvalues have signs + + − or − − + (in any order) so that
+the surfaces vT Mv = 0 and vT Nv = 0 are both double cones, vertices touching at v = 0
+as shown in Fig. 2(a). The cones have an elliptical cross section whose ellipticity is constant
+with distance from v = 0 in the linear approximation. Because two ellipses can intersect
+at either zero, two or four points (as shown in the lower part of Fig. 2(a)), there must be
+either zero, two or four C lines passing through the electric field zero. If one matrix, say
+M, is positive or negative definite (all positive or all negative eigenvalues), Re{˜E · ˜E} will
+solely increase or decrease in all outward directions from v = 0. Then, the constant-valued
+surface vT Mv = C becomes an ellipsoid, and vT Mv = 0 is satisfied only at v = 0 so that
+no C lines pass through the 3D vortex.
+To reveal the number of L lines that extend through the 3D electric field zero, we must
+calculate the electric field spin, given by,
+(5)
+SE ∝ Im{E∗ × E} = 2Re{E} × Im{E}.
+When the electric field is linearly polarised (SE = 0), the real and imaginary parts of E
+must be (anti)parallel. Under the approximation (3), this means,
+(6)
+Re{JE}v = λIm{JE}v,
+
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+7
+b
+a
+L-line
+C-line
+cone cross section
+on unit sphere
+Re{E · E} = 0
+E = 0
+Im{E · E} = 0
+2
+0
+4
+Figure 2. Electric polarisation singularities passing through a 3D electric
+field zero at a position r0. (a) Visualisation of why zero, two or four C
+lines must pass through r0. In a first-order approximation, the surfaces
+Re{E · E} = 0 (red) and Im{E · E} = 0 (blue) are double cones, and where
+they intersect, C lines exist. Two double cones intersect along two or four
+lines, or do not intersect at all, which is easy to see by considering the cones’
+cross sections on the unit sphere: ellipses which cross at zero, two or four
+points. (b) Six different examples of electric field zeros created at a position
+r0 (red circle), one per unique combination of C lines and L lines meeting
+there. The C lines are marked by blue regions where E · E ≈ 0 and the L
+lines by the green regions where Im{E∗ × E} ≈ 0. Each field zero is created
+in analytical simulations by designing the polarisation of ten plane waves
+with random wavevectors, wavelength 500 nm, to interfere destructively at
+r0. The plane waves have different polarisations and wavevectors for each
+example zero in (b).
+where λ is a positive or negative scalar. The directions of the L lines crossing through v = 0
+are given by the three eigenvectors of the matrix Im{JE}−1Re{JE}. Since this matrix is
+real-valued, either all three of these eigenvectors are real, corresponding to three L lines, or
+only one of them is real and is accompanied by a conjugate pair of complex eigenvectors. In
+that case, just one L line passes through the 3D zero because v cannot point in a complex
+direction.
+Summarising, either zero, two or four C lines and either one or three L lines always meet
+at r0 in a 3D electric field zero E(r0) = 0. An identical conclusion can be drawn for C lines
+and L lines of the magnetic field for the case of H(r0) = 0. In Fig. 2(b), an example of
+each of the six possible C line/L line combinations through a 3D zero is presented, the zeros
+
+8
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+created in the interference of ten plane waves. Each zero is enforced by separate ensembles
+of ten plane waves with random wavevector directions that are deliberately polarised to
+destructively interfere at a single point.
+2.4. Energy Flux Singularity. The flow of energy in a light field is described by the
+complex Poynting vector,
+1
+2E∗ × H.
+The real part of this vector (often itself called the
+‘Poynting vector’) corresponds to the time-averaged power transfer (sometimes known as
+active power) in the field, while reactive power (associated with oscillations in the transfer
+of power) is accounted for by the less-used imaginary part. We refer to these two real vectors
+as Pr and Pi,
+(7)
+Pr = 1
+2Re{E∗ × H}
+(8)
+Pi = 1
+2Im{E∗ × H}
+When either E or H is zero at a point r0, the complex Poynting vector vanishes, and its
+real and imaginary parts circulate in the space around the zero according to their first-order
+derivatives at r0. The real part Pr is divergence-less in free space where there is no absorption
+or energy generation, and must therefore be organised into a vector saddle point at r0. An
+example flow of active power around a 3D electric field zero created at r0 (E(r0) = 0,
+H(r0) ̸= 0) is given in the top row of panels in Fig. 3, where Pr is plotted on the xy, xz, and
+yz planes coinciding at r0. Although there is no net flow of active power in or out of the zero,
+Pr streamlines can be arranged in two topologically different ways depending on whether
+the signs of the eigenvalues of its first-order dyadic, Im{(JT
+E − JE)J∗
+E} (written electrically
+without prefactors), are + + − or + − −, corresponding to two possible topological orders
+of −1 or +1. One might notice that the imaginary Poynting vector Pi, which is plotted on
+the same planes for the same free space electric field zero at r0 in the lower row of panels of
+Fig. 3, is not divergence-free—in fact, it is physically possible for a source, sink or saddle of
+Pi to exist there, depending on whether E or H is zero. To see why, we first note that using
+Maxwell’s equations in free space (see supplemental information), the imaginary Poynting
+vector can be decomposed into a sum of two terms, one polarisation-independent and one
+polarisation-dependent, each containing electric and magnetic contributions,
+(9)
+Pi = − c2
+2ω ϵ0Re{(JT
+E − JE)E∗}
+= c2
+2ω µ0Re{(JT
+H − JH)H∗}
+= c2
+4ω
+�
+−1
+2ϵ0∇(E∗ · E) + 1
+2µ0∇(H∗ · H)
+�
++ c2
+4ω Re{ϵ0JEE∗ − µ0JHH∗}.
+The first term in Eq. (9) represents the difference in gradient of the electric and magnetic
+energy density of the light field, while polarisation-dependent behaviour of Pi derives from
+the second term since JEE∗ and JHH∗ contain inter-component multiplication. In certain
+cases such as a uniformly polarised standing wave, the second term is zero and the gradient
+of the difference in electric and magnetic energy density determines the direction of reactive
+power flow. Because E∗ · E = |E|2 is a positive real quantity, a 3D zero in E is a source for
+
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+9
+Pi
+Pr
+x
+y
+0.08λ
+x
+z
+y
+z
+y
+x
+x
+y
+z
+z
+Figure 3. Flow of the real (Pr, red) and imaginary (Pi, teal) parts of the
+Poynting vector, 1
+2E∗ × H, on the xy, xz and yz planes coinciding with an
+electric field zero at position r0 (blue circle). The real Poynting vector is
+divergence-free, meaning a vector saddle point of Pr is set up at r0. The
+imaginary Poynting vector is not necessarily divergence-free and can be or-
+ganised in a sink at r0 when E(r0) = 0 (a source is not possible unless
+the magnetic field is zero). Results are generated by designing the polari-
+sation of ten plane waves with random propagation directions to interfere
+completely at r0.
+the vector ∇(E∗ · E) (and likewise for H). Depending on how the polarisation-independent
+and polarisation-dependent terms combine in Eq. (9), the imaginary Poynting vector could
+have non-zero divergence at r0. Note that there is a difference in sign between the electric
+and magnetic terms in Eq. (9), meaning Pi behaves differently for E(r0) = 0, H(r0) ̸= 0
+and H(r0) = 0, E(r0) ̸= 0 and E(r0) = H(r0) = 0 3D zeros. To understand the flow of
+Pi through an optical field zero, we assume a non-dual electric field zero (E(r0) = 0 and
+H(r0) ̸= 0) and make a first-order approximation of Pi, this time referring to the relevant
+linear transformation matrix as the first-order dyadic of the imaginary Poynting vector,
+D(Pi), which is defined identically to JE in Eq. (2) with Pi and its components in place of
+E. Our approximate imaginary Poynting vector is,
+(10)
+˜Pi = D(Pi)v
+
+10
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+where v = r − r0. The dyadic D(Pi) = (∇ ⊗ Pi)T evaluated at r0 is, using the electric
+representation of Pi in Eq. (9) (top line),
+(11)
+D(Pi) = − c2
+2ω ϵ0Re{(JT
+E − JE)J∗
+E}.
+There are no second order derivatives of E in Eq. (11) because E(r0) = 0. Surprisingly,
+D(Pi) cannot have three positive eigenvalues, as justified in the supplemental information.
+The result is that at a 3D electric field zero, Pi is organised into one of two types of sad-
+dle with topological degree 1 or −1, or a sink with topological degree −1, never a source.
+When H(r0) = 0 and E(r0) ̸= 0, the opposite is true because of the dual-asymmetry of the
+imaginary Poynting vector: Pi can form a saddle or source at r0 but not a sink.
+2.5. Orbital Current Singularity. When divided by c2, the real Poynting vector Eq. (7)
+turns into a momentum density, the kinetic momentum density, which, using Maxwell’s
+equations for time-harmonic fields, can be split in to a well-known sum of separate orbit
+and spin contributions [37], [38]. For instance, by substituting (with prefactors) the curl of
+E for H, the kinetic momentum density can be written as,
+(12)
+Π =
+1
+2c2 Re{E∗ × H}
+= 1
+2ω ϵ0Im{E∗ · (∇)E} + 1
+2ω ϵ0∇ × 1
+2Im{E∗ × E},
+where A · (∇)B = Ax∇Bx + Ay∇By + Az∇Bz = JT
+BA, with JB being the Jacobian
+of B defined identically to Eq. (2) (the decomposition is explained in more detail in the
+supplemental information). The first decomposed term is po
+E, the orbital contribution to
+the kinetic momentum density, called the canonical momentum density, imparted by the
+electric field only,
+(13)
+po
+E = 1
+2ω ϵ0Im{E∗ · (∇)E} = 1
+2ω ϵ0Im{JT
+EE∗}.
+Eq. (12) can also be written purely in terms of H and by averaging these equivalent repre-
+sentations of Π, the dual-symmetric canonical momentum density is obtained,
+(14)
+po = 1
+4ω Im{ϵ0E∗ · (∇)E + µ0H∗ · (∇)H}.
+This momentum density definition contains both the electric and magnetic field’s influence,
+and produces the total orbital angular momentum of the field within a volume when r×po is
+integrated. Naturally, the electric and magnetic contributions to (14) become zero whenever
+E = 0 and H = 0. This means that, in a 3D electric field zero positioned at r0, the direction
+of the electric contribution po
+E is undefined and should circulate around r0 in some fashion.
+Of course, while the total canonical momentum density at r0 is not zero when only E = 0,
+we could draw the same conclusions we make here for Eq. (14) rather than Eq. (13) near a
+dual 3D vortex (E(r0) = H(r0) = 0). Note that by normalising E, the argument to Im{}
+in Eq. (13) defines the local electric wavevector [25],
+(15)
+ke
+loc = −ie∗ · (∇)e,
+where e =
+E
+√
+E∗·E. The real part of ke
+loc is the local phase gradient of the electric field, while
+
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+11
+Re{kloc} on yz plane
+xplane = -25 nm
+xplane = -95 nm
+xplane = 0 nm
+xplane = +25 nm
+|Re{kloc}| < 0.1*k
+r0
+Figure 4. Vortex pseudo-line (red) of the real electric local wavevector,
+Re{ke
+loc}, passing though an electric field zero at position r0 (blue circle),
+that is E(r0) = 0 and H(r0) ̸= 0. The red line indicates regions of space
+where |Re{ke
+loc}| < 0.1k, where k = 2π
+λ and λ = 500 nm. The line is roughly
+oriented along the x axis and the electric local wavevector is plotted on four
+different yz planes. The three planes which coincide with the line are −25
+nm, 0 nm, +25 nm in the x direction away from r0, showing clear vortex-
+like circulation of momentum around the axis of the red line. On the fourth
+plane −95 nm away from r0, the vortex-like circulation of Re{ke
+loc} has lost
+some definition, highlighting that Re{ke
+loc} is not exactly zero along a line,
+and only appears line-like near to the E field zero (the only location where
+Re{ke
+loc} is exactly zero is at r0, because Re{ke
+loc} vanishes at points, not
+along lines). Results are generated from interference of ten plane waves
+with random wavevectors, wavelength λ = 500 nm, deliberately polarised
+to create a 3D electric field zero at r0.
+
+12
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+Im{ke
+loc} points in the direction of decreasing electric field intensity. A three-dimensional,
+real vector, Re{kE
+loc} (and therefore canonical momentum density) can vanish at localised
+points in space with non-zero electric field, where a saddle-like circulation of Re{kE
+loc} sur-
+rounds [39], similar to the top row of panels in Fig. 3. But when the electric field vanishes
+and the direction of Re{kE
+loc} is automatically undefined, a different behaviour emerges.
+To understand why, we once again make a first-order approximation, this time of po
+E,
+capturing the electric canonical momentum very near to a 3D electric field zero at r0 in its
+dyadic D(po
+E),
+(16)
+˜po
+E = D(po
+E)v,
+where v = r − r0. The dyadic D(po
+E) = (∇ ⊗ po
+E)T at a general point in space is given by,
+(17)
+D(po
+E) = 1
+4ω ϵ0Im{JT
+EJ∗
+E} + 1
+4ω ϵ0Im{E∗
+xHess(Ex) + E∗
+yHess(Ey) + E∗
+zHess(Ez)},
+where Hess(A) is the Hessian matrix of the scalar field A,
+(18)
+Hess(A) =
+�
+�
+�
+∂2A
+∂x2
+∂2A
+∂x∂y
+∂2A
+∂x∂z
+∂2A
+∂y∂x
+∂2A
+∂y2
+∂2A
+∂y∂z
+∂2A
+∂z∂x
+∂2A
+∂z∂y
+∂2A
+∂z2
+�
+�
+� .
+As E approaches zero, the trace-less matrix D(po
+E) is dominated by the first term in Eq. (17)
+and if evaluated at a location r0 where E(r0) = 0, the linear approximation of po
+E responds
+only to the properties of the matrix in the first term of Eq. (17), Im{JT
+EJ∗
+E}. This is an
+anti-symmetric matrix which always has one zero and two purely imaginary eigenvalues,
+meaning that in the direction of the one real eigenvector of D(po
+E) at r0, the approximated
+electric canonical momentum does not increase at all, producing a zero-momentum line. The
+imaginary eigenvalues of D(po
+E) twists po
+E into a surrounding vortex-like structure. This
+special type of vector field singularity is called a circulation. Fundamentally, the canonical
+momentum should only be zero at confined points in general 3D fields, so this apparent
+vortex line is only preserved locally to the electric field zero at r0, dissolving with distance
+as higher-order derivatives of po
+E become significant (it is, in fact, just a very elongated null
+point of po
+E). The direction of the vortex pseudo-line in the vicinity of the electric field zero
+is also given by the curl of the orbital current,
+(19) D = ∇×po
+E ∝ Re{∇Ex}×Im{∇Ex}+Re{∇Ey}×Im{∇Ey}+Re{∇Ez}×Im{∇Ez}.
+We visualise this feature in Fig. 4, where a 3D electric field zero is created at a point r0
+by deliberately polarising ten plane waves, each with random wavevectors, to destructively
+interfere at r0. The real part of the electric local wavevector, Re{ke
+loc}, the real part of
+Eq. (15), is calculated and the region of space where |Re{ke
+loc}| < 0.1k (k = 2π
+λ ) is revealed
+by a red line approximately 0.1λ in length. The electric local wavevector is proportional to
+po
+E and shows the direction of canonical momentum carried by the electric field. This red
+line is not continuous; Re{ke
+loc} actually vanishes only at r0 but it increases in magnitude
+so slowly in a certain direction (the direction of the real eigenvector of Im{JT
+EJ∗
+E}) that a
+line-like structure of |Re{ke
+loc}| ≈ 0 exists very near to r0, stirring the electric canonical
+momentum into a local vortex. This is shown by the four yz planes on which Re{ke
+loc} is
+plotted in Fig. 4. The real part of the electric local wavevector forms a swirl around the
+red line, a swirl losing definition if the plotting plane is too far from r0. This remarkable
+
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+13
+structure always appears when all three electric field components are zero together at a point.
+2.6. Spin Current. In the decomposition of the kinetic momentum density, Eq. (12), the
+second term is called the spin momentum. It is proportional (and should not be confused
+with) the curl of the spin angular momentum of the electric, magnetic or electromagnetic field
+depending on the representation. Like before, we will focus on the electric representation of
+the decomposed kinetic momentum density, referring to the electric spin momentum with
+ps
+E,
+(20)
+ps
+E = 1
+2ω ϵ0∇ × 1
+2Im{E∗ × E} = − 1
+2ω ϵ0Im{JEE∗}.
+The electric spin momentum is a divergence-free vector whose dyadic D(ps
+E) has three non-
+zero eigenvalues when evaluated in the position of an electric field zero, organising ps
+E into
+one of two types of 3D vector saddle point, just like the real Poynting vector in Fig. 3.
+Expressing, in Eq. (20), the electric spin momentum with the electric field Jacobian reveals
+that only a difference in sign and orientation of JE separates ps
+E from the electric canon-
+ical momentum po
+E, given by Eq. (13). This means that, in a dual electric-magnetic zero,
+E(r0) = H(r0) = 0, where JE is symmetric from Maxwell’s equations, the spin and canoni-
+cal momentum dyadics are equal and opposite, D(ps
+E) = −D(po
+E) (this also means that the
+dyadic of the real Poynting vector is zero). In a first-order approximation of both ps
+E and
+po
+E near r0 in this case, a zero-line exists in exactly the same place for both vectors, and
+around it, ps
+E and po
+E have vortex-like circulation with opposite handedness to each other.
+2.7. Spin Angular Momentum. The dual spin angular momentum, created by the rota-
+tion of the electric and magnetic field vectors, is given by [40],
+(21)
+S = 1
+4ω Im{ϵ0E∗ × E + µ0H∗ × H}.
+The electric and magnetic parts individually describe the ellipticity of the electric and mag-
+netic polarisation ellipses, pointing in the perpendicular direction to the ellipse plane. Once
+more for simplicity, we will focus on the singularity in the electric field spin angular momen-
+tum, SE =
+1
+4ωIm{ϵ0E∗ × E}, left in a 3D electric field zero positioned at r0. The total spin
+angular momentum, Eq. (21), is not zero if only E(r0) = 0, but we could draw similar con-
+clusions for S as we do here for SE when the electric and magnetic fields are simultaneously
+zero at r0.
+Decomposing SE using Maxwell’s equations, we can write its first-order dyadic at r0 in
+terms of the light field Jacobian matrices (see supplemental material),
+(22)
+D(SE) =
+1
+4ω2 ϵ0Re{(JT
+H − JH)J∗
+E}.
+We note that Eq. (22), describing the spatial derivatives of the electric field spin only, de-
+pends on the magnetic field Jacobian matrix JH, which is automatically symmetric whenever
+E = 0 from Maxwell’s equations. The consequence is that JT
+H − JH = 0 and all elements
+of D(SE) at r0 are zero when E(r0) = 0. Higher-order derivatives of SE (Hessian matrices
+for each component) need to be calculated to fully understand the flux of the electric spin
+angular momentum in the neighbourhood of a 3D zero in E.
+
+14
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+Matrix at r0
+Characteristic
+JE
+3D complex Jacobian of the electric field at r0 (Eq. (2))
+1.
+Im{JE}−1Re{JE}
+Number of real eigenvalues is the number of L lines passing through r0
+2.
+Re{JT
+EJE}
+Eigenvectors are the principle axes of the double cone Re{E · E} = 0.
+Number of intersections of this double cone with that of matrix 3 are the number of C lines.
+3.
+Im{JT
+EJE}
+Eigenvectors are the principle axes of the double cone Im{E · E} = 0.
+Number of intersections of this double cone with that of matrix 2 are the number of C lines.
+4.
+Im{JT
+EJ∗
+E}
+Direction of real eigenvector (there is only one) is the axis of the electric local wavevector vortex.
+Imaginary eigenvectors give the handedness of momentum circulation.
+5.
+−Im{JEJ∗
+E}
+Proportional to first-order dyadic of spin current (Eq. (20)).
+Eigenvalue signs give the type of minimum at r0
+6.
+Im{(JT
+E − JE)J∗
+E}
+Proportional to first-order dyadic of real Poynting vector (active power flow).
+Eigenvalue signs give the type of minimum at r0
+7.
+−Re{(JT
+E − JE)J∗
+E}
+Proportional to first-order dyadic of imaginary Poynting vector (reactive power flow).
+Eigenvalue signs give the type of minimum at r0
+Table 1. Summary of the seven dyadics (numbered) which classify the
+vector field singularities organised by a 3D electric field zero.
+2.8. Summary Table. Here, in Table 1, we summarise the seven dyadics which classify
+the number of crossing C lines and L lines, the flux of the real and imaginary parts of the
+Poynting vector, the spin current, and the orientation of the canonical momentum vortex
+pseudo-line existing at a 3D electric field zero, E(r0) = 0 while H(r0) ̸= 0. To characterise
+a magnetic field zero, the matrices can be written magnetically by substituting JE for JH
+(and changing the ‘−’ sign in front of matrix 7 to a ‘+’), in which case matrices 1, 2, and
+3 characterise magnetic polarisation singularities, and matrix 4 and 5 the magnetic local
+wavevector and spin current respectively. In the case of a dual 3D zero, E(r0) = H(r0) = 0,
+matrices 6 and 7 are zero because both JE and JH are symmetric.
+3. Discussion
+Three-dimensional optical field zeros are co-dimension 6 entities which, unlike axial zeros
+in beams, are completely localised, the optical field growing brighter in all outward directions.
+Although they rarely occur naturally in light (requiring three additional parameters beyond
+spatial x, y, z due to their codimension), 3D zeros can be deliberately created in plane wave
+interference or in the near fields of light-scattering matter [21] to reveal the unusual features
+they imprint in the light field’s energy, wavevector and polarisation structures. Both with
+mathematical argument and by creating field zeros in plane wave interference, we showed
+that whenever the electric or magnetic field is zero at a point r0, then some combination of
+zero, two or four C lines, lines of pure circular polarisation, and one or three L lines, lines of
+pure linear polarisation of the field in question, intersect at r0. Likewise, an imprint is made
+at r0 in the surrounding flux of the parts of the complex Poynting vector 1
+2E∗ × H, the local
+wavevector, the spin momentum and spin angular momentum, each organised in a vector
+source, sink or saddle point. The signs of the eigenvalues of the first-order dyadics of each
+quantity at r0 reveal this. Of particular interest is the canonical momentum: while typically
+
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+15
+vanishing at confined points in space, a zero in E or H at r0 twists the canonical momentum
+imparted by that null-containing field into a sub-wavelength, vortex-like structure around
+an axis with an easily calculated direction. We say it is a sub-wavelength object because,
+although it resembles the twisted vortex structures of well-known doughnut beams, it is
+not preserved with increasing distance from r0.
+In the combination of the way energy
+flows through r0 and the number of intersecting polarisation singularities, any 3D field zero
+inscribes one of a discrete number of topologically unique signatures in the electromagnetic
+field. We identify seven dyadics whose spectra could classify all physically possible imprints
+of 3D optical field zeros.
+It is tempting to speculate that a surface enclosing an electric or magnetic field point
+zero might, in addition to the quantities already identified, possess a nonzero topological
+Chern number due to a nontrivial geometric phase 2-form (Berry curvature) resulting from
+the neighbouring polarisation pattern. The appropriate expression for the geometric phase
+2-form is the curl of the local wavevector Eq. (15),
+(23)
+V = ∇ × ke
+loc
+Near an electric field zero, V is anti-symmetric; integrating over a small sphere centred on
+the field zero gives zero. We showed that in its neighbourhood, a 3D zero in E constructs
+a local wavevector vortex with an identifiable axis along which |V| is very large.
+It is
+interesting that even when the complete vector characteristics of light are considered, a
+linear momentum vortex line still persists when all three field components are zero at a
+confined point. This vector field vortex is an analogue to a phase vortex in a complex scalar
+field, with a key difference being that the vector field vortex line is not continuous. Although
+the electromagnetic zero has some topological effects as we described in this paper, it is not
+so strong as to endow a surface around it with a nonzero Chern number.
+We have shown that, despite being unstable to perturbation, 3D zeros of the electric and
+electromagnetic field have topological properties generalising those of scalar vortices and
+polarisation singularities. Further studies might indicate how these properties behave under
+perturbation. We hope that by highlighting the unusual properties of 3D field zeros, we
+can inspire new applications that may be otherwise unachievable with traditionally used,
+lower-dimensional dark spots, such as those in beams or simple standing waves.
+4. Methods
+3D electric field zeros were created in analytical simulations of ten monochromatic inter-
+fering plane waves. In all simulations, ten random wavevectors (all of the same magnitude
+k = 2π
+λ ) were generated, and for each, two orthogonal polarisation basis vectors were defined,
+representing the two electric field degrees of freedom of a plane wave propagating in that
+direction. The ten plane waves were then polarised deliberately to destructively interfere
+and leave a 3D electric field zero at a single confined point, r0, following the procedure given
+in [21]. Let eikj·rˆej,1 and eikj·rˆej,2 be the two orthogonal polarisation states (degrees of free-
+dom) of the electric field of the jth plane wave with unit amplitude at the origin (j ranges
+from 1 to 10, kj is the jth plane wave’s random wavevector with magnitude |kj| =
+2π
+λ ,
+and ˆej,1 and ˆej,2 are two orthogonal unit vectors satisfying ˆej,1 · ˆej,2 = 0, ˆej,1 · kj = 0,
+ˆej,2 · kj = 0). In total, we have twenty available polarisation degrees of freedom, and by
+propagating each plane wave, we can calculate the electric field that each individual degree
+of freedom develops in the position of a desired electric field zero, r0. Now, we multiply
+
+16
+3D ZEROS IN ELECTROMAGNETIC FIELDS
+each degree of freedom by a complex scalar, so that the jth plane wave has components
+xj,1eikj·rˆej,1 and xj,2eikj·rˆej,2. Adding together all scaled degrees of freedom, evaluated at
+r = r0, we have a linear system of three equations, one per component of the total field
+at r0, with complex variables xj,1 and xj,2 representing the amplitude of the orthogonal
+components of the jth plane wave phasor. Setting to zero all three total electric field com-
+ponents at r0, we may solve the system of equations to find the polarisation components of
+each plane wave required for complete destructive interference at r0. Since only three scalar
+conditions are enforced (Ex = 0, Ey = 0 and Ez = 0 for the total field at r0) by twenty
+degrees of freedom, the system is under-determined and seventeen possible solutions exist
+for a 3D zero at r0. Any one of these solutions may be chosen to realise the zero, or, as we
+do, the solutions may be combined in a linear sum with random complex amplitudes. A 3D
+zero could be produced with as few as four plane waves (in fact, a zero could be enforced by
+only two plane waves, but it would not be three-dimensional), though the total field would
+appear less random.
+References
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+Review Letters 102 (2009).
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+3.
+O’Holleran, K., Padgett, M. J. & Dennis, M. R. Topology of optical vortex lines formed
+by the interference of three, four, and five plane waves. Optics Express 14, 3039 (2006).
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+Dennis, M. R., King, R. P., Jack, B., O’Holleran, K. & Padgett, M. J. Isolated optical
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+by Spin-Selective Metasurface Holograms. Advanced Optical Materials 7 (2019).
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+ported by European Research Council Starting Grant ERC2016-STG-714151-PSINFONI.
+6. Author Contribution
+A.J.V. conducted mathematical analyses and simulations; M.R.D. gave direction to and
+supervised the research; F.J.R-F. supervised the research. All authors wrote the manuscript;
+A.J.V. wrote the first draft.
+7. Competing Interests
+The Authors declare no competing interests.
+

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+Locomotion-Action-Manipulation:
+Synthesizing Human-Scene Interactions in Complex 3D Environments
+Jiye Lee
+Hanbyul Joo
+Seoul National University
+{kay2353,hbjoo}@snu.ac.kr
+Figure 1. Our system, LAMA, produces high-quality and realistic 3D human motions that include locomotion, scene interactions, and
+manipulations given a 3D environment and designated interaction cues.
+Abstract
+Synthesizing interaction-involved human motions has
+been challenging due to the high complexity of 3D environ-
+ments and the diversity of possible human behaviors within.
+We present LAMA, Locomotion-Action-MAnipulation, to
+synthesize natural and plausible long term human move-
+ments in complex indoor environments. The key motivation
+of LAMA is to build a unified framework to encompass a
+series of motions commonly observable in our daily lives,
+including locomotion, interactions with 3D scenes, and ma-
+nipulations of 3D objects. LAMA is based on a reinforce-
+ment learning framework coupled with a motion matching
+algorithm to synthesize locomotion and scene interaction
+seamlessly under common constraints and collision avoid-
+ance handling. LAMA also exploits a motion editing frame-
+work via manifold learning to cover possible variations
+in interaction and manipulation motions. We quantitatively
+and qualitatively demonstrate that LAMA outperforms ex-
+isting approaches in various challenging scenarios. Project
+page: https://lama-www.github.io/.
+1. Introduction
+In our daily lives, we can easily observe that humans do
+not live in isolation nor in voids, but continuously interact
+with a complex environment surrounded by many objects.
+Notably, humans perform such a diverse set of daily life
+actions effortlessly. Imagine that we visit a new indoor en-
+vironment (e.g., a hotel room) we have never been before.
+It is expected that we can still easily figure out how to move
+from rooms to rooms, how to sit on a chair, how to open the
+doors of closets, and so on. However, endowing machines
+or virtual humans with such abilities is still a largely unex-
+plored area, despite its importance.
+Synthesizing scene interactions within real-life 3D envi-
+ronments has been a challenging research problem due to
+its complexity and diversity. Human movements in real life
+consists of various types of behaviors, including locomotion
+with avoiding cluttered areas, diverse interactions with 3D
+scenes, and sophisticated object-manipulations. In particu-
+lar, the spatial constraint that arises from real-life 3D envi-
+ronments where many objects are cluttered makes motion
+synthesis highly constrained and complex, and various pos-
+sible arrangements of 3D environments make generalization
+difficult. As human-scene interactions cover a wide range of
+technical challenges, previous approaches have focused on
+sub-problems, such as (1) modeling static poses [17,24,49,
+64,69,71,72] or (2) human object interactions with a single
+target object or interaction type [10, 47, 53–55, 66, 67, 70].
+Recent methods [15,59,60] extend to synthesizing dynamic
+interaction motions in cluttered real-world 3D scenes. How-
+ever, the performance of these methods are fundamentally
+limited due to the lack of 3D ground truth data that contains
+both human motions and paired 3D environments.
+1
+arXiv:2301.02667v1  [cs.CV]  9 Jan 2023
+
+In this paper, we present LAMA, Locomotion-Action-
+MAnipulation, to synthesize natural and plausible long term
+human motions in complex indoor environments. The key
+motivation of LAMA is to build a unified framework to
+include locomotion, interactions with 3D scenes, and ma-
+nipulations of 3D objects, which are the series of motions
+commonly observable in our daily lives. LAMA is based
+on a reinforcement learning framework coupled with a mo-
+tion matching algorithm to synthesize locomotion and scene
+interaction seamlessly while adapting to complicated 3D
+scenes with collision avoidance handling. The reinforce-
+ment learning framework interprets the 3D information of
+the given scene and optimally traverses among the motion
+capture database via motion matching. As an advantage, our
+system does not require any “scene-paired” datasets where
+human movements are captured with the surrounding 3D
+environments simultaneously, which is rarely available. To
+further cover the numerous variations of interaction mo-
+tions, we also exploit an autoencoder based motion editing
+approach to learn the motion manifold space [20] in which
+the editing is performed. Through extensive quantitative
+and qualitative evaluations against existing approaches, we
+demonstrate that our method outperforms previous methods
+in various challenging scenarios.
+Our contributions are summarized as follows: (1) we
+present the first method to generate realistic long term mo-
+tions combined with locomotion, interaction with scene,
+and manipulation in complicated cluttered scenes; (2) we
+propose a novel, unified framework that synthesizes loco-
+motion and human-scene interactions in a seamless man-
+ner, by introducing scene interpretation terms to a reinforce-
+ment learning based approach to automatically generate op-
+timal transitions; and (3) our outputs show the state-of-the-
+art motion synthesis quality with longer duration (more than
+10 sec) than previous methods.
+2. Related Work
+Generating Human-Scene Interactions. Generating
+natural human motion has been a widely researched topic
+in the computer vision community. Early methods focus on
+synthesizing or predicting human movements by exploiting
+neural networks [11,13,35,35,38,46,56,58]. However, these
+approaches primarily address the synthesis of human mo-
+tion itself, without taking into account the surrounding 3D
+environments. Recent approaches begin to tackle modeling
+and synthesizing human interactions within 3D scenes, or
+with objects. Most of the researches focus on statically pos-
+ing humans within the given 3D environment [16,24,69,71],
+by generating human scene interaction poses from vari-
+ous types of input including object semantics [17], im-
+ages [21,23,64,65,68], and text descriptions [49,72].
+More recently, there have been approaches to synthesize
+dynamic human object interactions (e.g., sitting on chairs,
+Encoder
+Decoder
+Task-Adaptive Motion Editing
+Motion Generation
+Action 
+Controller
+3D Scene
+Interaction 
+Cue     
+Action 
+Posture
+Motion
+Synthesizer
+Optimization
+Figure 2. Overview of LAMA.
+carrying boxes). Starke et al. [53] introduce an autoregres-
+sive learning framework with object geometry-based envi-
+ronmental encodings to synthesize various human-object
+interactions. Later work [15, 70] extends this by synthe-
+sizing motions conditioned with variations of objects and
+contact points. Other approaches [47, 54, 55, 66, 67] focus
+on generating natural hand movements for manipulation,
+which is extended by including full body motions [54].
+Physics-based character control to synthesize human object
+interactions has been also explored in [8,10,39,47,66]. Al-
+though these approaches cover a wide range of human ob-
+ject interactions, most of them solely focus on the relation-
+ship between human and the target object without long-term
+navigation in cluttered 3D scenes.
+More recent approaches include generating natural hu-
+man scene interactions within a complex 3D scene clut-
+tered with many objects [6, 59–61], closely related to ours.
+These methods are trained using human motion datasets
+paired with 3D scenes, which require both ground truth mo-
+tions and simultaneously captured 3D scenes for supervi-
+sion. Due to such difficulties, some methods exploit syn-
+thetic datasets [6,61] or data fitted from depth videos [60].
+In previous approaches [15,59], navigation to move through
+cluttered environments is often performed by a separate
+module via a path planning algorithm (e.g., A∗ algorithm)
+by approximating the volume of a human as a cylinder. This
+path planning based methods approximate the spatial infor-
+mation of the scene and the human body and therefore have
+limitations under highly complex conditions.
+Motion Synthesis and Editing. Synthesizing natural
+human motions by leveraging motion capture data has also
+been a long-researched topic in computer graphics. Some
+approaches [26,37] construct motion graphs, where plausi-
+ble transitions are inserted as edges and motion synthesis is
+done by traversing through the constructed graph. Similar
+approaches [31, 51] connect motion patches to synthesize
+interactions in a virtual environment or multi-person inter-
+actions. Due to its versatility and simplicity, a number of
+variations have been made on the graph based approach,
+such as motion grammar [22] which enforces traversing
+rules in the motion graph. Motion matching [5, 9] can also
+be understood as a special case of motion graph traver-
+sal, where the plausible transitions are not precomputed but
+searched during runtime. Recent advances in deep learning
+allow to leverage motion capture data for motion manifold
+2
+
+rendel
+reset
+prev
+play
+nextlearning [19, 20, 52]. Autoregressive approaches based on
+variational autoencoders (VAE) [36, 46] and recurrent neu-
+ral networks [14,29,41] are also used to forecast future mo-
+tions based on past frames. These frameworks are general-
+ized to synthesizing a diverse set of motions including lo-
+comotion on terrains [19] mazes [36], action-specified mo-
+tions [46], and interaction-involved sports [29, 41]. Neural
+network-based methods are also reported to be successful
+in various motion editing tasks such as skeleton retarget-
+ing [2], style transfer [3,20], and inbetweening [14].
+Reinforcement learning (RL) has also been successful in
+combination with both data-driven and physics-based ap-
+proaches for synthesizing human motions. Combined with
+data-driven approaches, these RL frameworks serve as a
+control module that generates corresponding motions to a
+given user input by traversing through motion graphs [28],
+latent space [34, 36, 57], and precomputed transition ta-
+bles [30]. Deep reinforcement learning (DRL) has been
+widely used recently in physics simulation as well to syn-
+thesize physically plausible movements with a diverse set
+of motor skills [4,32,41,43–45,62].
+3. Method
+3.1. Overview
+Our system, dubbed as LAMA, outputs a sequence of
+human poses M = {mt}T
+t=1 by taking the 3D surrounding
+cues W and desired interaction cues Φ, as inputs:
+M = LAMA(W, Φ).
+(1)
+The output posture at time t, mt = (p0, r1, ..., rJ) ∈
+R3J+3, is represented by a concatenated vector of global
+root position p0 ∈ R3 and local joint orientations of J
+joints where each j-th joint is in angle-axis representations
+rj ∈ so(3). Throughout our system, the skeleton tree struc-
+ture and joint offsets are fixed and shown in Fig. 3 (a). We
+represent the 3D environments W = {wi} as a set of 3D
+object and environment meshes, including the background
+scene mesh and other object meshes targeted for manip-
+ulation. The interaction cues, Φ = [φ1, φ2, ...φn], are an
+ordered list of desired interaction inputs φi = {qj}j∈Ji
+where qj ∈ R3 indicates desired positions of j-th joint,
+and Ji is a set of specified joints for interaction (in prac-
+tice, few joints such as root 1 or end-effectors). Examples of
+the 3D environment W and interaction inputs φi are shown
+in Fig. 5 (a). Intuitively, φi specifies the expected positions
+of selected joints of the human character. Note that we do
+not specify the exact timing of the interaction, as the timing
+is automatically determined by our action controller. More
+details are addressed in Sec. 3.4.
+To synthesize locomotion, interaction, and manipulation
+together, LAMA is designed via a three-level system com-
+1For root, orientation in angle-axis representation is also included in φ.
+Figure 3. (a) Skeleton with joints and box nodes. (b) Automatically
+detected collision points (colored as red).
+posed of the action controller A and the motion synthesizer
+S, followed by a manifold-based motion editor E. By taking
+3D scene cues W and desired interaction cues Φ as input,
+the action controller A makes the use of a reinforcement
+learning (RL) framework by training the control policy π to
+sample an action at time t, π(at|st, W, Φ), where at con-
+tains the plausible next action cues including predicted ac-
+tion types and short-term future forecasting. st is the state
+cues to represent the current status of human characters in-
+cluding its body posture, surrounding scene occupancy, and
+current target interaction cue, which can be computed via
+a function ψ, st = ψ(mt−1, mt, W, Φ). Intuitively, action
+controller A predicts the plausible next action cues at by
+considering the current character-scene state st. The gener-
+ated action signals at from the action controller A is pro-
+vided as the input for the motion synthesizer S, which then
+determines the posture at the next time step mt+1, i. e.,
+S(mt, at) = mt+1. Afterwards, the character’s next state
+can be computed again via st+1 = ψ(mt, mt+1, W, Φ),
+which is input to the action controller recursively.
+Followed by the initial motion generation part from A
+and S, our system furthermore applies a motion editor
+E(M) = ˜M, where ˜M = { ˜Mt}T
+t=1 is the edited motions
+to further express the motions involving complex human-
+object interactions such as manipulation (e.g. moving ob-
+jects, opening doors). Fig. 2 shows the overview of LAMA.
+3.2. Scene-Aware Action Controller
+Based on reinforcement learning, our action controller
+A enables the character to perform locomotion and desired
+actions with fulfilling the interaction cues Φ and avoiding
+collisions in the 3D environment W. A is a trained control
+policy π where π(at|st, W, Φ). Different from previous
+approaches where navigation and scene-object interactions
+(e.g., sitting) are performed by separate modules [15, 59],
+our RL-based framework performs both in a unified way
+with a common objective by automatically determining the
+transition from navigation to specific actions. As a key ad-
+vantage, LAMA can be robustly generalized to challenging
+unseen 3D clutters in long-term human motion synthesis
+and also outperforms previous methods by avoiding colli-
+sions throughout the whole process, including navigation
+3
+
+Joint
+- Node
+Jointand interaction.
+State. The state st = ψ(mt−1, mt, W, Φ) at time t is
+a feature vector representing the current status of the hu-
+man character. st = (sbody
+t
+, sscene
+t
+, sinter
+t
+) is composed of
+body configuration sbody, 2D scene occupancy sscene, and
+desired current target interaction sinter. Body configuration
+sbody = {r, ˙r, θup, h, pe} includes r, ˙r ∈ RJ′×6 that are the
+joint rotations and velocities respectively for the J′ joints
+excluding the root in 6D representations [73], θup ∈ R that
+is the up vector of the root (represented by the angle w.r.t
+the Y-axis), h ∈ R that is the root height from the floor,
+and pe ∈ Re×3 that is the end-effector positions in person-
+centric coordinate (where e is the number of end-effectors).
+sscene = {gocc, groot} includes scene occupancy informa-
+tion in 2D floor plane, as shown in Fig. 4. gocc ∈ Rn2 rep-
+resents the 2D occupancy grid on the floor plane of neigh-
+boring n cells around the agent and groot ∈ R2 denote the
+current 2D global root position of the character in the dis-
+cretized grid plane. sinter is an element of Φ and represents
+the interaction cue the character is currently targeting, that
+is sinter = φi.
+Action. Given the current status of the character st,
+the control policy π outputs the feasible action at
+=
+(atype
+t
+, afuture
+t
+, aoffset
+t
+). atype
+t
+provides the probabilities
+of next action type among all possible actions, determining
+the transition timing between actions (e.g., from locomo-
+tion to sitting). afuture
+t
+predict future motion cues such as
+plausible root position for the next 10, 20, and 30 frames.
+aoffset
+t
+is intended to update the raw motion data searched
+from the motion database in motion synthesizer module S.
+Intuitively, our learned control policy generates an optimal
+posture offset aoffset
+t
+which is applied to the closest plau-
+sible raw posture in the database. This enables the character
+to perform more plausible scene-aware human poses, allow-
+ing our system to be generalized to any unseen 3D scenes
+given a limited amount of motion capture data. More details
+are addressed in Sec. 3.3.
+3.3. Motion Synthesizer
+Given the current motion output mt and actions sig-
+nals at from the action controller A as inputs, the mo-
+tion synthesizer produces the next plausible character pos-
+ture: S(mt, at) = mt+1. As the first step, motion synthe-
+sizer searches for the motion from a motion database that
+best matches the closest motion feature, then modifies the
+searched raw motion to be more suitable for the scene envi-
+ronment. To this end, motion synthesizer’s output mt+1 is
+in turn fed into the action controller recursively. We exploit
+a modified version of motion matching algorithm [5, 9, 18]
+for the first step of motion synthesis. In motion matching,
+motion synthesis is performed periodically by searching the
+most plausible next shot motion segments from a motion
+DB, and compositing them into a long connected sequence.
+Figure 4. Visual representation of the 2D occupancy grid near the
+root. Grid on the right represents top view of the grid. Blue in-
+dicates root position while gray represents the space is occupied.
+Occupied cells near the root are colored as black.
+Motion features. Motion feature represents the charac-
+teristic of each frame in the short motion segment and is
+computed as f(m) = {{pj}, { ˙pj}, θup, c, ofuture}. From
+a posture m, the positions and velocities pj, ˙pj
+∈ R3
+are extracted for the selected joints j ∈ {Head, Hand,
+Foot}, which are defined in a person-centric coordinate
+of m. θup ∈ R3 is the up-vector of the root joint, and
+c ∈ {0, 0.5, 1} indicates automatically computed foot con-
+tact cues of the left and right foot (0 for non-contact, 1 for
+contact, 0.5 for non-contact but close to the floor within
+a threshold). ofuture = {{pdt
+0 }, {rdt
+0 }} contains the cues
+for the short-term future postures, where pdt
+0 and rdt
+0 are
+the position and orientation of root joint at dt frames later
+from the current target frame. ofuture are computed in 2D
+XZ plane in person-centric coordinate of the current tar-
+get motion m, and thus pdt
+0 , rdt
+0
+∈ R2. The selected fu-
+ture frames are action-type specific, and for locomotion we
+extract 10, 20, and 30 frames in the future (at 30Hz) fol-
+lowing [9]. Intuitively, the motion feature extracts the target
+frame’s posture and temporal cues by considering neigh-
+boring frames 2. We pre-compute motion features for every
+frame of the motion clips in the motion database. Motion
+feature of the current state of the character, or the query
+feature, is also computed in the same way based on posture
+mt−1, mt and afuture
+t
+produced by the action controller,
+that is xt = f(mt−1, mt, atype
+t
+, afuture
+t
+). The component
+afuture
+t
+serves as ofuture in the query feature, which can be
+understood as the action controller providing cues for pre-
+dicted future postures.
+Motion searching and updating. The query motion fea-
+ture xt from the current character is computed as addressed
+above, and let the motion features in motion database de-
+noted as yk for the k-th clips in the DB. Motion searching
+finds the best matches in the motion database by computing
+the weighted euclidean distances between the query feature
+and DB features:
+k∗ = arg min
+k
+||wT
+f (xt − yk)||2,
+(2)
+where wf is a fixed weight vector to control the impor-
+2In practice, the input of feature extractor function f should take into
+account the motions of neighboring timesteps.
+4
+
+2D
+occupancy gridtance of feature elements. After finding the best match ˆmk∗
+from motion database, the motion synthesizer further up-
+dates it with the predicted motion offset aoffset
+t
+from at,
+that is τ( ˆmk∗+1, aoffset) = mt+1, where ˆmk∗+1 is the
+next plausible character posture and τ is an update function
+to update selected joints. In practice, the motion searching
+is performed periodically (e.g., every N-th frames) to make
+the synthesized motion temporally more coherent.
+3.4. Learning for Scene-Aware Action Controller
+In the reinforcement learning framework, the objective is
+to learn the optimal policy which maximizes the discounted
+cumulative reward. In our method, we design rewards to
+guide the agent to perform locomotion towards the target
+objects (e.g., sofa) and also perform desired interaction with
+the object (e.g., sitting). In particular, our RL-framework
+performs both navigation and interaction with common con-
+straints (e.g., smooth transitions, collision avoidance).
+Our reward function consist of the following terms:
+Rtotal = wtrRtr + wactRact + wregRreg,
+(3)
+where wtr, wact, and wreg are the weights to balance among
+reward terms. The trajectory reward Rtr is obtained when
+the character moves towards the desired interaction input φ
+while meeting the spatial constraints from the surrounding
+3D scene, described below:
+Rtr = rcoli · rpos · rroot, where
+(4)
+rcoli = exp
+�
+− 1
+σ2
+coli
+�
+b∈B
+wbρ(b, W)
+�
+,
+(5)
+rpos = exp
+�
+�−
+1
+σ2
+root
+�
+j∈J
+∥p0 − qj∥2
+�
+� ,
+(6)
+rvel =
+�
+1
+when ˙proot ≥ σth
+σvel∥ ˙p0∥2
+else.
+(7)
+The collision-avoidance reward rcoli penalizes collisions
+with 3D scenes. As depicted in Fig. 3 (a), body limbs in
+the skeletal structure are represented as a set of box-shaped
+nodes B with a fixed width, where each element b ∈ B is
+a 3D box representation of legs and arms (we exclude torso
+and head). The function ρ(b, W) detects the collision be-
+tween edges of a box-shaped node b with 3D scene meshes
+W and returns the number of intersection points. (Fig. 3
+(b)). wb is the weights to control importance of each limb b.
+The collision-avoidance reward is maximized when no pen-
+etration occurs, making the control policy π to find the opti-
+mal trajectory and pose offset to avoid physically implausi-
+ble collisions and penetrations. rpos are obtained when the
+agent moves to reach the targeting interaction cue φ, by en-
+couraging agent’s root position p0 to be closer to the target
+interaction cue {qj}. rvel encourages the character to move
+by penalizing when the root velocity ˙proot is less than a
+threshold σth. σcoli, σroot, and vel are weights to control
+the balance between terms.
+Action reward Ract enforces the synthesized motion to
+fulfill the given interaction cue φ = {qj}:
+Ract = rinter · r∆t · r∆v,
+where
+rinter = exp
+�
+�−
+1
+σ2
+inter
+�
+j∈J
+∥pj − qj∥2
+�
+� ,
+r∆t = exp
+�
+−σ2
+∆tCtr
+�
+,
+r∆v = exp
+�
+−σ2
+∆vCvel
+�
+,
+(8)
+where interaction reward term rinter is maximized when the
+performed action meets the positional constraints provided
+by interaction cues. Smoothness reward terms r∆t and r∆v
+minimizes the transition cost, which is based on the subpart
+of the feature distances defined in Eq. 2, where Ctr is the
+weighted feature distances of pj, θup, and c, and Cvel is
+from ˙p. These are intended to penalize the case where the
+character makes abrupt changes.
+Regularization reward Rreg penalizes the aoffset
+t
+exces-
+sively modifying the original posture brought from the mo-
+tion synthesizer, denoted as ˆmt, and maintains temporal
+consistency among frames.
+Rreg = exp
+�
+−
+1
+σ2reg
+�
+∥ ˆmt − mt∥2 + ∥mt − mt−1∥2��
+.
+It is reported that [33, 41] multiplying rewards with con-
+sistent goals are suitable for learning, as the reward is re-
+ceived when the conditions are simultaneously met. Fur-
+thermore, to accelerate learning, we use early termination
+conditions [43] and limited action transitions. The episode
+is terminated when the character moves out of the scene
+bounding box, or when the collision reward rcoli is under a
+certain threshold. Also, the action controller first checks in
+advance whether the action signal is valid when it makes
+transitions from locomotion to other actions. When the
+nearest feature distance of Eq. 2 in the motion synthesizer
+is over a certain threshold, the action controller discards the
+transition and continues navigating. The control policy is
+learned through Proximal Policy Optimization (PPO) algo-
+rithm [50].
+3.5. Task-Adaptive Motion Editing
+Interaction includes a massively diverse pool of mo-
+tions, and these variations cannot be fully handled by lim-
+ited amount of motion database. In order to cover such di-
+versity, we include a task-adaptive motion editing module
+in our motion synthesis framework. The goal of our edit-
+ing module E is (1) to edit motion M to fit into diverse
+5
+
+Figure 5. Visual representation of system input Φ, W and output
+motion sequence. On the left, interaction cues are shown as cyan
+spheres and arrows (indicating orientation). The right is the syn-
+thesized human motion ˜
+M.
+target object geometries (e.g., sitting on a chair with dif-
+ferent height), and (2) to generate additional hand move-
+ments for manipulation (e.g., grasping). In particular, in the
+case of manipulation, additional interaction cue φ can be
+provided to enforce an end-effector (e.g., a hand) to fol-
+low the desired trajectories to express the manipulation task
+on the target object, as shown in Fig 8 (left). The edited
+motion ˜M = E(M) should not only fulfill the sparsely
+given positional constraints, but also preserve the temporal
+consistency between frames and spatial correlations among
+joints in order to maintain its naturalness. We adopt the mo-
+tion manifold learning approach with convolutional autoen-
+coders [20] to compress motion to a latent vector within a
+motion manifold space. Motion editing is done by searching
+for an optimal latent vector among the manifold. For train-
+ing the autoencoder, motion sequence, which we denote as
+X converted from M, is represented as a time-series of hu-
+man postures by concatenating joint rotations in 6D repre-
+sentations [73], root height, root transform relative to the
+previous frame projected on the XZ plane, and foot contact
+labels. The encoder and decoder module are trained based
+on reconstruction loss, ||X − Ψ−1(Ψ (X)) ||2, where Ψ is
+the encoder and Ψ−1 is the decoder.
+The latent vector from the encoder z = Ψ(X) repre-
+sent the motion manifold space by preserving the spatio-
+temporal relationship among joints and frames within the
+motion sequence. As demonstrated in [20], editing motions
+in this manifold space ensures the edited motion to be re-
+alistic and temporally coherent. To this end, we find the
+optimal latent vector z∗ by minimizing a loss function L
+by constraining the outputs motions to follow the interac-
+tion constraint φ. We also include additional regularizers in
+L so that the output motion to maintain the foot locations
+and root trajectories to the original motions. See supp. mat.
+for more details on L. Finally, the edited motion ˜M can be
+computed via Ψ−1(z∗).
+4. Experiments
+We evaluate LAMA’s ability on synthesizing long-term
+motions with various human-scene and human-object inter-
+Method
+Plausibility
+Naturalness
+Slip
+Penetration
+FDtotal
+FDroot
+FDjoint
+Wang et al. [60]
+5.13
+3.88
+1.38
+0.45
+0.93
+Wang et al. [60]*
+24.8
+4.58
+1.44
+0.44
+1.00
+SAMP [15]
+10.5
+12.49
+1.25
+0.30
+0.95
+LAMA (ours)
+5.21
+1.52
+1.22
+0.31
+0.91
+Table 1. Baseline comparison Foot slip loss (cm, ↓) averaged over
+all frames. Penetration loss(percentage, ↓) is counted based on in-
+tersection points of the 3D environment and the skeleton. Natural-
+ness score is based on fr´echet distance (FD ↓). Wang et al. with an
+asterisk indicates without post-processing optimization.
+actions involved. We exploit an extensive set of quantitative
+metrics and perceptual study to evaluate the physical plau-
+sibility and naturalness of the synthesized motion.
+Dataset. For constructing the database for the motion
+synthesizer, motion capture data are selectively collected
+and refined from Ubisoft La Forge [14], COUCH [70],
+and SAMP [15]. All the data used in this system are mo-
+tion capture data (in bvh format) with no scene or ob-
+ject related information, and are retargeted into a unified
+skeletal structure with MotionBuilder. We use PROX [16]
+and Matterport3D [7] datasets for 3D environment and
+SAPIEN [63] object meshes for manipulation. Our code and
+pre-processed data will be publicly released.
+Implementation Details. The policy and the value net-
+work of the action controller module consists of 4 and
+2 fully connected layers of 256 nodes, respectively. The
+encoder and decoder of the task-adaptive motion editing
+module consist of three convolutional layers. Adam opti-
+mizer [25] is used for training and optimization. We use
+Nvidia RTX 3090 for training the action controller and the
+motion editing module. It takes 10 to 80 minutes to learn
+a single control policy, where the training time mainly de-
+pends on how difficult the interaction cues are to achieve.
+For optimization in the motion editing module, it takes 3 to
+4 minutes for 500 epochs. See supp. mat. for more detail.
+4.1. Experimental Setup
+Evaluation metrics. Quantifying motion synthesis quality
+is challenging due to the lack of ground-truth data or offi-
+cial evaluation metrics. We try to quantify them in terms of
+physical plausibility and naturalness.
+• Physical plausibility: We use contact and penetration
+metrics to evaluate the physical plausibility of the synthe-
+sized motions. Contact loss penalizes the foot movement
+when the foot is in contact. Since foot contact is a critical
+element in dynamics, contact-based metric is closely related
+in determining the physical plausibility of motions. Pene-
+tration loss (“Penetration” in Table 1) measures implausible
+cases when the body penetrates the objects in the scene. We
+compute penetration metric by counting frames where the
+6
+
+interaction cue Φ2
+interaction cue ΦFigure 6. Comparison with LAMA (left) and LAMA without col-
+lision reward (right). As shown in the right, without collision re-
+ward the character fails to avoid collisions with obstacles (marked
+as red).
+intersection points (Sec. 3.4) goes over a certain threshold. 3
+• Naturalness: We measure the naturalness of the synthe-
+sized motions by measuring the Fr´echet distance, as re-
+ported in [15, 35, 40] between the synthesized motion and
+motions from motion capture data. Features are extracted
+from motion sequences and the Fr´echet distance is com-
+puted with the extracted features. We measure the natural-
+ness of character root movements FDroot, including root ori-
+entation and velocity, and character joint rotations FDjoint.
+Baselines. We compare our LAMA with the state-of-the-art
+approaches as well as variations of ours.
+• Wang et al. [60] is the state-of-the-art long term mo-
+tion synthesis method for human-scene interactions within
+a given 3D scene. We use the author’s code for evaluation.
+As Wang et al. uses optimization to post-process the synthe-
+sized motion to improve foot contact and reduce collisions,
+we both compare Wang et al. with and without optimization.
+• SAMP [15] generates interactions which can be general-
+ized not only for object variations but also random starting
+points within a given 3D scene. SAMP explicitly exploits a
+path planning module to navigate through cluttered 3D en-
+vironments.
+• Ablative baselines We perform ablation studies on the ac-
+tion controller and task-adaptive motion editing module. We
+perform ablation studies on the scene reward rcoli, and ac-
+tion offset aoffset
+t
+to present the contribution of both terms
+on our system’s capability to generate scene-aware motions.
+We also compare our method without the transition reward
+r∆t and r∆v terms (Sec. 3.4) in the action controller. Fi-
+nally, we demonstrate the strength of our task-adaptive mo-
+tion editing module to edit motions naturally (Sec. 3.5) by
+comparing with inverse kinematics (IK).
+4.2. Comparisons with Previous Work
+Quantitative Evaluation. We compare methods in 6
+different scenarios from various 3D scenes in the PROX
+dataset [16]. Foot contact is automatically labeled based on
+310 for legs and 7 for arms
+Figure 7. Comparison with LAMA (left) and LAMA without ac-
+tion offset (right). The character in original LAMA moves forward
+while tilting its arms to avoid collision with walls, while in LAMA
+without action offset does not.
+positional velocity of the foot joint. Foot slip metric is mea-
+sured by foot joint positions. To compute penetration metric
+in a fair way, SMPL-X outputs of Wang et al. and SAMP are
+converted to box-shaped skeletons as in ours and intersec-
+tion point are counted. Table 1 shows the results.
+As shown, our LAMA outperforms Wang et al both in
+naturalness and physical plausibility. It is noted that Wang
+et al performs optimization as post-processing to explic-
+itly minimize foot slip, and yet LAMA still shows on-par
+performance against it (and better in all other metrics).
+Compared with SAMP, our method shows much better re-
+sults in plausibility metrics (both Slip and Penetration),
+and shows slightly better performance in naturalness. Apart
+from SAMP which relies on a separate navigation mod-
+ule, our RL-based action controller handles collisions in the
+same way of scene-interaction and shows much better per-
+formance in in complex and cluttered 3D scenes.
+A Human Study. To further validate our results, we
+compare the quality of our output over other baselines,
+Wang et al. and SAMP, through A/B testing from human
+observers. For the study, we choose 5 scenarios from dif-
+ferent indoor scenes, and render the results of each method
+using the exactly same view and 3D characters, so that they
+cannot be distinguished from the appearance side. We build
+two separate sets, where in each set the result videos of
+our method are shown with each competitor side by side
+in a random order. Human observers are asked to choose a
+motion clip that is more human-like and plausible in the
+given 3D scene. We perform each set of tests with non-
+overlapping 15 participants. See our supp. mat. for more
+details about the study setup. As the result, the outputs of
+our method are preferred by the majority (more than 50%
+voting) in all cases. By considering all votes independently,
+our method are preferred 80.0% over SAMP and 97.3%
+over Wang et al.’s work. In particular, we found that our
+method greatly outperform the competing methods in terms
+of the naturalism of foot stepping, transition between loco-
+motion and action, and collision avoidance with the scenes.
+See our supp. videos for more results.
+7
+
+LAMA
+LAMA w/o collision rewardoffset
+LAMA w/o a
+LAMA
+LAMAFigure 8. (a) Comparison with LAMA (top) and LAMA without
+manifold and replaced with IK (bottom) of a character opening the
+toilet lid. (b) Comparison with LAMA (top) and LAMA without
+motion editing (bottom) in sitting.
+4.3. Ablation Studies
+Ablation Studies on Action Controller. We quantita-
+tively compare the original LAMA and the LAMA without
+collision reward rcoli. We intend to demonstrate the role of
+rcoli that enforces the action controller to search for optimal
+actions for generating motions without collisions. Ablation
+studies are done in 5 PROX scenes. In the original LAMA,
+penetrations occur in only 1.1% of the frames among the
+whole motion sequences, while the ratio is 15.7% in LAMA
+without collision reward. The result supports that the colli-
+sion reward rcoli enforces the action controller to compute
+optimal actions for synthesizing body movement according
+to the spatial constraint of the given 3D scene. Example re-
+sults are shown in Fig. 6.
+We also compare the contribution of other components
+in the action controller module in generating natural inter-
+actions. As seen in Fig. 7, with the action controller without
+aoffset
+t
+the character fails to avoid penetration with objects
+or walls, as the raw motion from the motion database does
+not have any information of the scene. This demonstrates
+that action offset also plays a role in generating detailed
+scene-aware poses even from raw motion capture data.
+Moreover, the results with the action controller without
+smoothness rewards r∆t and r∆v are not smooth enough,
+showing unnatural movements such as jerking. These abla-
+tion studies justify the advantages of our reward terms.
+Ablation Studies on Task-Adaptive Motion Editing.
+We ablate our motion editing module by replacing it with
+an alternative approach via Inverse-Kinematics (IK). An ex-
+ample result is shown in Fig. 8 (left). For manipulation, the
+results with IK show jerky and awkward motions because
+the temporal and inter-joint correlations in natural human
+motions are not reflected in IK, while original LAMA with
+task-adaptive motion editing module shows much natural
+motions. Our motion editing module can also be used to
+Figure 9. Examples of synthesized manipulation motions. The tar-
+get object for manipulation is colored as orange. Top is a motion
+sequence of walking and opening a toilet lid, and the bottom is a
+sequence of walking and opening doors. The character is colored
+purple at start and aqua at the end.
+further adjust the character movements in different object
+geometries, going over the limit of the motion database. As
+seen in Fig 8 (right), the motion editing module enables the
+character to properly sit in chairs with various sizes.
+5. Discussion
+In this paper, we present a method to synthesize locomo-
+tion, scene-interaction, and manipulation in a unified sys-
+tem. Leveraging a RL framework with motion matching,
+our method enables to produce natural and plausible hu-
+mans motions in complex and cluttered 3D environments
+only with a limited amount of motion-only datasets. Our
+method has been thoroughly evaluated in diverse scenar-
+ios, outperforming previous approaches [15, 60]. We also
+demonstrate the robustness and generalization ability of our
+system by covering a wide range of human interactions in
+many different 3D environments.
+While our RL-based method can be generalized to any
+unseen 3D environments, a new control policy has to be
+trained for each motion sequence. Combining RL with a
+supervised learning framework for better efficiency can be
+an interesting future research direction. Furthermore, al-
+though we assume a fixed skeletal information throughout
+the system, interaction motions may change depending on
+the character’s body shape and sizes. We leave synthesizing
+motions on varying body shapes as future work.
+Acknowledgments: This work was supported by SNU-
+Naver Hyperscale AI Center, SNU Creative-Pioneering Re-
+searchers Program, and NRF grant funded by the Korea
+government (MSIT) (No. 2022R1A2C209272411).
+8
+
+LAMA
+LAMA
+LAMA w/o motion editing
+LAMA w/o manifold + IK梦人庆A. Supplementary Video
+The supplementary video shows the results of our
+method, LAMA, on various scenarios. In the video, we
+show the human motion synthesis results on PROX [16],
+Matterport3D [7], and also our own home-brewed 3D scene
+produced by Polycam App [1] in an iPad pro. We use
+SAPIEN [63] object meshes for manipulation examples. As
+shown, our method successfully produces plausible and nat-
+ural human motions in many challenging scenarios. Our
+supplementary video contains several ablation studies of
+our method by showing the importance of collision reward
+rcoli in Eq. (4), transition reward (r∆t , r∆v) in Eq. (8), pos-
+ture offset aoffset
+t
+in Action Controller (Sec. 3.2), and our
+motion editing modules (Sec. 3.5) compared to the tradi-
+tional Inverse Kinematics (IK). We also show the compari-
+son with previous state-of-the arts [15, 59, 60] and demon-
+strate that our results produces better quality of motions
+with better collision avoidance performance in complicated
+3D scenes.
+B. Additional Details on Implementations
+B.1. Action Controller
+Implementation Details.
+For the action controller A and
+motion synthesizer module S, we use the animation library
+DART [27]. We also use a publicly available PPO imple-
+mentation [32, 41], where we remove the variable time-
+stepping functions stepping in [32] by following the origi-
+nal PPO algorithm. The details of the training regarding the
+policy and value network of the action controller are written
+in Table 2.
+Early Termination Conditions.
+As written in the main
+paper, the episode is terminated (1) when the character
+moves out of the scene bounding box; (2) when the colli-
+sion reward rcoli is under a certain threshold; or (3) when
+the root of the human character is located in the blocked
+(occupied) regions of the scenes in 2D grid space during
+the locomotion status.
+Name
+Value
+Learning rate of policy network
+2e-4
+Learning rate of value network
+0.001
+Discount factor (γ)
+0.95
+GAE and TD (λ)
+0.95
+Clip parameter (ϵ)
+0.2
+# of tuples per policy update
+30000
+Batch size for policy/value update
+512
+Table 2. Details on the hyper-parameters for learning the control
+policy of the action controller A.
+B.2. Motion Synthesizer
+Motion Database Information.
+As described in our
+main paper, we pre-process the motion segments by selec-
+tively collecting and clipping from Ubisoft La Forge [14],
+COUCH [70], and SAMP [15]. The length (in frames)
+of motion segments (“Seg. Length” in tables), number of
+motion segment (“Seg. Count” in tables), and the number
+of total frames (“Total Frames” in tables) are summarized
+in Table 3.
+Action-Specific Feature Definition.
+The motion feature,
+as defined in our main paper Sec 3.3, represents both the
+current state of the motion and a short term future move-
+ments: f(m) = {{pj}, { ˙pj}, θup, c, ofuture}. In particu-
+lar the action specific feature ofuture = {{pdt
+0 }, {rdt
+0 }}
+contains future motions so that the motion search process
+can take into account the future motion consistency, where
+pdt
+0 , rdt
+0 ∈ R2 are the position and orientation of root joint at
+dt frames later from the current target frame. For locomo-
+tion, we extract dt = 10, 20, and 30 frames in the future (at
+30Hz) following [9], as addressed in our main paper. For sit-
+ting, we specifically choose dt as the frame where the char-
+acter completes the sit-down motion. The major motivation
+of this design choice is encourage the motion synthesizer to
+search the motion clips with the desired target action.
+Computation Cost for Searching.
+The computation time
+for searching the motion database is done between 1-2 mil-
+liseconds in CPU, where we test on AMD Ryzen 5950X
+CPU. The number of search times varies and is dependent
+to the 3D scenes and desired motions. In one of our sce-
+narios, total 17 searches in locomotion(walk) and 14 in ac-
+tion(sit) were done. For locomotion, the searching time is
+average 1.743 milliseconds (standard deviation 0.46) and
+for action(sit) 1.103 milliseconds (standard deviation 0.63).
+B.3. Motion Editing via Motion Manifold
+Implementation Details.
+For the convolutional autoen-
+coder of task-adaptive motion editing, we use PyTorch [42],
+FairMotion [12], and PyTorch3d [48]. The autoencoder is
+trained with the Adam optimizer with learning rate 0.0001.
+We use 3 layers of 1D temporal-convolutions with kernel
+width of 25 and stride 2, and the channel dimension of each
+output feature is 256. The training datasets are summarized
+in Table 4. Note that we use different pre-processing steps
+between Motion editing module and Motion Synthesizer.
+Reconstruction Loss.
+The encoder Ψ and decoder Ψ−1
+are trained based on reconstruction loss Lrecon = ||X −
+Ψ−1(Ψ (X)) ||2, where:
+Lrecon = wcLcontact + wrLroot + wqLquat + wpLpos.
+(9)
+9
+
+Lcontact, Lroot, and Lquat are the MSE losses of foot con-
+tact labels, root status (height and transform relative to the
+previous frame projected on the XZ plane), and the joint
+rotations in 6D representations [73]. To penalize errors ac-
+cumulating along the kinematic chain, we perform forward
+kinematics (FK) and measure the global position distance of
+joints between original and reconstructed motion. As global
+positions of the joints are highly dependent on the root po-
+sitions, for the early epochs, the distance is measured based
+on root-centric coordinates to ignore the global location of
+roots, which we found empirically more stable.
+Motion Editing Loss
+For motion editing, the positional
+loss and regularization loss are defined as follows.
+L = wpLpos + wfLfoot + wrLroot,
+where
+Lpos =
+�
+j,qj∈φ
+∥pj − qj∥2, if φ exists at t
+Lfoot =
+�
+foot
+∥pe
+foot − pi
+foot∥2,
+Lroot = wr∥re
+xz − ri
+xz∥2 + w∆r∥˙re
+xz − ˙ri
+xz∥2.
+(10)
+pj denotes positions of joint j, and r, ˙r denotes root po-
+sitions and velocities respectively. Superscript e and i in-
+dicates whether it is from edited or initial motion, respec-
+tively. Subscript xz indicates the vector is projected onto
+the XZ plane. The loss term L enforces the edited motion
+to maintain contact and root trajectory (in the XZ plane) of
+the initial motion, while generating natural movements of
+the other joints to meet the sparse positional constraints.
+Generating Interaction Cue for Manipulation
+To syn-
+thesize character’s arm motions naturally interacting with
+the movements of articulated target objects, we produce
+desired interaction cues by producing the 3D trajectories
+of a chosen 3D position of the object at which the hand
+part of the character are expected to touch. Specifically,
+we apply the expected articulated motion of the 3D object
+model to produce the 3D trajectory of a chosen object ver-
+tex, v(Rt, Tt, θt), where Rt, Tt, are the global orientation
+and translation of the object and θt is the parameters for the
+object articulation (e.g., the hinge angle of the cover of a
+laptop) at time t. v(·) represents the 3D location of the cho-
+sen vertex v. To this end, we input the produced trajectory
+as the desired 3D interaction cue for a character’s joint (e.g.,
+a hand joint) assuming the joint is touching this object tra-
+jectory for manipulation φ = [v(Rt, Tt, θt)]t. Note that, in
+our visualization, we apply the desired articulated motions
+for the 3D object at each time, synced to the produced in-
+teraction cues.
+Label
+Seg. Length
+Seg. Count
+Total Frames
+Locomotion
+10
+11063
+11498
+Sit
+50 – 85
+5842
+14942
+Table 3. Details on pre-processed motion datasets per each action
+category for training our motion synthesizer S.
+Name
+Value
+Motion sequence length
+120
+Number of sequence (training)
+11397
+Number of sequence (validation)
+3135
+Number of sequence (test)
+2139
+Table 4. Details on pre-processed motion datasets for training our
+motion editing module M.
+C. More Details on Experiments
+C.1. Frechet Distance Features
+FDroot is computed by root feature vector, which is a con-
+catenated vector of root orientation in angle-axis represen-
+tation, root up vector, and root transform relative to the pre-
+vious frame. We note that all of the motions for comparison
+have the same up axis (y) and floor plane (xz). FDjoint is
+computed by joint feature vector, represented as joint orien-
+tations in angle-axis representation, excluding the root.
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