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+Scaling laws for single-agent reinforcement learning
+Jacob Hilton
+OpenAI
+[email protected]
+Jie Tang
+OpenAI
+[email protected]
+John Schulman
+OpenAI
+[email protected]
+Abstract
+Recent work has shown that, in generative modeling, cross-entropy loss improves
+smoothly with model size and training compute, following a power law plus
+constant scaling law. One challenge in extending these results to reinforcement
+learning is that the main performance objective of interest, mean episode return,
+need not vary smoothly. To overcome this, we introduce intrinsic performance,
+a monotonic function of the return defined as the minimum compute required to
+achieve the given return across a family of models of different sizes. We find that,
+across a range of environments, intrinsic performance scales as a power law in
+model size and environment interactions. Consequently, as in generative modeling,
+the optimal model size scales as a power law in the training compute budget.
+Furthermore, we study how this relationship varies with the environment and with
+other properties of the training setup. In particular, using a toy MNIST-based
+environment, we show that varying the “horizon length” of the task mostly changes
+the coefficient but not the exponent of this relationship.
+1
+Introduction
+Recent studies of how neural network performance varies with model size and training compute have
+found these relationships to be governed by smooth power laws [Kaplan et al., 2020, Henighan et al.,
+2020, Droppo and Elibol, 2021, Ghorbani et al., 2021]. These studies have focused primarily on
+generative modeling, in which the training objective is cross-entropy loss, and have found test loss to
+scale smoothly. In this work we seek to extend these results to reinforcement learning, in which there
+is generally no cross-entropy loss.
+In some reinforcement learning environments, there is still a performance metric that varies smoothly.
+For example, in competitive games, it is often possible to assign Elo ratings to players such that
+scaled differences in Elo ratings give approximate logit probabilities of victory. Recently it has been
+shown that, in the board games Hex [Jones, 2021], Connect Four and Pentago [Neumann and Gros,
+2022], the exponentiated Elo rating of a policy trained using AlphaZero [Silver et al., 2018] follows a
+power law in training compute (within a certain Elo range). We call metrics that follow such simple
+relationships natural performance metrics.
+However, in other reinforcement learning environments, there may be no obvious natural performance
+metric. For example, there may be no reason to expect the number of objects collected in a video
+game to vary smoothly, since crossing some threshold may require some challenging new capability.
+To overcome this difficulty, we introduce intrinsic performance, which is defined to be equal to
+training compute on the compute-efficient frontier of the tradeoff between model size and environment
+interactions. This causes the relationship between performance and training compute to follow a
+power law by definition, thereby making it possible to study the remaining relationships between
+performance, model size and environment interactions.
+We study these relationships across a range of environments: the easy and hard modes of environments
+from Procgen Benchmark [Cobbe et al., 2020]; a 1v1 version of Dota 2 [OpenAI et al., 2019]; and a toy
+environment based on MNIST [LeCun, 1998] for which we vary the “horizon length”. Across these
+arXiv:2301.13442v1  [cs.LG]  31 Jan 2023
+
+environments, we find intrinsic performance to scale as a power law in model size and environment
+interactions, in much the same way as the analogous quantities in generative modeling.
+One consequence of this scaling law is that, as in generative modeling, the optimal model size for
+a given training compute budget follows a power law. We study in detail how the coefficient and
+exponent of this relationship vary with properties of the training setup, including: the difficulty mode
+of environment, for Procgen; the horizon length of the task, for the MNIST-based environment; the
+period of training used to fit the power law; and whether the width or depth of the model is scaled.
+Contents
+1
+Introduction
+1
+2
+Scaling laws without cross-entropy loss
+3
+2.1
+Intrinsic performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+2.2
+The power law for intrinsic performance . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.3
+Optimal model size vs compute . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+3
+Experimental setup
+5
+4
+Results
+7
+4.1
+Optimal model size vs compute . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+4.2
+Effect of task horizon length . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+4.3
+Variability of exponents over training
+. . . . . . . . . . . . . . . . . . . . . . . .
+10
+4.4
+Scaling depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+4.5
+Natural performance metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+5
+Discussion
+13
+5.1
+Extrapolating sample efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+5.2
+Cost-efficient reinforcement learning . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+5.3
+Limitations
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+5.4
+Forecasting compute requirements . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+6
+Conclusion
+16
+A Curve-fitting methodology
+19
+B
+Hyperparameters
+21
+C Results in full
+24
+D Parameter and FLOP calculations
+27
+E
+Fitted constants
+28
+F
+Proof of the lemma
+32
+G Proof sketch of the proposition
+33
+2
+
+1014
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+5
+10
+15
+20
+25
+30
+Mean episode return
+StarPilot, hard
+(a) Using the usual metric of mean episode return.
+1014
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+Parameters
+104.3
+104.6
+104.9
+105.2
+105.5
+105.8
+106.1
+106.4
+106.7
+107.0
+StarPilot, hard
+(b) Using intrinsic performance instead.
+Figure 1: Learning curves as a function of total training compute for StarPilot, an environment from
+Procgen Benchmark, using CNNs of different widths. Mean ±1 sample standard deviation over three
+seeds shown.
+2
+Scaling laws without cross-entropy loss
+2.1
+Intrinsic performance
+In generative modeling, cross-entropy test loss scales smoothly with training compute, following a
+power law plus constant scaling law [Henighan et al., 2020]. However, in reinforcement learning
+(RL), there is generally no cross-entropy loss, and the usual objective of mean episode return need
+not scale so smoothly.
+For example, consider StarPilot, a side-scrolling shooter from Procgen Benchmark [Cobbe et al.,
+2020]. The agent receives a reward of 1 for destroying each enemy, and the episode continues until
+either the agent is destroyed, or the agent reaches the end of the level and obtains a bonus reward
+of 10. There is no reason to expect mean episode return in this game to scale smoothly. Indeed, it
+takes some ability with aiming and dodging to reach a mean episode return of 5 or 10, but not much
+additional skill to reach a mean episode return of 15 or 20. This irregular difficulty profile is reflected
+in the uneven shape of learning curves for this environment (see Figure 1(a)).
+It may be tempting to conclude that the scaling law methodology cannot be applied to such an
+environment. However, in generative modeling, there are smooth scaling laws that do not depend on
+test loss per se. For example, the model size that achieves the minimum test loss for a given compute
+budget scales as a power law with compute. In order to study such relationships in the context of
+RL, we would like a performance metric that behaves like test loss, i.e., some monotonic function
+of the return that scales as a power law with compute. We achieve this with our notion of intrinsic
+performance by simply using compute itself as our performance metric.
+Definition. A scalable model family is collection of models trained in a uniform way, parameterized
+by the model size and the total compute used in training. Given a scalable model family, the intrinsic
+performance of an arbitrary policy is the minimum compute required to train a model of any size in
+the family to reach the same return (averaged over random seeds).
+Another way of explaining this definition is to consider learning curves as a function of compute
+for a family of models of different sizes, as in Figure 1. The maximum performance over all model
+sizes defines the compute-efficient frontier. When using the usual metric of mean episode return (as
+in Figure 1(a)), the compute-efficient frontier need not follow any particular trend. However, when
+using intrinsic performance instead (as in Figure 1(b)), the efficient frontier is mapped onto the line
+3
+
+1014
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+5
+10
+15
+20
+25
+30
+Mean episode return
+StarPilot, hard
+(a) Using mean episode return.
+1014
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+Parameters
+104.3
+104.6
+104.9
+105.2
+105.5
+105.8
+106.1
+106.4
+106.7
+107.0
+Learning
+curve
+Power law
+fit
+Power law
+asymptote
+Efficient
+frontier
+Efficient
+points
+StarPilot, hard
+(b) Using intrinsic performance.
+Figure 2: Learning curves as a function of total training compute for StarPilot, together with their
+power law fits. The asymptotes show the E → ∞ limits of the power law fits, representing the
+predicted performance at convergence. The efficient points show where the power law fits are tangent
+to the efficient frontier. Mean over three seeds shown.
+y = x by definition. This reveals the regularity of the learning curves, which, as we shall see next,
+now follow a power law trend.
+We describe in detail how we compute intrinsic performance in Appendix A.
+2.2
+The power law for intrinsic performance
+Our main empirical result is that intrinsic performance I scales approximately as a power law with
+model parameters N and environment interactions E,
+I−β =
+�Nc
+N
+�αN
++
+�Ec
+E
+�αE
+,
+(1)
+where αN, αE, β, Nc and Ec are positive constants.
+This is essentially the same as the corresponding scaling law for language models [Kaplan et al.,
+2020, equation (1.6)], but with test loss replaced by I−β. Although it appears that we have introduced
+an additional exponent β, the intrinsic definition of I means that β is actually determined by αN and
+αE (see Lemma 1).
+The intuition behind this equation is that, when the number of interactions is not bottlenecked
+(E → ∞), I scales as a power law in N, and when model size is not bottlenecked (N → ∞), I
+scales as a power law in E.
+2.3
+Optimal model size vs compute
+An important implication of equation (1) is that the optimal model size for a given compute budget
+scales as a power law in that compute budget.
+More precisely, we assume that total training compute is proportional to NE (ignoring the compute
+required to run the environment, at least for now). Hence, for a given compute budget, there is a
+trade-off between N and E (the optimum of which defines a point on the compute-efficient frontier).
+What we will now show is that, under equation (1), the optimal value of N scales as a power law in
+the compute budget, with an exponent that we will specify.
+4
+
+Since training compute is proportional to NE, for convenience we choose units of compute such
+that training compute equals NE exactly (although in plots we will continue to display compute in
+FLOPs). This implies that I = NE along the compute-efficient frontier.
+Lemma 1. If I satisfies equation (1) and I = NE along the compute-efficient frontier, then the
+compute-efficient frontier is described by the equation
+αN
+�Nc
+N
+�αN
+= αE
+�Ec
+E
+�αE
+.
+(2)
+Moreover, once αN and αE are chosen, β and NcEc are determined:
+1
+β =
+1
+αN
++ 1
+αE
+and
+1
+NcEc
+=
+�
+1 + αN
+αE
+�
+1
+αN �
+1 + αE
+αN
+�
+1
+αE .
+For a proof, see Appendix F.
+Substituting equation (2) into equation (1), it follows that along the compute-efficient frontier,
+N = Nc
+�
+1 + αN
+αE
+�
+1
+αN C
+1
+1+ αN
+αE ,
+where C := NE. In other words, for a given compute budget C, the optimal model size N scales as
+N ∝ C
+1
+1+ αN
+αE .
+3
+Experimental setup
+We ran experiments using variety of RL environments:
+• Procgen Benchmark [Cobbe et al., 2020]: CoinRun, StarPilot and FruitBot in both easy
+and hard modes, separately varying CNN width and depth.
+• Dota 2 [OpenAI et al., 2019]: a 1v1 version of the game, varying LSTM size.
+• MNIST: an RL environment in which the agent has to correctly label a handwritten digit
+from MNIST [LeCun, 1998], using hyperparameters to artificially alter the “horizon length”
+of the task, varying CNN width.
+All our experiments used a variant of either the PPO algorithm [Schulman et al., 2017] or its close
+cousin PPG [Cobbe et al., 2021], along with the Adam optimization algorithm [Kingma and Ba,
+2014].
+The remainder of this section discusses further details of our experimental setup. Hyperparameters
+for all our experiments are given in Appendix B.
+3.1
+Procgen Benchmark
+For our Procgen Benchmark experiments, we used CoinRun, StarPilot and FruitBot. We chose these
+environments because they have lower-variance learning curves than other Procgen environments,
+and because CoinRun’s binary reward enabled us to study the scaling of natural performance metrics
+(see Section 4.5). We used both the easy and hard difficulty modes of these environments to see if
+this would have an effect on the scaling constants.
+We used PPG-EWMA [Hilton et al., 2021] with a fixed KL penalty objective [Cobbe et al., 2021],
+and trained for 200 million environment interactions.
+We used the CNN architecture from IMPALA [Espeholt et al., 2018] and conducted both width-
+scaling and depth-scaling experiments. For our width-scaling experiments, we varied the total number
+of parameters from
+1
+64 of the default to 8 times the default, rounding to integer numbers of channels.
+For our depth-scaling experiments, we varied the number of residual blocks per stack from 1 to 64,
+and used 1
+4 of the default width since the default number of residual blocks per stack was only 2.
+5
+
+3.2
+Dota 2
+For our Dota 2 experiments, we used a 1v1 version of the game to save computational expense.
+Following OpenAI et al. [2019], we used PPO, but we adjusted the asynchronous setup to ensure that
+training used only on-policy data with no data reuse. We used 8 parallel GPU workers and trained for
+between 13.6 billion and 82.6 billion environment interactions.
+We used an LSTM architecture and varied the width of the network, with the sizes of the embedding
+and hidden state varying from 8 to 4096.
+3.3
+MNIST
+Our MNIST environment samples a handwritten digit from the MNIST training set uniformly and
+independently random at each timestep, and provides an immediate reward of 1 for a correct label
+and 0 for an incorrect label. There are no episode boundaries, and so we measure mean training
+accuracy instead of mean episode return.
+The use of immediate rewards with no episode boundaries allows the horizon length of the task
+to be artificially controlled by varying the hyperparameters of our method advantage estimation,
+GAE [Schulman et al., 2015]. First, we set the GAE credit assignment parameter λ to 1, so that the
+algorithm assigns credit for each reward to all previous actions, instead of assigning more immediate
+credit. Then we vary the GAE discount rate γ, so that the algorithm discounts future rewards at this
+rate. In separate experiments, we set γ = 1 −
+2
+h+1 for different values of the “horizon length” h
+ranging from 1 to 256. (This equation is equivalent to saying that an exponentially-weighted moving
+average with decay parameter γ has the same center of mass as the interval [0, h − 1].)
+We used PPO-EWMA [Hilton et al., 2021] with rollouts of length 512 (twice as long as our maximum
+value of h), and trained for 225 environment interactions.
+We used a simple CNN architecture with ReLU activations and the following layers: a 5 × 5
+convolutional layer with 40 channels, 2×2 max pooling, a 3×3 convolutional layer with 80 channels,
+2 × 2 max pooling, and a dense layer with 1,000 channels. We scaled the width of this network by
+varying total number of parameters from
+1
+64 of the default to 8 times the default. We used separate
+policy and value function networks because we did not expect there to be much transfer between the
+two objectives, since the environment samples digits independently.
+3.4
+Learning rates
+Although we would not expect our qualitative results to change much, our quantitative results
+such as scaling exponents depend crucially on using well-tuned hyperparameters. By far the most
+important hyperparameter to tune in our setup is the Adam learning rate, whose optimal value can
+vary substantially with model size and compute budget.
+When varying model size, we found that a good heuristic is to keep the Adam learning rate propor-
+tional to the initialization scale. For our width-scaling experiments, this means keeping the Adam
+learning rate proportional to 1/
+√
+width, since we use Kaiming He initialization [He et al., 2015]. For
+our Procgen depth-scaling experiments, which use a residual network, it means keeping the Adam
+learning rate proportional to 1/
+�
+depth
+1
+L , where L is the number of layers per residual block (L = 2
+in our case), since we use an initialization similar to Fixup initialization [Zhang et al., 2019]. For
+Procgen and MNIST, we tuned the learning rate at one model size and followed this heuristic to select
+the learning rate for the other model sizes. For Dota 2, we tuned the learning rate separately for each
+model size, but this amounted to following approximately the same heuristic.
+When varying the compute budget for a given model size, it can actually be necessary to use separate
+training runs for each compute budget, each with its own learning rate schedule, rather than taking
+different snapshots at different points of the same training run [Hoffmann et al., 2022]. Unfortunately,
+due to the challenge of carefully tuning learning rate schedules for RL and the expense of multiplying
+the number of training runs, we took the latter approach. To mitigate the impact of this, we found a
+learning rate schedule that seemed to work well for a variety of compute budgets, which we explain
+in Appendix B.1. Nevertheless, the values of our scaling exponents should be considered uncertain
+because of this.
+6
+
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+αN
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+αE
+1
+1 + αN/αE
+= 0.8
+1
+1 + αN/αE
+= 0.7
+1
+1 + αN/αE
+= 0.6
+Procgen (width)
+CoinRun
+StarPilot
+FruitBot
+Easy, single seed
+Easy, mean return
+Hard, single seed
+Hard, mean return
+Dota 2
+1v1
+Reference
+αN/αE = const
+MNIST horizons
+1
+2
+4
+8
+16
+32
+64
+128
+192
+256
+αN vs αE
+Figure 3: Fitted values of αN and αE. For Procgen, we also show the values fitted using each of
+the 3 random seeds, to show the variation due to the choice of random seed. The dotted lines show
+contours for
+1
+1+αN/αE , the exponent for the scaling of optimal model size with compute.
+4
+Results
+Our main result is that our power law for intrinsic performance, equation (1), holds across envi-
+ronments and model sizes, at least after an initial transient period of training (which we discuss in
+more detail in Section 4.3). This result is supported by the closeness of the power law fit to our
+learning curves, as shown in Figure 2 for StarPilot and in Appendix C for all our environments. Our
+methodology for fitting this power law is described in Appendix A.
+It is interesting to study the sensitivity of the exponents αN and αE, which govern the scaling
+behavior of I with N and E (and determine the other exponents of interest). The fitted values of
+these exponents for the different environments are shown in Figure 3. The numerical values of all of
+the fitted constants may be found in Appendix E.
+Although our measurements of these exponents are uncertain, due to the limitations discussed in
+Section 5.3, we make a number of observations:
+• The primary determinant of αN and αE is the domain (Procgen, Dota 2, or MNIST), which
+we expect is a consequence of the fact that so many experimental details are shared within
+each domain.
+• Within MNIST, increasing the horizon seems to lower αE, but as we explain in Section 4.2,
+this effect is confounded by a measurement problem caused by under-training.
+• Within Procgen, the easy and hard modes of each Procgen game tend to have closer
+exponents to one another than to other Procgen games. We believe that this is because
+identifying visual features is a core part of Procgen, and the two modes of each game have
+very similar observation distributions.
+7
+
+10−7
+10−6
+10−5
+10−4
+10−3
+10−2
+Compute (PF-days)
+103
+104
+105
+106
+107
+Parameters
+Procgen (width)
+CoinRun
+StarPilot
+FruitBot
+Easy
+Hard
+Dota 2
+1v1
+Generative modeling
+Language
+(Hoffmann et al.)
+Language
+(Kaplan et al.)
+Image 32x32
+(Henighan et al.)
+MNIST horizons
+1
+2
+4
+8
+16
+32
+64
+128
+192
+256
+Optimal model size vs compute
+Figure 4: Optimal model size vs compute for all our environments. Note that the individual points,
+which correspond to the sizes of models that we trained, are themselves obtained from a power law
+best fit. Hence the fact that the lines pass through the points exactly is automatic and does not indicate
+goodness of fit.
+• The Procgen difficulty mode does not obviously have any particular effect on the scaling
+exponents. We hypothesize that humans tend to judge a task as easier when a near-perfect
+score can be achieved with less compute, even if it takes a lot of additional compute to eke
+out the final few points. Conversely, it does not seem to matter to the RL algorithm exactly
+how the score maps on to intrinsic performance (i.e., the compute required).
+4.1
+Optimal model size vs compute
+As explained in Section 2.3, our power law for intrinsic performance implies that, for a given compute
+budget, the optimal model size scales as a power law with exponent
+1
+1+αN/αE .
+Figure 4 shows these inferred relationships for our different environments, along with some generative
+modeling relationships taken from the literature. The full equations for these relationships are
+provided in Appendix E.
+The exponent
+1
+1+αN/αE varied between around 0.40 and 0.65 for Procgen and 0.66 and 0.80 for
+MNIST, and was around 0.76 for Dota 2. By comparison, the corresponding exponent for language
+modeling, which was carefully measured by Hoffmann et al. [2022], is around 0.50. Previous work
+by Kaplan et al. [2020] and Henighan et al. [2020] measured this exponent less carefully but using a
+methodology that more closely matches our own, and found an exponent of around 0.73 for language
+0.65 for 32x32 images.
+An intriguing conjecture, which is also suggested by theoretical considerations [Bahri et al., 2021],
+is that the exponent of this relationship would be around 0.5 in every domain if it were measured
+carefully enough (i.e., with optimal hyperparameters and enough random seeds). Given the limitations
+of our experiments, we consider our results to be inconclusive on this question.
+Nevertheless, it is clear that the scaling coefficient of this relationship varies significantly between
+domains. With the exception of our toy MNIST environment, the optimal model size for RL for
+8
+
+a given compute budget is consistently smaller than for generative modeling, in some cases by
+multiple orders of magnitude. We believe that this is because RL tasks have a longer horizon length
+than generative modeling in some sense, and explore this hypothesis with our MNIST environment
+in Section 4.2. Another possibility is that the arithmetic intensity (i.e., the number of FLOPs per
+parameter in a forward pass) of the architecture is a confounder, which we discuss in more depth in
+Section 4.4.
+4.2
+Effect of task horizon length
+As explained in Section 3.3, for our MNIST experiments, we artificially altered the “horizon length”
+of the task by setting the GAE credit assignment parameter λ to 1 and varying the GAE discount rate
+γ.
+The expected effect of varying γ in this context is given by the following theoretical result.
+Proposition 1. Consider an MDP with independent timesteps (by which we mean that each st is
+identically distributed and independent of st−1 and at−1, and episodes never terminate). Suppose we
+train a model with parameters θ on this MDP using Vanilla Policy Gradient,1 estimating advantages
+using GAE with γ = 1 −
+2
+h+1 and λ = 1, and working with separate policy and value function
+networks. Then the covariance matrix of the policy gradient is approximately
+Σθ + Πθ
+�
+h + 1
+h − 2
+�
+for some symmetric positive semi-definite matrices Σθ and Πθ that do not depend on h.
+For a proof sketch, see Appendix G.
+Intuitively, this result says that gradient variance may be decomposed into two pieces: one piece that
+is inherent to the task (the Σθ term), and one piece that comes from imperfect credit assignment (the
+Πθ term). For example, when h = 1 (i.e., γ = 0), credit is correctly assigned to the previous action
+only, and hence the second term vanishes. Ignoring the 1
+h term (since h ≥ 1), we may stylize this
+result as: gradient variance is an affine function of h (i.e., a linear function with an intercept).
+This can be directly translated into a statement about sample efficiency, since multiplying the gradient
+variance by some factor c can be exactly compensated for by multiplying the batch size by c, which
+multiplies the number of samples used by c. Hence in order to reach a given performance level,
+the number of environment interactions required should be an affine function of h. This affine
+function will come from integrating certain functionals of Σθ and Πθ over the course of training,
+and will therefore depend both on the model architecture and on the choice of performance level.
+To test this prediction, we looked at the number of environment interactions required to reach a
+1% failure rate (i.e., 99% training accuracy) on MNIST as a function of the horizon length h. Our
+results are shown in Figure 5, along with affine fits. As expected, the number of interactions closely
+follows an affine function of the horizon length, although the fit is less good for shorter horizons and
+larger models. At very short horizons, the number of interactions even decreases with the horizon
+length, suggesting a hyperparameter issue (perhaps a suboptimal learning rate schedule, or reward
+normalization implicitly decreasing the KL penalty and entropy bonus).
+The implication of this for our optimal model size vs compute scaling law is that once h becomes large
+enough, further increasing h should lead to a proportional increase the compute budget corresponding
+to each given optimal model size, without changing the scaling exponent of this relationship. This
+is because the intercept term of the affine function will eventually become dominated by the term
+involving h, and so the number of environment interactions required to reach a given performance
+level will eventually scale approximately proportionally to h. (For small values of h, however, the
+relationship between the two components of the covariance matrix of the policy gradient may have a
+more complex dependence on model size.)
+This effect is visible in Figure 4, where the main impact of increasing the horizon length is to shift
+the optimal model size vs compute curve to the right. The curve also gets shallower as the horizon
+1Vanilla Policy Gradient is a primitive version of PPO, explained here: https://spinningup.openai.
+com/en/latest/algorithms/vpg.html
+9
+
+0
+50
+100
+150
+200
+250
+Horizon length h
+1
+2
+3
+4
+5
+6
+Interactions
+×106
+Parameters
+104.8
+105.1
+105.4
+105.7
+106.0
+106.3
+106.6
+106.9
+107.2
+107.5
+Value
+Affine fit
+Interactions required to reach a 1% failure rate, MNIST
+Figure 5: Sample efficiency for MNIST as a function of the horizon length h, for all our model sizes.
+length is increased, but this effect is confounded by a measurement problem caused by under-training,
+which we explain in more detail in Section 4.3.
+Our MNIST environment is useful because our it allows us to vary the task horizon length in a fine-
+grained, quantifiable way by varying γ. But our analysis of this environment relies on the assumption
+of independent timesteps, which does not hold in most environments (and in particular removes the
+need for exploration). Nevertheless, our results are suggestive of a more general explanation for the
+large differences in optimal model size for a given compute budget between different environments:
+that different environments have different task horizon lengths in a more general sense. We speculate
+that, in this more general sense, task horizon length is influenced by how long rewards are delayed
+for relative to the actions the agent is currently learning (which may increase throughout training as
+the agent learns skills with feedback loops that are less and less tight), and that γ determines only an
+upper bound on the task horizon length.
+4.3
+Variability of exponents over training
+Although our power law for intrinsic performance holds across environments and model sizes, we
+only obtain a good fit by excluding an initial transient period of training. Put another way, the scaling
+constants vary over the course of training.
+This phenomenon is clearest with with our MNIST environment, since we were able to use many
+random seeds to reduce variance. Recall that in this environment, the agent observes a randomly
+sampled MNIST training set digit each timestep, and the horizon length of the task is artificially
+controlled using the GAE discount rate γ, as explained in Section 3.3. We fitted our power law to
+three different periods of training for this environment: an early period (216–219 interactions), a
+middle period (219–222 interactions), and a late period (222–225 interactions).
+Figure 6 shows the fitted values of αN and αE for these different periods of training. We found αE
+to be significantly lower during the early and middle periods of training, especially for the shorter
+horizon lengths.
+In order to accurately measure the scaling constants for optimal model size vs compute, it is best to
+use a period of training during which the learning curves reach the compute-efficient frontier, since
+otherwise the measurement is an extrapolation. As shown in Figure 7, this is always in the late period
+10
+
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+αN
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+αE
+1
+1 + αN/αE
+= 0.9
+1
+1 + αN/αE
+= 0.8
+1
+1 + αN/αE
+= 0.7
+MNIST periods
+Early
+Middle
+Late
+MNIST horizons
+1
+2
+4
+8
+16
+32
+64
+128
+192
+256
+αN vs αE, MNIST
+Figure 6: Fitted values of αN and αE for MNIST
+with different horizons, using different periods of
+training to fit the power laws. The horizon h is
+defined by γ = 1 −
+2
+h+1, where γ is the discount
+rate.
+1013
+1014
+1015
+1016
+1014
+1016
+MNIST, horizon 1, late period
+Parameters
+104.8
+105.1
+105.4
+105.7
+106.0
+106.3
+106.6
+106.9
+107.2
+107.5
+1013
+1014
+1015
+1016
+1013
+1014
+1015
+Intrinsic performance (FLOPs)
+MNIST, horizon 256, late period
+1012
+1013
+1014
+1015
+Compute (FLOPs)
+1012
+1013
+MNIST, horizon 1, middle period
+Learning
+curve
+Power law
+fit
+Efficient
+frontier
+Efficient
+points
+Figure 7: Learning curves as a function of total
+training compute for MNIST, using different hori-
+zons and different periods of training, together
+with their power law fits. Mean over the middle-
+performing 16 of 20 random seeds shown.
+of training, if at all. For this reason, we use the late period of training for all of our results on MNIST
+outside of this section.
+Figure 7 also shows that, for the longer horizon lengths, the learning curves of the larger models
+did not reach the compute-efficient frontier even during the late period of training. Hence our
+measurements of
+1
+1+αN/αE , the exponent for the scaling of optimal model size with compute, are
+likely underestimates for these longer horizon lengths.
+For our other environments, we found that it was enough to exclude only the first
+1
+64 of training
+in order for our power law for intrinsic performance to be a good fit around the compute-efficient
+frontier. This is similar to what is needed for the corresponding law for language [Kaplan et al., 2020,
+Figure 4, right]. Nevertheless, it is possible that the measurement problem identified in this section
+affects some of our other results.
+4.4
+Scaling depth
+Most of our experiments involved scaling the width of our networks, but for Procgen, we also tried
+scaling the depth, as explained in Section 3.1. We found that our power law for intrinsic performance
+still held, but with more noise than the width-scaling experiments, as a consequence of using fewer
+model sizes. The fitted values of αN and αE for the depth-scaling experiments lay in a similar region
+to the width-scaling experiments, but there were no clear relationships between the depth-scaling
+exponents for the different environments, nor between the width-scaling and depth-scaling exponents
+11
+
+10−5
+10−4
+10−3
+10−2
+Compute (PF-days)
+104
+105
+106
+Parameters
+Optimal model size vs compute, Procgen
+(a) Using parameters as the measure of model size.
+10−5
+10−4
+10−3
+10−2
+Compute (PF-days)
+106
+107
+108
+FLOPs per forward pass
+Generative modeling
+Language
+(Hoffmann et al.)
+Language
+(Kaplan et al.)
+Image 32x32
+(Henighan et al.)
+Procgen
+CoinRun
+StarPilot
+FruitBot
+Easy
+Hard
+Width
+Depth (*)
+Optimal model size vs compute, Procgen,
+arithmetic intensity-adjusted
+(b) Using FLOPs per forward pass instead of parameters.
+Figure 8: Comparison of optimal model size vs compute for our Procgen width- and depth-scaling
+experiments. (*) It is important to understand how parameters and FLOPs were counted to interpret
+the depth-scaling results. This is explained in detail in Appendix D.
+for a given environment. Plots of our results may be found in Appendix C, and the numerical values
+of the fitted constants may be found in Appendix E.
+The main difference between our width-scaling and depth-scaling results is that the optimal model
+size for a given compute budget was significantly smaller for our depth-scaling experiments, but
+this was an artifact of how we counted parameters and FLOPs. As explained in Appendix D, we
+only included the part of the network being scaled in our parameter and FLOP calculations, which
+meant excluding the final dense layer of the network for our depth-scaling experiments, but not our
+width-scaling experiments. If this layer had been included in our depth-scaling calculations, it would
+have accounted for between 16% and 90% of the parameters but only 2% or fewer of the FLOPs,
+depending on the depth.
+Interestingly, as shown in Figure 8, the optimal model size vs compute scaling laws for our width-
+and depth-scaling experiments become much more similar if we measure model size using FLOPs
+per forward pass rather than parameters. This is because excluding the final dense layer from the
+parameter and FLOP calculations significantly increases the arithmetic intensity (i.e., FLOPs per
+parameter in a forward pass) as calculated for the depth-scaling experiments. This suggests that,
+when comparing models with very different arithmetic intensities, FLOPs per forward pass may
+be a better measure of model size than parameters (or perhaps arithmetic intensity should even be
+considered as an additional independent variable).
+4.5
+Natural performance metrics
+Although in general there may be no obvious performance metric that scales smoothly with model
+parameters and environment interactions, motivating our use of intrinsic performance, there may still
+be such a metric in some environments. We call such metrics natural performance metrics, and we
+were able to find them in a couple of our environments:
+• CoinRun: In the CoinRun environment from Procgen Benchmark, the episode return is
+always either 10 or 0, corresponding to whether or the agent successfully collects the coin at
+the end of the level. We found the fail-to-success ratio F := 10−R
+R
+, where R is the mean
+episode return, to be a natural performance metric for CoinRun. This is similar to the failure
+rate 1 − R
+10, since R is close to 10 for most of training, but provides a slightly better fit
+early in training, since it does not have an upper bound of 1. Note that the logarithm of the
+12
+
+1014
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+10−2
+10−1
+Fail-to-success ratio
+Easy
+Hard
+Learning
+curves
+Power law
+fitted to
+I−β
+(arbitrary
+function
+of ratio)
+Fail-to-
+success
+ratio
+CoinRun, efficient frontier fits
+Figure 9: Comparison of the efficient frontier fits
+for CoinRun, using intrinsic performance and the
+fail-to-success ratio.
+1014
+1016
+1018
+1020
+Compute (FLOPs)
+−5
+0
+5
+10
+15
+20
+25
+TrueSkill
+Learning
+curves
+Power law
+fitted to
+I−β
+(arbitrary
+function
+of T)
+e−αT T
+Dota 2, efficient frontier fits
+Figure 10: Comparison of the efficient frontier
+fits for Dota 2, using intrinsic performance and
+exponentiated scaled TrueSkill.
+fail-to-success ratio can also be thought of as the logit function (inverse sigmoid) of the
+failure rate.
+• Dota 2: Dota 2 is a two-player game, and so the performance of a policy must be measured
+by comparing it to other policies. The standard method for this is the TrueSkill rating
+system,2 in which differences in rating between policies correspond to win probabilities
+when the policies are played against one another, similarly to the Elo rating system. We
+found TrueSkill to be a natural performance metric for Dota 2.
+Specifically, we found that our power law for intrinsic performance, equation (1), still roughly held
+with the left-hand side replaced by a suitable function of the natural performance metric. For CoinRun,
+we used the fail-to-success ratio directly, but discarded data from early in training where this ratio
+was above 0.5. For Dota 2, we used e−αT T , where T is TrueSkill and αT is a fitted constant, which
+was needed because the scale of T is arbitrary.
+Figures 9 and 10 compare the efficient frontier fits for intrinsic performance and for the natural
+performance metric, for CoinRun and Dota 2 respectively. The fits match closely, except for Dota 2 at
+higher levels of TrueSkill. We conjecture that Dota 2 has an analog of an irreducible loss [Henighan
+et al., 2020], representing the maximum attainable TrueSkill for the family of models we trained.
+We explored introducing an additional fitted constant T ∗ for this maximum attainable TrueSkill, and
+using either of the functional forms e−αT T − e−αT T ∗ and (T ∗ − T)αT . However, it was unclear
+to us which of these forms made the most theoretical sense, and we were unsure whether we could
+justify the extra degree of freedom given the lack of data at higher levels of TrueSkill.
+The fitted constants for all of these alternative power laws for both CoinRun and Dota 2 are given in
+Appendix E. Interestingly, for CoinRun, the values of the scaling exponent for the fail-to-success
+ratio F in terms of intrinsic performance I, corresponding to the slopes of the lines in Figure 9, are
+similar between the two difficulty modes: F ∝ I−0.40 in easy mode and F ∝ I−0.48 in hard mode.
+5
+Discussion
+5.1
+Extrapolating sample efficiency
+We may use our power law for intrinsic performance, equation (1), to extrapolate sample efficiency
+to unseen model sizes N and environment interactions E. For example, in Figure 11, we show the
+2https://en.wikipedia.org/wiki/TrueSkill
+13
+
+0.0
+0.5
+1.0
+1.5
+2.0
+Interactions
+×108
+5
+10
+15
+20
+25
+30
+Mean episode return
+Parameters
+104.3
+104.6
+104.9
+105.2
+105.5
+105.8
+106.1
+106.4
+106.7
+107.0
+Learning
+curve
+Power law fit
+Power law
+N → ∞
+limit
+Sample efficiency, StarPilot, hard
+Figure 11: Learning curves for StarPilot (hard
+mode, scaling width), together with their power
+law fits, and the N → ∞ limit of the power law.
+10−7
+10−6
+10−5
+10−4
+10−3
+10−2
+Compute (PF-days)
+103
+104
+105
+106
+107
+Parameters
+Procgen (width)
+CoinRun
+StarPilot
+FruitBot
+Easy
+Hard
+Dota 2
+1v1
+GM (various)
+MNIST horizons
+1–256
+Optimal model size vs compute, Ne = 105
+Figure 12: Optimal model size vs compute, taking
+into account a hypothetical compute cost per en-
+vironment interaction equal to that of a model of
+size Ne = 105. See Figure 4 for the full legend.
+extrapolated learning curve for StarPilot in the infinite-width limit. This reaches the final performance
+of our largest model in about half the number of environment interactions. Note, however, that
+without a natural performance metric, we cannot extrapolate to unseen performance levels.
+It is natural to ask how this extrapolated infinite-width limit compares to human sample efficiency. On
+StarPilot (slowed down to 3 frames per second), a human can reach a mean episode return of around
+20 after a few episodes, whereas the extrapolated infinitely-wide model takes 18 million interactions,
+around 10,000 times as many. This is not really a fair comparison though, because much of the
+challenge in Procgen is to learn to identify basic visual features, which humans are already able to do.
+For Dota 2, we crudely estimate that it would take a human around 50–500 hours of gameplay to
+reach the performance of the extrapolated infinitely-wide LSTM after 5 billion interactions, a factor
+of 100–1,000 in sample efficiency. This comparison may be fairer, because Dota 2 has a structured
+observation space and is more challenging than StarPilot, although it still draws on many pre-existing
+human intuitions. Of course, our models were all trained from scratch, and we should expect this
+factor to be smaller for models that have been pre-trained to learn useful representations.
+5.2
+Cost-efficient reinforcement learning
+In the reinforcement learning literature, sample efficiency is usually taken to be the primary metric
+of algorithmic progress. This can be thought of as focusing on the cost of running the environment,
+but not the algorithm. At the other extreme, we have so far focused on the computational cost of the
+algorithm, but not on the cost of the environment. However, it is straightforward to now take both
+into account. To do this, let Ne be the cost of the environment, measured in terms of the number of
+parameters in a model with the same cost per interaction. Thus the total cost of both the algorithm
+and the environment is proportional to (N + Ne) E.
+The cost-efficient frontier is now described by the following generalization of equation (2):
+�
+1 + Ne
+N
+�
+αN
+�Nc
+N
+�αN
+= αE
+�Ec
+E
+�αE
+.
+Substituting this into our power law given by equation (1), it follows that along the cost-efficient
+frontier,
+C =
+�
+1 + Ne
+N
+� �
+1
+1 + αN
+αE
+�
+1 + Ne
+N
+�
+�
+1
+αN +
+1
+αE � N
+Nc
+�1+ αN
+αE ,
+14
+
+where C := (N + Ne) E. Thus for a given budget C, the optimal model size N scales as the same
+power law in C as before once N ≫ Ne, and it is only efficient to take N ≪ Ne when C is very
+small. This validates and makes precise the rule-of-thumb that it is usually inefficient to use a model
+that is much cheaper to run than the environment, at least when training from scratch.
+To illustrate this relationship, Figure 12 shows the optimal model size vs compute relationship from
+Figure 4, but incorporating a fixed hypothetical compute cost associated with each environment
+interaction.
+5.3
+Limitations
+Our experiments have several limitations:
+• As explained in Section 3.4, we did not use separate training runs for each compute budget,
+each with their own learning rate schedule, which can be necessary to accurately measure
+scaling exponents [Hoffmann et al., 2022]. We tried to mitigate this by using a learning rate
+schedule that worked well for a variety of compute budgets, as explained in Appendix B.1,
+but this may not have been enough.
+• As explained in Section 4.3, the variability of exponents over training gives rise to a
+measurement problem. We mitigated this to some extent by excluding data from early in
+training when fitting our power law, but this does not fully correct for the fact that some of
+our models were under-trained relative to the compute-efficient frontier.
+• We did not carefully optimize the aspect ratios of our models, instead scaling width and
+depth separately. More generally, suboptimal hyperparameters or other problems with our
+training setups could have lead to errors in our measurements of scaling constants.
+• Learning curves in reinforcement learning are often very high-variance, adding significant
+noise to power law fits. We mitigated this to some extent by choosing environments with
+relatively low-variance learning curves and using multiple random seeds, but a lot of variance
+still remained.
+As a result of these limitations, we do not think conclusions that depend on the precise fitted values of
+our scaling constants can be drawn with confidence, although we consider our mitigations sufficient
+for more qualitative conclusions. We are excited for future work to fix these limitations, explore
+new domains, and more carefully disentangle the effects of the choice of algorithm, architecture and
+hyperparameters as well as properties of the environment.
+5.4
+Forecasting compute requirements
+The scaling of optimal model size with compute is a key input into the biological anchors framework
+for forecasting transformative artificial intelligence [Cotra, 2020]. In this framework, the human brain
+is used as a biological anchor for estimating the number of parameters in a transformative model, and
+optimal model size vs compute scaling laws are used to forecast the total compute required to train
+such a model. In this section we summarize the main implications of our work for this framework.
+Scaling exponents for reinforcement learning lie in a similar range to generative modeling. The
+exponent for the scaling of optimal model size with compute,
+1
+1+αN/αE , varied between around 0.4
+and 0.8 for our environments, a range that encompasses previous measurements of this exponent for
+generative modeling. However, as discussed in Section 5.3, we do not think our measurements of this
+exponent should be taken literally, due to the limitations of our experiments. The results of Hoffmann
+et al. [2022] and Bahri et al. [2021] suggest the possibility that this exponent would be around 0.5 in
+every domain if it were measured carefully enough, and we consider our results to be inconclusive on
+this question.
+Scaling coefficients for reinforcement learning vary by multiple orders of magnitude. The
+coefficient for the scaling of optimal model size with compute, Nc
+�
+1 + αN
+αE
+�
+1
+αN , varied substantially,
+enough that we do not think this variation is attributable only to the limitations of our experiments.
+For example, the scaling exponents for MNIST (with a horizon length of 1) and Dota 2 are very
+similar, but a model of the same size needs to be trained for around 2,000 times longer on Dota 2
+than on MNIST to be compute-efficient. By comparison, Henighan et al. [2020] found generative
+15
+
+modeling to require around 20 times as much training on 32x32 images than on language. Moreover,
+our analysis of the effect of the task horizon length gives a plausible mechanism for this variation.
+Arithmetic intensity may confound scaling coefficients. As discussed in Section 4.4, the coefficient
+for the scaling of optimal model size with compute can be affected by the arithmetic intensity (i.e.,
+the number of FLOPs per parameter in a forward pass) of the model. This alone does not explain the
+large variation in this coefficient between MNIST and Dota 2, for example, but it may explain some
+of the other variation. We hypothesize that, when comparing models with very different arithmetic
+intensities, due to parameter sharing or methods such as mixture of experts, it may be better to
+measure model size in FLOPs per forward pass rather than in parameters.
+Sample efficiency is an affine function of the task horizon length. We study the effect of the
+task horizon length using a toy MNIST-based environment in Section 4.2. Both theoretically (see
+Proposition 1) and empirically, the number of samples required to reach a given level of performance
+grows with the horizon length as an affine function (i.e., a linear function with an intercept) that
+depends on both the model size and the target performance level. However, our analysis makes a
+simplifying assumption of independent timesteps, which does not hold in most environments. In
+particular, we do not analyze the need for curricula and/or exploration to solve tasks for which it is
+challenging to obtain useful feedback. Instead, we simply assume that the algorithm pays attention to
+rewards over a longer time horizon, making credit assignment harder.
+This result validates and refines the analysis of Cotra [2020], who defined the “effective horizon
+length” as a quantity that scales linearly with training data requirements, incorporating not only the
+horizon length as we define it, but also reward sparsity, noise and so on. Our result specifically isolates
+the explicit horizon length, showing that training data requirements are a sum of two components,
+at least in our toy setting: one corresponding to a version of the task in which the horizon ends
+immediately, and another that is proportional to the horizon length. This implies that, for a given fixed
+task, continuing to increase the horizon length will eventually lead to a proportional increase in the
+compute budget corresponding to a given optimal model size, without changing the exponent of this
+scaling law. However, this will only happen once the first component has become negligible, and it is
+unclear whether there are realistic tasks of different horizon lengths for which this first component is
+negligible in practice.
+We are excited for future work to study other aspects of the “effective horizon length”, such as
+reward sparsity and noise, as well as studying the explicit horizon length in environments that are less
+artificial. It is not entirely clear how to quantify these properties in general, and they could potentially
+affect scaling exponents as well as scaling coefficients, if for example they change over the course of
+training.
+Measuring scaling exponents precisely is challenging. The biological anchors framework uses
+the scaling of optimal model size with compute to perform a substantial extrapolation, making it
+particularly sensitive to the exponent of this relationship. This makes it challenging to measure this
+exponent with sufficient precision. In addition to the challenges raised by Hoffmann et al. [2022]
+involving learning rate schedules, we hope that others will benefit from learning about the other
+challenges we faced, which are summarized in Section 5.3.
+6
+Conclusion
+We have shown how to extend scaling laws to single-agent reinforcement learning using the notion of
+intrinsic performance. Across a range of environments, intrinsic performance scales as a power law
+in model size and environment interactions, and hence the optimal model size scales as a power law in
+the training compute budget. We have studied how this relationship is affected by various properties
+of the training setup, including the horizon length of the task, and have discussed the implications of
+this for the biological anchors framework for forecasting transformative artificial intelligence.
+7
+Acknowledgments
+Thanks to Mira Murati, Karl Cobbe, Chris Hesse, David Farhi, Paul Christiano, Jared Kaplan, Long
+Ouyang and Ajeya Cotra for discussions, ideas, help, advice, support and inspiration that have greatly
+benefited this project.
+16
+
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+arXiv:1912.06680, 2019.
+J. Schulman, P. Moritz, S. Levine, M. Jordan, and P. Abbeel. High-dimensional continuous control
+using generalized advantage estimation. arXiv preprint arXiv:1506.02438, 2015.
+J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov. Proximal policy optimization
+algorithms. arXiv preprint arXiv:1707.06347, 2017.
+D. Silver, T. Hubert, J. Schrittwieser, I. Antonoglou, M. Lai, A. Guez, M. Lanctot, L. Sifre, D. Ku-
+maran, T. Graepel, et al. A general reinforcement learning algorithm that masters chess, shogi, and
+Go through self-play. Science, 362(6419):1140–1144, 2018.
+17
+
+H. Zhang, Y. N. Dauphin, and T. Ma. Fixup initialization: Residual learning without normalization.
+arXiv preprint arXiv:1901.09321, 2019.
+18
+
+A
+Curve-fitting methodology
+In this section we discuss our methodology for computing intrinsic performance and fitting the power
+law constants, which require some care. Code for our full procedure, along with its application to
+our experiments, may be found in this Colab notebook: https://colab.research.google.com/
+drive/1PzwZyXsi9jRdVCj1GJrS8JdOPBQ7LHZV.
+Recall that the intrinsic performance of a policy is the minimum compute required to train a model of
+any size in the same family to reach the same return (averaged over random seeds). The naive way to
+compute this would be to train models of many different sizes, and to take the best-performing model
+size for each possible compute budget. However, it may not be feasible to train models of enough
+different sizes to get a reasonable level of granularity, while using enough different random seeds
+sufficiently to reduce the high variance of learning curves.
+To cope with this, we compute intrinsic performance and fit the power law constants together. This
+allows us to make use of all the data from each learning curve, instead of just a single point from
+each one. We do this by jointly fitting the power law constants and a monotonic function f to
+f (R)−β =
+�Nc
+N
+�αN
++
+�Ec
+E
+�αE
+,
+where R is the mean episode return (or another performance metric such as TrueSkill), N is the
+number of model parameters, and E is the number of environment interactions. By also requiring the
+relationships between the constants from Lemma 1 to hold, this provides us both with the power law
+constants, and with the desired function f satisfying f (R) = I, where I is intrinsic performance.
+We perform this fit by using a black-box optimization algorithm such as CMA-ES to fit αN, αE
+and Nc, which determine β and Ec, with monotonic regression3 in the inner loop to fit f, using the
+squared error of the regression as the black-box loss function. We actually fit log (f) rather than f in
+order to obtain a good fit to I on a logarithmic scale, and we weight the data in proportion to 1
+E so
+that each interval is given equal weight on a logarithmic scale. In our Colab notebook, this routine is
+performed by the function fit_coeffs.
+This procedure seems to work well off-the-shelf, typically converging to a unique local minimum.
+However:
+• When there is a lack of data or the data is very noisy, the local minimum may not be a global
+minimum, and the procedure can diverge to a degenerate solution.
+• It is necessary to first smooth learning curves so that they are mostly monotonic, to prevent
+the monotonic regression from overfitting. In our Colab notebook, we use the function
+smooth, which uses standard errors to automatically choose smoothing parameters (although
+note that we used slightly different smoothing parameters for MNIST).
+• As discussed in Section 4.3, it is important to exclude data from early in training.
+Our full procedure is therefore as follows.
+• Smooth learning curves. Plot the smoothed curves on a logarithmic scale to check the
+monotonicity and fit, and adjust the smoothing parameters if necessary.
+• Exclude data from early in training, balancing the need for data against how much the early
+data skews the fit. Typically at least the first
+1
+64 of training should be excluded.
+• Fit the power law constants and f using the black-box optimization with monotonic regres-
+sion routine.
+• Plot the fit to check the routine did not diverge. If it did, re-run routine, or constrain the
+constants and re-run, or include more data in step 2. If none of these fixes the divergence,
+then it may be necessary to collect more data.
+• Check the fit is not overly skewed by data from early in training. If it is, exclude more data
+in step 2.
+3https://en.wikipedia.org/wiki/Isotonic_regression
+19
+
+This procedure led us to exclude the first 3 million environment interactions for Procgen, the first 2
+billion environment interactions for Dota 2, and the first 216, 219 or 222 environment interactions for
+MNIST depending on the period of training being considered, as discussed in Section 4.3.
+A.1
+Fitting to natural performance metrics
+As discussed in Section 4.5, as well as fitting our power law with I−β on the left-hand side, as in
+equation (1), we also fit it using various other expressions, such as e−αT T , where T is TrueSkill and
+αT is a fitted constant. When doing this, we adopt the convention that the constraints on β and Ec
+from Lemma 1 should continue to hold. This necessitates introducing an additional multiplier, and
+instead fitting
+Tce−αT T =
+�Nc
+N
+�αN
++
+�Ec
+E
+�αE
+for example, where Tc is a fitted constant. Doing this allows us to continue interpret the left-hand
+side of this equation as I−β.
+To fit equations of this form, we continue use the same black-box optimization method, and simply
+replace the monotonic regression by another method of fitting log (f). For example, we may fit
+f (T)−β = Tce−αT T
+by using linear regression to fit log (f). (Recall that β is already determined by αN and αE.)
+The function from our Colab notebook, fit_coeffs, provides options for fitting various functional
+forms for f, although it can sometimes be slow. (This is because it sometimes uses black-box
+optimization again in the inner loop for ease of implementation, even though this could be collapsed
+into the outer loop if speed were important.)
+20
+
+B
+Hyperparameters
+Our default hyperparameters for Procgen, Dota 2 and MNIST are given in Tables 1, 2 and 3
+respectively. We modified these defaults in two ways:
+• We adjusted the Adam step size as the model was scaled, as explained in Section 3.4.
+• For Procgen and MNIST, we incorporated a batch ramp and learning rate schedule, as
+explained in Section B.1.
+Table 1: Default PPG-EWMA hyperparameters for Procgen.
+Hyperparameter
+Value
+PPO
+Parallel environments
+1024
+Timesteps per rollout (T)
+256
+Minibatches per epoch
+8
+Adam step size (α)
+5 × 10−4
+Value function coefficient
+0.5
+Entropy coefficient
+0.01
+PPO clipping parameter (ϵ)
+Not used
+PPO KL penalty coefficient (β)
+1
+GAE discount rate (γ)
+0.999
+GAE bootstrapping parameter (λ)
+0.95
+Reward normalization?
+Yes
+Advantage normalization?
+Yes
+Total environment interactions
+200 million
+PPG
+Policy iterations per phase (Nπ)
+32
+Policy phase policy epochs (Eπ)
+1
+Policy phase value function epochs (EV )
+1
+Auxiliary phase epochs (Eaux)
+6
+Auxiliary phase minibatches per epoch
+16Nπ
+Auxiliary phase cloning coefficient (βclone)
+1
+PPG-EWMA
+Proximal policy EWMA decay rate (βprox)
+8
+9
+Batch ramp
+Initial batch size multiplier
+1
+32
+Table 2: PPO hyperparameters for Dota 2.
+Hyperparameter
+Value
+Parallel environments
+6144
+Timesteps per rollout (T)
+512
+Minibatches per epoch
+32
+Epochs (E)
+1
+Adam step size (α)
+10−4 to 10−3
+PPO clipping parameter (ϵ)
+0.2
+PPO KL penalty coefficient (β)
+Not used
+GAE bootstrapping parameter (λ)
+0.95
+Total environment interactions
+13.6–82.6 billion
+21
+
+Table 3: Default PPO-EWMA hyperparameters for MNIST in terms the horizon length h, which
+varied from 1 to 256.
+Hyperparameter
+Value
+PPO
+Parallel environments
+16
+Timesteps per rollout (T)
+512
+Minibatches per epoch
+8
+Epochs (E)
+1
+Adam step size (α)
+1 × 10−3
+Value function coefficient
+0.5
+Entropy coefficient
+0.01
+PPO clipping parameter (ϵ)
+Not used
+PPO KL penalty coefficient (β)
+1
+GAE discount rate (γ)
+1 −
+2
+h+1
+GAE bootstrapping parameter (λ)
+1
+Reward normalization?
+Yes
+Advantage normalization?
+Yes
+Total environment interactions
+225
+PPO-EWMA
+Proximal policy EWMA decay rate (βprox)
+8
+9
+Batch ramp
+Initial batch size multiplier
+√
+h
+64
+B.1
+Batch ramp and learning rate schedule
+As explained in Section 3.4, it was important to use a well-tuned learning rate schedule, and to use
+a schedule that works well for a variety of compute budgets. It was also important to use a batch
+ramp, i.e., to start with a small batch size and increase it over the course of training, because the
+critical batch size is smaller at the start of training, and we needed training to still be sample-efficient
+for small compute budgets. Without a batch ramp, we would have needed to adjust our power law,
+equation (1), in much the same way as the corresponding law for language [Kaplan et al., 2020,
+equation (1.6)], which uses Smin (S), the minimum number of optimization steps as estimated using
+a power law fit to the gradient noise scale.
+Note, however, that increasing the batch size has a very similar effect to lowering the learning rate.
+To simplify matters, we used PPO-EWMA and PPG-EWMA, which are batch size-invariant [Hilton
+et al., 2021], allowing us to have almost the same effect as increasing the batch size by instead
+lowering the learning rate and increasing the center of mass of the proximal policy EWMA. We then
+considered only the batch size schedule, whether implemented explicitly or implicitly via these other
+hyperparameters.
+To explore promising schedules, we implemented a greedy adaptive batch size algorithm, which tries
+doubling the batch size and switches if that performs better, or else backtracks and stays with the
+current batch size. We experimented with this on StarPilot’s easy difficulty setting, using model sizes
+spanning a factor of around 2048. We found our algorithm to fairly consistently choose a schedule
+that can be well-approximated by the power law
+B = max
+�
+Bmin, E0.84
+80
+�
+,
+where B is the batch size in interactions, E is the total number of interactions so far, and Bmin = 256
+was our initial batch size.
+Having fit this power law schedule on one Procgen environment, we tested it on several different
+Procgen environments, and found it to consistently outperform our usual fixed batch size both at the
+start and end of training. (Curiously, our schedule sometimes underperformed the fixed batch size in
+the middle of training. We believe this may be explained by the smaller initial batch size causing the
+entropy to fall too quickly at the start of training, highlighting a pitfall of the greedy approach.) In
+particular, we were able to use the same schedule on both the easy and hard difficulty settings. Our
+22
+
+usual fixed batch size, on the other hand, was larger for the hard setting, corresponding to the fact
+that it was tuned to longer training runs.
+The same schedule also worked well on our MNIST environment at every horizon length, although it
+was necessary to tune Bmin. Using too small a value for Bmin seemed to result in an instability which
+could not always be recovered from. We found the optimal Bmin to vary based on the horizon length
+h, and we took Bmin = 16
+√
+h (though taking Bmin to have the form A0 + A1h would probably have
+made more theoretical sense in hindsight, given the results of Section 4.2). If trying our schedule
+on other environments, we suggest tuning Bmin to ensure stability at the start of training, but it is
+probably less important to tune the power law constants.
+We used this batch size schedule for both our Procgen and MNIST experiments (although it would
+probably have been better to fully re-fit the schedule for MNIST). We implemented this using a batch
+size multiplier, explicitly reducing the batch size when the multiplier was less than 1, and changing
+the learning rate and center of mass of the proximal policy EWMA instead when the multiplier was
+greater than 1. With Procgen, for which we used PPG-EWMA, we also changed the number of policy
+iterations per phase, Nπ, in proportion to the batch size, since we thought the number of optimization
+steps per phase should remain constant, and we rounded the batch size multiplier to the nearest power
+of two, with minimum and maximum multipliers of
+1
+32 and 4 (corresponding to batch sizes of 1024
+and 131072 respectively).
+For Dota 2, we did not use a batch size schedule, since those experiments were carried out before we
+investigated batch size schedules.
+23
+
+C
+Results in full
+All the data from our experiments may be accessed using this Colab notebook: https://colab.
+research.google.com/drive/1PzwZyXsi9jRdVCj1GJrS8JdOPBQ7LHZV.
+This also includes
+code for analyzing this data, including model size and compute calculations, intrinsic performance
+and power law fitting, and generating all the plots in this paper.
+Figures 13, 14, 15 and 16 show learning curves as a function of total training compute, together with
+their power law fits, for all of our experiments. On the left of each figure we show mean episode
+return (or failure rate for CoinRun and MNIST, or TrueSkill for Dota 2), with error bars showing
+mean ±1 sample standard deviation over the random seeds. On the right of each figure, we show
+intrinsic performance, with error bars hidden for clarity.
+1015
+1017
+Compute (FLOPs)
+10−2
+10−1
+Failure rate
+CoinRun, easy
+1015
+1017
+Compute (FLOPs)
+10−1
+Failure rate
+CoinRun, hard
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+Intrinsic performance (FLOPs)
+CoinRun, easy
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+CoinRun, hard
+Parameters
+104.3
+104.6
+104.9
+105.2
+105.5
+105.8
+106.1
+106.4
+106.7
+107.0
+1015
+1017
+Compute (FLOPs)
+10
+20
+30
+40
+50
+60
+Mean episode return
+StarPilot, easy
+1015
+1017
+Compute (FLOPs)
+5
+10
+15
+20
+25
+30
+Mean episode return
+StarPilot, hard
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+StarPilot, easy
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+StarPilot, hard
+Learning
+curve
+Power law
+fit
+Power law
+asymptote
+Efficient
+frontier
+Efficient
+points
+1015
+1017
+Compute (FLOPs)
+5
+10
+15
+20
+25
+30
+Mean episode return
+FruitBot, easy
+1015
+1017
+Compute (FLOPs)
+0
+5
+10
+15
+20
+25
+Mean episode return
+FruitBot, hard
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+Intrinsic performance (FLOPs)
+FruitBot, easy
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+FruitBot, hard
+Procgen, width
+Figure 13: Learning curves as a function of total training compute for our Procgen width-scaling
+experiments, together with their power law fits. Left half: mean episode return or failure rate, mean
+±1 sample standard deviation over three seeds shown. Right half: intrinsic performance, mean only
+shown.
+24
+
+1015
+1017
+Compute (FLOPs)
+10−2
+10−1
+Failure rate
+CoinRun, easy
+1015
+1017
+Compute (FLOPs)
+10−1
+Failure rate
+CoinRun, hard
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+Intrinsic performance (FLOPs)
+CoinRun, easy
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+CoinRun, hard
+Parameters
+103.9
+104.1
+104.4
+104.6
+104.9
+105.2
+105.5
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+20
+30
+40
+50
+60
+Mean episode return
+StarPilot, easy
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+5
+10
+15
+20
+25
+Mean episode return
+StarPilot, hard
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+StarPilot, easy
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+StarPilot, hard
+Learning
+curve
+Power law
+fit
+Power law
+asymptote
+Efficient
+frontier
+Efficient
+points
+1015
+1017
+Compute (FLOPs)
+5
+10
+15
+20
+25
+30
+Mean episode return
+FruitBot, easy
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+0
+5
+10
+15
+20
+25
+Mean episode return
+FruitBot, hard
+1015
+1017
+Compute (FLOPs)
+1014
+1015
+1016
+1017
+Intrinsic performance (FLOPs)
+FruitBot, easy
+1015
+1016
+1017
+1018
+Compute (FLOPs)
+1015
+1016
+1017
+1018
+Intrinsic performance (FLOPs)
+FruitBot, hard
+Procgen, depth
+Figure 14: Learning curves as a function of total training compute for our Procgen depth-scaling
+experiments, together with their power law fits. Left half: mean episode return or failure rate, mean
+±1 sample standard deviation over three seeds shown. Right half: intrinsic performance, mean only
+shown.
+1014
+1016
+1018
+1020
+Compute (FLOPs)
+−5
+0
+5
+10
+15
+20
+25
+TrueSkill
+1014
+1016
+1018
+1020
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+1017
+1018
+1019
+Intrinsic performance (FLOPs)
+Parameters
+102.7
+104.5
+105.1
+105.7
+106.3
+106.9
+108.1
+Learning
+curve
+Power law
+fit
+Power law
+asymptote
+Efficient
+frontier
+Efficient
+points
+Dota 2
+Figure 15: Learning curves as a function of total training compute for Dota 2, together with their
+power law fits. Only one random seed was used. Left: TrueSkill. Right: intrinsic performance.
+25
+
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+Failure rate
+Horizon 1
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+Failure rate
+Horizon 2
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 1
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 2
+Parameters
+104.8
+105.1
+105.4
+105.7
+106.0
+106.3
+106.6
+106.9
+107.2
+107.5
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+Failure rate
+Horizon 4
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+Failure rate
+Horizon 8
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 4
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 8
+Learning
+curve
+Power law
+fit
+Power law
+asymptote
+Efficient
+frontier
+Efficient
+points
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+Failure rate
+Horizon 16
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+Failure rate
+Horizon 32
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 16
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 32
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+Failure rate
+Horizon 64
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+10−2
+Failure rate
+Horizon 128
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 64
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+1016
+Intrinsic performance (FLOPs)
+Horizon 128
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+10−2
+Failure rate
+Horizon 192
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+10−3
+10−2
+Failure rate
+Horizon 256
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+Intrinsic performance (FLOPs)
+Horizon 192
+1013
+1014
+1015
+1016
+Compute (FLOPs)
+1013
+1014
+1015
+Intrinsic performance (FLOPs)
+Horizon 256
+MNIST, late period
+Figure 16: Learning curves as a function of total training compute for MNIST, together with their
+power law fits, for the late period of training (222–225 environment interactions). Left half: failure
+rate, mean ±1 sample standard deviation over the middle-performing 16 of 20 random seeds shown.
+Right: intrinsic performance, mean only shown.
+26
+
+D
+Parameter and FLOP calculations
+In counting parameters and FLOPs, we apply the following principles:
+• We only include the part of the network that is being scaled (ignoring things like embedding
+parameters), since we consider that to be the bottleneck.
+• We use round numbers (ignoring negligible contributions such as as biases and activations),
+for simplicity.
+• We include both rollout and optimization FLOPs (including any additional overhead of
+PPO-EWMA).
+• We treat an add-multiply as 2 FLOPs.
+For example, we treat the forward pass of a dense layer as taking 2 FLOPs per batch item per
+parameter, and a convolutional layer as taking 2houtwout FLOPs per batch item per parameter. We
+treat a backward pass as taking 2× the FLOPs of a forward pass.
+For the Procgen width-scaling experiments, we ignore the first convolution, since it scales as width
+(instead of as width squared), and has few parameters. Similarly, for the depth-scaling experiments,
+we ignore the final dense layer, since we only vary the number of convolutional layers. Unfortunately,
+as discussed in Section 4.4, the final dense layer contains many parameters, which skews our constants.
+In both cases, we include both the policy and value networks, which are separate with identical
+architectures. We use PPG-EWMA with 1 policy epoch and 6 auxiliary epochs, totaling 9 forward
+and 7 backward passes per interaction.
+For the Dota experiments, we ignore the embedding layer, considering only the LSTM. Since each
+interaction was used only once, we count 2 forward passes and 1 backward pass per interaction (1
+forward pass for the rollout, and 1 forward-backward pass for optimization).
+For the MNIST experiments, we ignore the first convolution, as for the Procgen width-scaling
+experiments. However, we only include the policy network, since the task of the value network is
+trivial (due to timesteps being independent). We use PPO-EWMA with 1 epoch, totaling 3 forward
+passes and 1 backward pass per interaction.
+The numerical results of these calculations are as follows.
+• Procgen, scaling width: for the width multiplier w = 2−3, 2−2.52−2, . . . , 22.5, we count
+1242112w2 parameters and 2652897280w2 FLOPs per interaction.
+• Procgen, scaling depth: for the number of residual blocks b = 1, 2, 4, . . . , 64, we count
+5184b + 1944 parameters and 61046784b + 81395712 FLOPs per interaction.
+• Dota 2: for the LSTM size s = 8, 64, 128, 256, 512, 1024, 4096, we count 8s2 parameters
+and 64s2 FLOPs per interaction.
+• MNIST: for the width multiplier w = 2−3, 2−2.52−2, . . . , 22.5, we count 3948800w2
+parameters and 95648000w2 FLOPs per interaction.
+Note that one of our modeling assumptions is that the number of FLOPs per interaction is proportional
+to the number of parameters, but this is not true for our Procgen depth-scaling experiments. In other
+words, the number of FLOPs per param-interact, which is used to convert compute from units of
+parameters × interactions to units of FLOPs, is not constant. However, this number differs by at most
+40% from the mean of this number over the different depths, and so we simply used the mean when
+doing this conversion.
+27
+
+E
+Fitted constants
+In this section we provide the constants αN, αE and Nc, together with the values of β and Ec derived
+using Lemma 1, for our fitted power laws for intrinsic performance I as given by equation (1). We
+also provide Imin and Imax, the minimum and maximum intrinsic performance obtained during the
+span of interaction counts considered; our model is not able to predict mean episode return outside
+this range. Recall that the units of I are parameters × interactions; the conversion to FLOPs may be
+performed using the values given in Appendix D.
+We also provide the derived equations for optimal model size N vs compute C in PF-days. By
+substituting equation (2) for the compute-efficient frontier into equation (1), these are given by
+N = Nc
+�
+1 + αN
+αE
+�
+1
+αN � C × 1015 × 24 × 3600
+FLOPs per param-interact
+�
+1
+1+ αN
+αE
+for
+Nmin ≤ N ≤ Nmax.
+We take Nmin and Nmax to be the minimum and maximum model sizes we tested whose power law
+fit intersects the compute-efficient frontier somewhere between Imin and Imax.
+For our comparison to generative modeling, we use these equations for optimal model size N vs
+compute C in PF-days:
+• Language [Hoffmann et al., 2022]: N = (
+C
+1.4×10−18 )0.5
+• Language [Kaplan et al., 2020]: N = (
+C
+3.3×10−13 )0.73
+• Image 32x32 [Henighan et al., 2020]: N = (
+C
+1.6×10−13 )0.65
+Further fitted constants, such as for single seeds, for different spans of interaction counts (see Section
+4.2), and fitted to natural performance metrics (see Section 4.5), may be found in this Colab notebook:
+https://colab.research.google.com/drive/1PzwZyXsi9jRdVCj1GJrS8JdOPBQ7LHZV.
+E.1
+Procgen, scaling width
+The fitted constants for our Procgen width-scaling experiments are as follows.
+Environment
+αN
+αE
+β
+Nc
+Ec
+Imin
+Imax
+CoinRun, easy
+0.542
+0.462
+0.249
+2.53 × 10−2
+2.49 × 100
+4.83 × 1010
+2.55 × 1014
+CoinRun, hard
+0.759
+0.576
+0.328
+1.55 × 10−1
+8.00 × 10−1
+6.07 × 1010
+3.45 × 1014
+StarPilot, easy
+0.318
+0.604
+0.208
+2.25 × 10−4
+2.02 × 102
+4.88 × 1010
+1.95 × 1015
+StarPilot, hard
+0.453
+0.533
+0.245
+4.55 × 10−3
+1.31 × 101
+5.43 × 1010
+1.09 × 1015
+FruitBot, easy
+0.527
+0.350
+0.210
+9.17 × 10−2
+4.46 × 10−1
+5.24 × 1010
+1.67 × 1014
+FruitBot, hard
+0.478
+0.346
+0.201
+1.14 × 10−1
+2.96 × 10−1
+6.00 × 1010
+7.26 × 1014
+These imply the following equations for optimal model size N vs compute C in PF-days.
+• CoinRun, easy: N = 4.615 × 106 × C0.4600 for 19408 ≤ N ≤ 310528
+• CoinRun, hard: N = 6.881 × 106 × C0.4315 for 43668 ≤ N ≤ 587092
+• StarPilot, easy: N = 6.383 × 107 × C0.6549 for 19408 ≤ N ≤ 4968448
+• StarPilot, hard: N = 1.668 × 107 × C0.5404 for 19408 ≤ N ≤ 1242112
+• FruitBot, easy: N = 2.243 × 106 × C0.3994 for 19408 ≤ N ≤ 174672
+• FruitBot, hard: N = 6.631 × 106 × C0.4201 for 43668 ≤ N ≤ 587092
+As discussed in Section 4.5, for CoinRun, we also fit power laws using the fail-to-success ratio F,
+excluding data for which F > 0.5. As explained in Section A.1, we replaced I−β with F
+Fc , where Fc
+is a fitted constant. The fitted constants for these power laws are as follows.
+28
+
+Difficulty
+αN
+αE
+β
+Nc
+Ec
+Imin
+Imax
+Easy
+0.899
+1.007
+0.475
+1.00 × 10−2
+2.33 × 101
+2.55 × 1010
+2.60 × 1014
+Hard
+0.833
+0.776
+0.402
+4.69 × 10−2
+3.80 × 100
+5.14 × 1011
+7.38 × 1014
+Difficulty
+Fc
+Easy
+3.88 × 104
+Hard
+2.52 × 104
+These imply the following relationships between I and F.
+• Easy: I = 4.57 × 109 × F −
+1
+0.475
+• Hard: I = 9.15 × 1010 × F −
+1
+0.402
+They also imply the following equations for optimal model size N vs compute C in PF-days.
+• Easy: N = 1.216 × 107 × C0.5285 for 19408 ≤ N ≤ 587092
+• Hard: N = 1.148 × 107 × C0.4822 for 77632 ≤ N ≤ 1242112
+E.2
+Procgen, scaling depth
+The fitted constants for our Procgen depth-scaling experiments are as follows.
+Environment
+αN
+αE
+β
+Nc
+Ec
+Imin
+Imax
+CoinRun, easy
+0.351
+0.469
+0.201
+2.64 × 10−4
+1.26 × 102
+5.43 × 109
+3.72 × 1013
+CoinRun, hard
+0.336
+0.581
+0.213
+1.02 × 10−4
+4.47 × 102
+6.58 × 109
+6.24 × 1013
+StarPilot, easy
+0.800
+0.821
+0.405
+9.65 × 10−3
+1.87 × 101
+1.70 × 1010
+5.52 × 1013
+StarPilot, hard
+0.380
+0.381
+0.190
+2.87 × 10−3
+9.11 × 100
+1.58 × 1010
+5.21 × 1013
+FruitBot, easy
+0.539
+0.564
+0.276
+2.92 × 10−3
+2.77 × 101
+9.58 × 109
+3.76 × 1013
+FruitBot, hard
+0.401
+0.463
+0.215
+1.23 × 10−3
+3.26 × 101
+1.34 × 1010
+4.64 × 1013
+These imply the following equations for optimal model size N vs compute C in PF-days. Note,
+however, that:
+• As discussed in Section 4.4, we exclude the final dense layer, which would have accounted
+for between 16% and 90% of the parameters, depending on the depth. This skews the
+leading constants here.
+• As discussed in Appendix D, we also ignored the variation in the number of FLOPs per
+param-interact between models of different depths, leading to errors of up to 40%.
+• CoinRun, easy: N = 1.390 × 106 × C0.5723 for 7128 ≤ N ≤ 43416
+• CoinRun, hard: N = 3.962 × 106 × C0.6337 for 7128 ≤ N ≤ 167832
+• StarPilot, easy: N = 2.202 × 106 × C0.5063 for 7128 ≤ N ≤ 167832
+• StarPilot, hard: N = 1.410 × 106 × C0.5007 for 7128 ≤ N ≤ 84888
+• FruitBot, easy: N = 1.172 × 106 × C0.5110 for 7128 ≤ N ≤ 84888
+• FruitBot, hard: N = 1.671 × 106 × C0.5359 for 7128 ≤ N ≤ 84888
+29
+
+E.3
+Dota 2
+As explained in Sections 4.5 and A.1, we fit power laws to I−β, Tce−αT T , Tc
+�
+e−αT T − eαT T ∗�
+and Tc (T ∗ − T)αT , where I is intrinsic performance, T is TrueSkill, and αT , Tc and T ∗ are fitted
+constants. The fitted constants for these different functional forms are as follows.
+Fit to
+αN
+αE
+β
+Nc
+Ec
+Imin
+Imax
+I−β
+0.186
+0.593
+0.141
+1.98 × 10−8
+1.04 × 106
+6.83 × 1011
+1.79 × 1018
+Tce−αT T
+0.180
+0.486
+0.131
+3.53 × 10−8
+3.33 × 105
+4.62 × 1011
+2.24 × 1017
+Tc(e−αT T − eαT T ∗)
+0.181
+0.560
+0.137
+2.07 × 10−8
+8.32 × 105
+6.31 × 1011
+1.77 × 1018
+Tc(T ∗ − T)αT
+0.183
+0.569
+0.138
+2.06 × 10−8
+8.82 × 1005
+6.71 × 1011
+1.23 × 1018
+Fit to
+αT
+Tc
+T ∗
+I−β
+-
+-
+-
+Tce−αT T
+0.0572
+2.16 × 10−2
+-
+Tc(e−αT T − eαT T ∗)
+0.0402
+2.40 × 10−2
+35.43
+Tc(T ∗ − T)αT
+2.84
+2.14 × 10−7
+54.01
+As discussed in Section 4.5, we have less confidence in the last two functional forms, which is
+reflected in the very different estimates for T ∗, which represents the maximum attainable TrueSkill
+for the family of models we trained.
+These imply the following relationships between I and T for the last three fits.
+• Tce−αT T :
+I = 4.93 × 1012 × 1.5462T
+• Tc(e−αT T − eαT T ∗):
+I = 6.49 × 1011 ×
+�
+1.0410−T − 1.0410−35.43�−
+1
+0.137
+• Tc(T ∗ − T)αT :
+I = 1.48 × 1048 × (54.01 − T)− 2.84
+0.138
+They also imply the following equations for optimal model size N vs compute C in PF-days.
+• I−β:
+N = 2.703 × 107 × C0.7617 for 512 ≤ N ≤ 2097152
+• Tce−αT T :
+N = 1.607 × 107 × C0.7302 for 512 ≤ N ≤ 524288
+• Tc(e−αT T − eαT T ∗):
+N = 2.305 × 107 × C0.7552 for 512 ≤ N ≤ 2097152
+• Tc(T ∗ − T)αT :
+N = 2.385 × 107 × C0.7567 for 512 ≤ N ≤ 2097152
+E.4
+MNIST
+The fitted constants for our MNIST experiments are as follows. As discussed in Section 4.3, these
+constants are for the late period of training (222–225 environment interactions). Recall also that the
+horizon h is such that the interval [0, h − 1] has the same center of mass as an exponentially-weighted
+moving average with decay parameter γ, i.e., γ = 1 −
+2
+h+1.
+30
+
+Horizon
+αN
+αE
+β
+Nc
+Ec
+Imin
+Imax
+1
+0.263
+1.050
+0.210
+9.79 × 10−6
+9.43 × 103
+1.79 × 1011
+1.00 × 1015
+2
+0.265
+0.979
+0.208
+1.32 × 10−5
+6.30 × 103
+1.87 × 1011
+9.66 × 1014
+4
+0.284
+0.791
+0.209
+4.21 × 10−5
+1.50 × 103
+1.94 × 1011
+4.19 × 1014
+8
+0.276
+0.826
+0.207
+2.83 × 10−5
+2.33 × 103
+1.80 × 1011
+6.24 × 1014
+16
+0.252
+0.830
+0.193
+1.59 × 10−5
+3.78 × 103
+1.62 × 1011
+7.69 × 1014
+32
+0.263
+0.856
+0.201
+1.73 × 10−5
+3.83 × 103
+1.59 × 1011
+7.47 × 1014
+64
+0.307
+0.736
+0.217
+7.27 × 10−5
+8.40 × 102
+1.64 × 1011
+4.16 × 1014
+128
+0.315
+0.769
+0.224
+6.27 × 10−5
+1.08 × 103
+1.45 × 1011
+3.64 × 1014
+192
+0.330
+0.688
+0.223
+1.22 × 10−4
+4.86 × 102
+1.33 × 1011
+2.08 × 1014
+256
+0.358
+0.681
+0.235
+2.11 × 10−4
+3.04 × 102
+1.33 × 1011
+1.53 × 1014
+These imply the following equations for optimal model size N vs compute C in PF-days.
+• Horizon 1:
+N = 1.586 × 1010 × C0.7999 for 61700 ≤ N ≤ 15795200
+• Horizon 2:
+N = 1.309 × 1010 × C0.7871 for 61700 ≤ N ≤ 15795200
+• Horizon 4:
+N = 5.507 × 109 × C0.7357 for 61700 ≤ N ≤ 3948800
+• Horizon 8:
+N = 6.406 × 109 × C0.7493 for 61700 ≤ N ≤ 7739648
+• Horizon 16: N = 7.787 × 109 × C0.7671 for 61700 ≤ N ≤ 7739648
+• Horizon 32: N = 7.535 × 109 × C0.7652 for 61700 ≤ N ≤ 7739648
+• Horizon 64: N = 2.746 × 109 × C0.7053 for 61700 ≤ N ≤ 3948800
+• Horizon 128: N = 2.681 × 109 × C0.7092 for 61700 ≤ N ≤ 3948800
+• Horizon 192: N = 1.376 × 109 × C0.6757 for 61700 ≤ N ≤ 987200
+• Horizon 256: N = 9.876 × 108 × C0.6553 for 61700 ≤ N ≤ 987200
+31
+
+F
+Proof of the lemma
+Proof of Lemma 1. We may write I (N, E) as a function of N and compute C := NE:
+I (N, C)−β =
+�Nc
+N
+�αN
++
+�EcN
+C
+�αE
+.
+The compute-efficient frontier is defined by the value of N that maximizes I (N, C) for each C.
+Equivalently, since β > 0, this value of N minimizes I (N, C)−β, and so it satisfies
+∂
+∂N
+�
+I (N, C)−β�
+= 0.
+Differentiating and multiplying through by N, this equation becomes
+−αN
+�Nc
+N
+�αN
++ αE
+�EcN
+C
+�αE
+= 0.
+Eliminating C, this is exactly equation (2), as required.
+By assumption, we also have I (N, E) = NE along the compute-efficient frontier. Substituting (2)
+into I (N, E), this equation becomes
+�
+1 + αN
+αE
+� �Nc
+N
+�αN
+= (NE)−β .
+(3)
+Thus both equations (2) and (3) are power law relationships between N and E that hold along the
+compute-efficient frontier, so we may simply equate exponents and constants. Equating exponents,
+αN
+αE
+= αN
+β − 1
+and hence
+1
+β =
+1
+αN
++ 1
+αE
+,
+as required. Equating constants,
+�αN
+αE
+�
+1
+αE N
+αN
+αE
+c
+E−1
+c
+=
+�
+1 + αN
+αE
+� 1
+β
+N
+αN
+β
+c
+,
+and hence
+1
+NcEc
+=
+�
+1 + αN
+αE
+�
+1
+αN +
+1
+αE � αE
+αN
+�
+1
+αE =
+�
+1 + αN
+αE
+�
+1
+αN �
+1 + αE
+αN
+�
+1
+αE ,
+as required.
+32
+
+G
+Proof sketch of the proposition
+A formal statement and proof of Proposition 1 would require a formal analysis of Vanilla Policy
+Gradient, which is beyond the scope of this work. Instead, we provide a proof sketch in which we
+make approximations informally.
+Proof sketch of Proposition 1. The horizon length h only affects the algorithm via GAE, which in
+the case λ = 1 produces the value function targets and advantage estimates
+ˆVt := rt + γrt+1 + · · · + γT −trT = rt + γ ˆVt+1
+and
+ˆAt := ˆVt − V (st) = rt − V (st) + γ ˆVt+1,
+where V is the value function. Since timesteps are independent, γ ˆVt+1 is independent of st and at,
+and so should be thought of as noise. The value function will quickly learn to incorporate the mean
+of this noise, and so
+V (st) ≈ V 0 (st) + E
+�
+γ ˆVt+1
+�
+,
+where V 0 (st) is the “immediate reward value function” that would have been obtained had we
+used the value function targets ˆV 0
+t := ˆVt − E
+�
+γ ˆVt+1
+�
+. Writing ϵ := γ ˆVt+1 − E
+�
+γ ˆVt+1
+�
+for the
+zero-mean component of γ ˆVt+1, we obtain
+ˆV 0
+t = rt + ϵ
+and
+ˆAt ≈ rt − V 0 (st) + ϵ.
+In other words, the entire impact of varying h is that it changes the variance of the noise term ϵ added
+to the value function targets and advantage estimates.
+Let us now analyze the policy gradient, which equals
+ˆEt
+�
+∇θρt (θ) ˆAt
+�
+≈ ˆEt
+�
+∇θρt (θ)
+�
+rt − V 0 (st) + ϵ
+��
+,
+where ρt (θ) :=
+πθ(at|st)
+πθold(at|st). Since ϵ is independent of st and at and E [ϵ] = 0, the covariance matrix
+of this decomposes as
+Σθ + ΦθVar [ϵ] ,
+where Σθ is the covariance matrix of ∇θρt (θ)
+�
+rt − V 0 (st)
+�
+, and Φθ := E
+�
+∇θρt (θ) ∇T
+θ ρt (θ)
+�
+is
+the uncentered covariance matrix of ∇θρt (θ).
+Note that V 0 (st) simply estimates E [rt], which does not depend on h. The variance of V 0 (st) does
+depend on h via the addition of ϵ to the value function targets, but this additional variance is small
+compared to the variance of ϵ itself. We may therefore treat Σθ as approximately independent of h.
+It remains to express Var [ϵ] in terms of h. We assume that T is large enough compared to h that we
+may take T → ∞. (In our experiments, we use rollouts of length 512 and h ≤ 256.) Thus
+Var [ϵ] = Var
+�
+γ ˆVt+1
+�
+=
+�
+γ2 + γ4 + γ6 + . . .
+�
+Var [rt]
+=
+γ2
+1 − γ2 Var [rt]
+= 1
+4
+�
+h + 1
+h − 2
+�
+Var [rt] .
+Hence the covariance matrix of the policy gradient is approximately
+Σθ + Πθ
+�
+h + 1
+h − 2
+�
+,
+where Σθ and Πθ := 1
+4Var [rt] Φθ are symmetric positive semi-definite matrices that do not depend
+on h, as required.
+33
+

-NFLT4oBgHgl3EQfuy_h/content/tmp_files/2301.12157v1.pdf.txt

@@ -0,0 +1,476 @@
+Simple Realistic Model of Spin Reorientation in 4f-3d
+Compounds
+Alexander Moskvin*, Evgenii Vasinovich, Anton Shadrin
+Ural Federal University, Ekaterinburg, Russia
+Abstract: Spin reorientation is an important phenomenon of rare-earth perovskites, orthoferrites
+and orthochromites. In this study, we consider a simple but realistic microscopic theory of the
+spontaneous spin-reorientation transitions induced by the 4f-3d interaction, more specifically, the
+interaction of the main Kramers doublet or non-Kramers quasi-doublet of the 4f ion with an effective
+magnetic field induced by the 3d sublattice. The obtained results indicate that the cause of both the
+temperature and the character of the spin-reorientation transition is a competition between the second
+and fourth order spin anisotropy of the 3d sublattice, the crystal field for 4f ions, and the 4f-3d
+interaction.
+Keywords: 4f-3d interaction; (quasi)doublets; spin reorientation
+1 Introduction
+Rare-earth orthorhombic perovskites, orthoferrites RFeO3 and orthochromites RCrO3 (where
+R is a rare-earth ion and yttrium), exhibit many important features such as weak ferro- and
+antiferromagnetism, magnetization reversal, anomalous circular magnetooptics, and the phenomenon
+of the spontaneous spin reorientation. The spin reorientation (SR) is one of their unique properties
+that have attracted a lot of attention back in the 70s of the last century [1, 2], though their exact
+microscopic origin is still a challenge to theorists and experimentalists.
+The revival of interest in the mechanism of the spontaneous spin reorientation and
+magnetic compensation in rare-earth perovskites in recent years is related with the discovery
+of the magnetoelectric and the exchange bias effect, which can have a direct application in
+magnetoelectronics. Along with the emergence of new experimental studies (see, e.g., Refs. [3, 4]),
+there also appeared theoretical works claiming to modify the mean-field theory of the spontaneous
+spin-reorientation transitions [5] or to scrutinize the microscopic mechanism responsible for
+spin reorientations and magnetization reversal [6]. In fact, these results are not directly related
+to the microscopic theory of the spontaneous spin reorientation in rare-earth orthoferrites and
+orthochromites. For instance, the authors of the most recent paper [6] did not take into account
+*[email protected]
+1
+arXiv:2301.12157v1  [cond-mat.str-el]  28 Jan 2023
+
+a number of interactions, such as the fourth-order anisotropy for the 3𝑑 sublattice of orthoferrites
+and the crystal field for 𝑅-ions, which play a fundamental role in determining the spontaneous
+spin reorientation. The spin anisotropy of the second order in the 3𝑑 sublattice of orthorhombic
+orthoferrites and orthochromites is generally not reduced to an effective uniaxial form as adopted in
+Ref. [6]. Furthermore, the density functional theory does not allow in principle to give an adequate
+description of such effects of higher orders of perturbation theory as spin anisotropy or antisymmetric
+exchange [7].
+In this paper, we present the results of a simple but realistic microscopic model of the spontaneous
+spin reorientation in rare-earth orthoferrites and orthochromites, which takes into account all the main
+relevant interactions. This model was developed back in the 80s of the last century [8], but has not
+been published until now.
+2 Model formulation
+The most popular examples of systems with the spontaneous SR transitions are magnets based
+on 3𝑑 and 4𝑓 elements such as rare-earth orthoferrites RFeO3, orthochromites RCrO3, intermetallic
+compounds RCo5, RFe2 etc. In all cases, an important cause of the spontaneous SR is the 4𝑓 − 3𝑑
+interaction. Usually this interaction is taken into account by introducing an effective field of the
+magnetically ordered 3𝑑 sublattice acting on the 4𝑓 ions.
+To consider the contribution of the rare-earth sublattice to the free energy at low temperatures, we
+are developing a model which takes into account either the well isolated lower Kramers doublet of the
+4𝑓 ions (with an odd number of the 4𝑓 electrons) or the well isolated two lower Stark sublevels with
+close energies that form a quasi-doublet.
+Within the framework of such “single-doublet” approximation we consider the spontaneous SR
+transition in orthorhombic weak ferromagnets RFeO3 and RCrO3, where the free energy per ion can
+be represented as follows
+Φ(𝜃) = 𝐾1 cos 2𝜃 + 𝐾2 cos 4𝜃 − 𝑘𝑇 ln 2 cosh ∆(𝜃)
+2𝑘𝑇 ,
+(1)
+where 𝐾1, 𝐾2 are the first and second anisotropy constants of the 3𝑑 sublattice, which are temperature
+independent (at least in the SR region), 𝜃 is the orientation angle of the antiferromagnetic, or N´eel
+vector G of the 3𝑑 sublattice (e.g. in the 𝑎𝑐 plane), and ∆(𝜃) is the lower doublet (quasi-doublet)
+splitting of the 4𝑓 ion in a magnetic field induced by the 3𝑑 sublattice.
+Theoretical estimations [8–10] of the different contributions to the first constants of the magnetic
+anisotropy for orthoferrites RFeO3 point to a competition of several main mechanisms with relatively
+regular (Dzyaloshinskii-Moriya (DM) coupling, magnetodipole interaction) or irregular (single-ion
+anisotropy, SIA) dependence on the type of R-ion. For instance, the microscopic theory predicts an
+unexpectedly strong increase in values of the constant 𝐾1(𝑎𝑐) for LuFeO3 as compared with YFeO3.
+The SIA contribution to 𝐾1(𝑎𝑐) partially compensates for the large contribution of the DM interaction
+in YFeO3, whereas in LuFeO3, they add up. This result is confirmed by experimental data on the
+2
+
+measurement of the threshold field 𝐻𝑆𝑅 of spin reorientation Γ4 → Γ2 (𝐺𝑥 → 𝐺𝑧) in the orthoferrite
+Lu0.5Y0.5FeO3, in which 𝐻𝑆𝑅 = 15 T as compared to 𝐻𝑆𝑅 = 7.5 T in YFeO3 [10]. Thus, one can
+estimate 𝐾1(𝑎𝑐) in LuFeO3 as around three times as much as 𝐾1(𝑎𝑐) in YFeO3.
+Let us pay attention to recent works on the determination of the parameters of the spin Hamiltonian
+in YFeO3 from measurements of the spin-wave spectrum by the inelastic neutron scattering [11,
+12] and terahertz absorption spectroscopy [13]. However, these authors started with a simplified
+spin-Hamiltonian that took into account only Heisenberg exchange, DM interaction, and single-
+ion anisotropy. Obviously, disregarding the magnetic dipole and exchange-relativistic anisotropy, the
+“single-ion anisotropy” constants found by the authors are some effective quantities that are not directly
+related to the SIA.
+Unfortunately, despite numerous, including fairly recent, studies of the magnetic anisotropy of
+orthoferrites, we do not have reliable experimental data on the magnitude of the contributions of
+various anisotropy mechanisms.
+As shown by theoretical calculations [8,9,14] the constants 𝐾2 of the fourth order spin anisotropy
+rather smoothly decrease in absolute value, changing by no more than two times on going from La to
+Lu. But the most interesting was the conclusion about the different signs of these constants, positive
+for the 𝑎𝑐 and 𝑏𝑐 planes and negative for the 𝑎𝑏 plane, thus indicating a different character of spin-
+reorientation transitions in the corresponding planes, i.e., second-order transitions in the 𝑎𝑐 and 𝑏𝑐
+planes and first-order transitions in the 𝑎𝑏 plane [2]. Indeed, all currently known spin-reorientation
+transitions of the Γ4 −Γ2 (𝐺𝑥 −𝐺𝑧) type in orthoferrites RFeO3 (R = Pr, Nd, Sm, Tb, Ho, Er, Tm, Yb)
+are smooth, with two characteristic temperatures of the second-order phase transitions to be a start
+and finish of the spin reorientation, and the only known jump-like first order SR transition for these
+crystals is the SR transition Γ4 − Γ1 (𝐺𝑥 − 𝐺𝑦) in the 𝑎𝑏 plane in DyFeO3 [2]. A unique example
+that confirms the conclusions about the sign of the second anisotropy constant is a mixed orthoferrite
+Ho0.5Dy0.5FeO3 [2] in which two spin-reorientation transitions 𝐺𝑥 − 𝐺𝑦 (𝑇 = 46 K) and 𝐺𝑦 − 𝐺𝑧
+(18 ÷ 24 K) are realized through one phase transition of the first order in the 𝑎𝑏 plane and two phase
+transitions of the second order in the 𝑏𝑐 plane, respectively.
+The splitting value ∆(𝜃) for the Kramers doublet in a magnetic field H has the well-known form
+∆(𝜃) = 𝜇𝐵
+[︀
+(𝑔𝑥𝑥𝐻𝑥 + 𝑔𝑥𝑦𝐻𝑦)2 + (𝑔𝑥𝑦𝐻𝑥 + 𝑔𝑦𝑦𝐻𝑦)2 + 𝑔2
+𝑧𝑧𝐻2
+𝑧
+]︀1/2 ,
+(2)
+where it is taken into account that for the 4𝑓 ions in RFeO3 the ˆ𝑔-tensor (with the local symmetry 𝐶𝑠)
+has the form
+ˆ𝑔 =
+⎛
+⎜
+⎝
+𝑔𝑥𝑥
+𝑔𝑥𝑦
+0
+𝑔𝑥𝑦
+𝑔𝑦𝑦
+0
+0
+0
+𝑔𝑧𝑧
+⎞
+⎟
+⎠ .
+(3)
+The effective field H for the SR transition 𝐺𝑥 → 𝐺𝑧 in the 𝑎𝑐 plane can be represented as follows
+𝐻𝑥 = 𝐻(0)
+𝑥 cos 𝜃, 𝐻𝑦 = 𝐻(0)
+𝑦
+cos 𝜃, 𝐻𝑧 = 𝐻(0)
+𝑧
+sin 𝜃,
+(4)
+3
+
+so in the absence of an external magnetic field, for ∆(𝜃) we have the rather simple expression:
+∆(𝜃) =
+(︂∆2
+𝑎 − ∆2
+𝑐
+2
+cos 2𝜃 + ∆2
+𝑎 + ∆2
+𝑐
+2
+)︂1/2
+,
+(5)
+where ∆𝑎,𝑐 are the doublet splitting for the cases of 𝜃 = 0 (𝐺𝑧-phase) and 𝜃 = 𝜋/2 (𝐺𝑥-phase)
+respectively. The dependence ∆(𝜃) from (5) is also valid in the case of quasi-doublet.
+A contribution of splitting ∆ to the free energy Φ(𝜃) for the rare-earth sublattice is usually
+considered in the “high-temperature” approximation, when 𝑘𝑇 ≫ ∆ and the influence of the 4𝑓
+sublattice are reduced only to renormalization of the first anisotropy constant 𝐾1:
+𝐾*
+1 = 𝐾1
+(︂
+1 − 1
+𝜏
+)︂
+,
+(6)
+where 𝜏 = 𝑇/𝑇𝑆𝑅 is the reduced temperature and 𝑇𝑆𝑅 = (∆2
+𝑎 − ∆2
+𝑐)/16𝑘𝐾1 is the characteristic
+transition temperature.
+Below we will consider a specific situation when 𝐾1 > 0 and ∆𝑎 > ∆𝑐, i.e. when the configuration
+𝐺𝑥 (𝜃 = 𝜋/2) is realized at high temperatures and a decrease in temperature can lead to the spin
+reorientation 𝐺𝑥 → 𝐺𝑧 or 𝐺𝑥 → 𝐺𝑥𝑧 (transition to an angular spin structure). The type of the phase
+transition of the spin reorientation in the “high-temperature” approximation is determined by the sign
+of the second constant 𝐾2: at 𝐾2 < 0 it will be realized by one first-order phase transition at 𝑇 = 𝑇𝑆𝑅,
+i.e. 𝜏 = 1, or at 𝐾2 > 0 by two second-order phase transitions at 𝜏𝑠 = (1 + 𝛾)−1 and 𝜏𝑓 = (1 − 𝛾)−1,
+where 𝜏𝑠 and 𝜏𝑓 are the reduced temperatures of the beginning and end of the SR phase transition and
+𝛾 = 4𝐾2/𝐾1.
+3 Analysis of the “single-doublet” model
+A behavior of a system described by the free energy (1) can be analyzed rigorously. The condition
+𝜕Φ/𝜕𝜃 = 0 reduces in this case to two equations:
+sin 2𝜃 = 0,
+(7)
+𝛼𝜇 + 𝛽𝜇3 = tanh 𝜇
+𝜏 ;
+(8)
+where the following notations are introduced:
+𝛼 = 1 − 𝛾 ∆2
+𝑎 + ∆2
+𝑐
+∆2
+𝑎 − ∆2
+𝑐
+, 𝛽 =
+2𝛾
+𝜇2
+𝑓 − 𝜇2
+𝑠
+, 𝜇 = ∆(𝜃)
+2𝑘𝑇𝑆𝑅
+, 𝜇𝑠 =
+∆𝑐
+2𝑘𝑇𝑆𝑅
+, 𝜇𝑓 =
+∆𝑎
+2𝑘𝑇𝑆𝑅
+.
+(9)
+This corresponds to three possible magnetic configurations:
+• The configuration 𝐺𝑥: 𝜃 = ±𝜋/2, stable at tanh 𝜇𝑠/𝜏 ≤ 𝛼𝜇𝑠 + 𝛽𝜇3
+𝑠 .
+• The configuration 𝐺𝑧: 𝜃 = 0, 𝜋, stable at tanh 𝜇𝑓/𝜏 ≥ 𝛼𝜇𝑓 + 𝛽𝜇3
+𝑓 .
+4
+
+• The angular configuration 𝐺𝑥𝑧: the temperature dependence of 𝜃(𝜏) is determined by solving
+the equation (8) (see Figure 1), the state is stable at 𝜕𝜇/𝜕𝜏 ≤ 0.
+The peculiar 𝜇-𝜏 phase diagram which represents solutions of the master equation (8) given a fixed
+value of the 𝛼 parameter and different value of the 𝛽 parameter is shown in Figure 1, where areas
+with different character of the SR transition are highlighted in different colors. For the solutions in
+the FO region, the SR goes through one first-order phase transition, in the SO region we arrive at one
+or two second-order phase transitions, in the MO1,2 regions we arrive at a “mixture” of the first and
+second-order phase transitions. All the lines 𝜇(𝜏) on the right side converge to
+√︀
+|𝛼/𝛽| at 𝜏 → ∞; on
+the left side, when 𝜏 → 0 the branch point 𝜇 =
+3
+2𝛼 is obtained at 𝛽 = − 4
+27𝛼3, and the point 𝜇 = 1/𝛼
+at 𝛽 = 0; all the solutions, where 𝜇 can reach zero, converge to 𝜏 = 1/𝛼.
+0
+1/α
+τ
+1/α
+3
+2 α
+μ
+α / β1
+α / β2
+α / β3
+FO
+MO1
+MO2
+SO
+Fig. 1: (Color online) The peculiar 𝜇-𝜏 phase diagram which represents solutions of the master
+equation (8) given a fixed value of the 𝛼 parameter and different value of the 𝛽 parameter (see text for
+detail).
+The character of the SR transition will be determined by the form of the solution of the equation
+(8) in the region 𝜇𝑠 ≤ 𝜇 ≤ 𝜇𝑓. Let us analyze this equation starting with the simplest case 𝐾2 = 0,
+i.e. 𝛼 = 1, 𝛽 = 0. In this case, the main equation transforms into the molecular field equation well
+known in the basic theory of ferromagnetism:
+𝜇 = tanh 𝜇
+𝜏 = 𝐵 1
+2
+(︁𝜇
+𝜏
+)︁
+,
+(10)
+where 𝐵1/2(𝑥) is the Brillouin function. The equation has only one non-trivial solution at 0 ≤ 𝜏 ≤ 1,
+0 ≤ 𝜇 ≤ 1, and the function 𝜇(𝜏) has the usual “Weiss” form. Thus, with the absence of the
+cubic anisotropy (𝐾2 = 0) in the “single-doublet” model the SR will be realized either through two
+second-order phase transitions at 𝜇𝑓 ≤ 1 (the complete spin-reorientation 𝐺𝑥 → 𝐺𝑧), or through one
+second-order phase transition at 𝜇𝑓 > 1, but in this case the SR will be incomplete, i.e. it will end
+with a transition to the angular spin structure 𝐺𝑥𝑧. The spin reorientation will begin at a temperature
+5
+
+𝑇𝑠 ≤ 𝑇𝑆𝑅 and 𝑇𝑠 is equal to 𝑇𝑆𝑅 only in the case 𝜇𝑠 = 0 (∆𝑐 = 0), which can be realized in the general
+case only for Ising ions (e.g. Dy3+ in DyFeO3). For this type of ions, the temperature dependence of
+the “order parameter” 𝜇 (in fact the splitting ∆(𝜃) of the doublet) in a close range of 𝑇𝑆𝑅 will be very
+sharp: 𝜇(𝑇) ∼ (𝑇 − 𝑇𝑆𝑅)−1/2. Nevertheless, the SR will be continuous and the temperature range of
+the SR ∆𝑇 = 𝑇𝑠 − 𝑇𝑓 at 𝜇 ≪ 1 can theoretically reach arbitrarily small values.
+Thus, the results of the rigorous analysis of the “single-doublet” model are fundamentally different
+from the conclusions of the simplified model (the “high-temperature” approximation), according to
+which for 𝐾2 = 0 the spin reorientation always occurs as the first-order phase transition at 𝑇 = 𝑇𝑆𝑅.
+For a positive second anisotropy constant (𝐾2 > 0, 𝛽 > 0), the main equation (8) has one non-
+trivial solution in the region 0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 𝜇0 at 𝛼 > 0, and one in the region 0 ≤ 𝜏 ≤ ∞,
+√︀
+|𝛼/𝛽| ≤ 𝜇 ≤ 𝜇0 at 𝛼 ≤ 0, where 𝜇0 is determined from the solution of the equation 𝛼𝜇0 +𝛽𝜇3
+0 = 1.
+The situation in this case is very similar to the previous one, i.e. the beginning of the SR will always
+be a second-order phase transition, and the reorientation will be complete (𝐺𝑥 → 𝐺𝑧) or incomplete
+(𝐺𝑥 → 𝐺𝑥𝑧). Note that under the condition (𝜇2
+𝑓 − 𝜇2
+𝑠)/(𝜇2
+𝑓 + 𝜇2
+𝑠) ≥ 𝛾, i.e. 𝛼 ≤ 0, the width of the
+reorientation region becomes very large, even if 𝜇𝑠 differs slightly from 𝜇𝑓.
+For Ising ions at ∆𝑐 = 0, the SR beginning temperature is determined in exactly the same way as
+in the “high-temperature” approximation 𝑇𝑠 = 𝑇𝑆𝑅/(1 + 𝛾).
+For a negative second anisotropy constant (𝐾2 < 0, 𝛽 < 0), the several fundamentally different
+solutions of the main equation (8) are possible. For 𝐾*
+2 ≥ 𝐾2, where 𝐾*
+2 is determined from the
+condition 𝛽 = − 1
+3𝛼3, i.e.
+2𝛾
+𝜇2
+𝑓 − 𝜇2
+𝑠
+= −1
+3
+(︃
+1 − 𝛾 𝜇2
+𝑓 + 𝜇2
+𝑠
+𝜇2
+𝑓 − 𝜇2
+𝑠
+)︃3
+,
+(11)
+there is one non-trivial solution of the equation (8) in the region 1/𝛼 ≤ 𝜏 < ∞, 𝜇 ≤
+√︀
+𝛼/𝛽, but
+here 𝜇(𝑇) decreases with decreasing temperature, i.e. 𝜕𝜇/𝜕𝜏 > 0. This solution is unstable and there
+is no fundamental possibility for a smooth rotation of spins, the SR is always realized through the
+first-order phase transition.
+In the intermediate range of values 𝐾2 (𝐾*
+2 < 𝐾2 < 0 or − 1
+3𝛼3 < 𝛽 < 0) the main equation
+has two non-trivial solutions, and for one of them 𝜕𝜇/𝜕𝜏 > 0 (corresponding to bigger values of 𝜇),
+and for the second 𝜕𝜇/𝜕𝜏 < 0 (corresponding to smaller values of 𝜇). It is convenient to consider
+separately three areas of variation 𝛽.
+1. − 4
+27𝛼3 < 𝛽 < 0:
+a) the first solution: 0 ≤ 𝜏 < ∞, 𝜇> ≤ 𝜇 <
+√︀
+|𝛼/𝛽|,
+b) the second solution: 0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 𝜇<,
+where 𝜇>, 𝜇< are the bigger and smaller positive solution of the equation 𝛼𝜇 + 𝛽𝜇3 = 1.
+2. 𝛽 = − 4
+27𝛼3:
+a) the first solution: 0 ≤ 𝜏 < ∞, 3/(2𝛼) ≤ 𝜇 <
+√︀
+|𝛼/𝛽|,
+b) the second solution: 0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 3/(2𝛼),
+moreover, in this case we have a branch point of the main equation solution at 𝜏 = 0, 𝜇 = 1.
+3. − 1
+3𝛼3 < 𝛽 < − 4
+27𝛼3:
+a) the first solution: 𝜏0 ≤ 𝜏 < ∞, 𝜇0 ≤ 𝜇 <
+√︀
+|𝛼/𝛽|,
+6
+
+b) the second solution: 𝜏0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 𝜇0,
+where the quantities 𝜇0, 𝜏0 correspond to the branch points of the main equation solutions.
+Illustrations of typical (a,b) and unconventional (c,d) SR transitions predicted by simple
+(quasi)doublet model are shown in Figure 2. The Figure 2a, built with 𝐾1 = 1, 𝛾 = 0.05, ∆𝑎 =
+30.84, ∆𝑐 = 14.82, which corresponds to 𝑇𝑆𝑅 = 45.73, 𝜇𝑠 = 0.162, 𝜇𝑓 = 0.337, 𝜏𝑠 = 1.04, 𝜏𝑓 =
+0.91, describes a typical smooth SR transition with two second-order phase transitions 𝐺𝑥 − 𝐺𝑥𝑧 at
+the beginning (𝜏𝑠) and 𝐺𝑥𝑧 − 𝐺𝑧 at the end (𝜏𝑓) of the spin reorientation.
+The Figure 2b, built with 𝐾1 = 1, 𝛾 = −0.1, ∆𝑎 = 33.19, ∆𝑐 = 27.1, which corresponds to
+𝑇𝑆𝑅 = 22.95, 𝜇𝑠 = 0.59, 𝜇𝑓 = 0.72, 𝜏𝑠 = 0.762, 𝜏𝑓 = 0.93, describes an abrupt first-order SR
+transition. For 𝜏 > 𝜏𝑓 there is the 𝐺𝑥-phase, which can remain stable up to 𝜏𝑠 when cooled. For 𝜏 < 𝜏𝑠
+there is the 𝐺𝑧-phase, which can remain stable up to 𝜏𝑓 when heated. The point 𝐴 marks a phase
+transition point when the phases 𝐺𝑥 and 𝐺𝑧 have equal energies.
+The Figure 2c, built with 𝐾1 = 1, 𝛾 = −0.222, ∆𝑎 = 6.72, ∆𝑐 = 1.63, which corresponds
+to 𝑇𝑆𝑅 = 2.65, 𝜇𝑠 = 0.307, 𝜇𝑓 = 1.266, 𝜏𝑠 = 0.778, 𝜏𝑓 = 0.523 and the Figure 2d, built with
+𝐾1 = 1, 𝛾 = −0.25, ∆𝑎 = 6.71, ∆𝑐 = 2.02, which corresponds to 𝑇𝑆𝑅 = 2.56, 𝜇𝑠 = 0.396, 𝜇𝑓 =
+1.31, 𝜏𝑠 = 0.73, 𝜏𝑓 = 0.545 describe unconventional "mixed"SR transitions. At 𝜏𝑠 there is the smooth
+second-order phase transition 𝐺𝑥 − 𝐺𝑥𝑧. At 𝜏 ≤ 𝜏𝑓 we have two stable phases 𝐺𝑧 and 𝐺𝑥𝑧: at those
+temperatures the sharp first-order phase transition 𝐺𝑥𝑧 − 𝐺𝑧 can happen, or the system could stay in
+the angular 𝐺𝑥𝑧-phase.
+(a)
+τ
+τs
+τf
+μ
+μs
+μf
+(c)
+τ
+τs
+τf
+μ
+μs
+μf
+(d)
+τ
+τs
+τf
+μ
+μs
+μf
+θ
+Φ
+θ
+Φ
+θ
+Φ
+τ > τf
+τ < τs
+A
+(b)
+τ
+τs
+τf
+μ
+μs
+μf
+A
+Fig. 2: Illustrations of typical (a,b) and unconventional (c,d) SR transitions predicted by simple
+(quasi)doublet model (see text for detail). The arrows indicate the direction of the antiferromagnetic
+vector G in the 𝑎𝑐 plane. The insets in panel (b) show the 𝜃-dependence of the free energy.
+7
+
+Thus, there are not only the smooth and abrupt SR transitions, a characteristic feature of the range
+of intermediate values 𝐾2 is the fundamental possibility of the existence of “mixed” SR transitions, in
+which the spins first smoothly rotate through a certain angle and then jump to the position with 𝜃 = 0.
+For this, it is sufficient that 𝜇𝑓 corresponds to a point on the upper branch of solutions, and 𝜇𝑠 to a point
+on the lower branch of solutions at 𝜏𝑓 < 𝜏𝑠. In this case, the spin reorientation begins with the single
+second-order transition 𝐺𝑥 → 𝐺𝑥𝑧 and then ends with the first-order phase transition 𝐺𝑥𝑧 → 𝐺𝑧.
+In contrast to the “high-temperature” approximation, the “single-doublet” model claims the nature
+of the phase transition is determined not simply by the sign of the second anisotropy constant, but
+also it depends on the ratio between 𝐾1, 𝐾2 and the doublet splitting in both phases. Nevertheless,
+if we apply the simplified model to describe the SR transition, we have to renormalize both the first
+and the second anisotropy constant, giving the last one sometimes a rather complicated temperature
+dependence, in particular with a change in sign when considering transitions of the “mixed” type.
+Of course, in this case Fe sublattice alone is not enough to provide the value of the effective second
+constant.
+4 Conclusion
+The model of the spin-reorientation transitions induced by the 4𝑓 − 3𝑑 interaction in rare-earth
+orthoferrites and orthochromites has been investigated. It is shown that both the temperature and
+the character of the spin-reorientation transition following from the solution of the transcendental
+equation (8) are the result of competition between the second and fourth order spin anisotropy of
+the 3𝑑 sublattice, the crystal field for 4f ions, and the 4𝑓 − 3𝑑 interaction. At variance with the
+“high-temperature” approximation, the “single-doublet” model, along with typical smooth and abrupt
+SR transitions, predicts the appearance of mixed-type SR transitions, with an initial second-order
+transition and a final abrupt first-order transition.
+Funding: The research was supported by the Ministry of Education and Science of the Russian
+Federation, project № FEUZ-2020-0054, and by Russian Science Foundation, project № 22-22-00682.
+References
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+Sc. Thesis, Lomonosov Moscow State University, Moscow, Russia, 1984. (In Russian)
+[9] Moskvin,
+A.
+Structure–Property
+Relationships
+for
+Weak
+Ferromagnetic
+Perovskites.
+Magnetochemistry 2021, 7, 111.
+[10] Kadomtseva, A.M.; Agafonov, A.P.; Lukina, M.M.; Milov, V.N.; Moskvin, A.S.; Semenov, V.A.;
+Sinitsyn, E.V. Nature of the Magnetic Anisotropy and Magnetostriction of Orthoferrites and
+Orthochromites. JETP 1981, 81, 700–706.
+[11] Hahn, S.E.; Podlesnyak, A.A.; Ehlers, G.; Granroth, G.E.; Fishman, R.S.; Kolesnikov, A.I.;
+Pomjakushina, E.; Conder, K. Inelastic neutron scattering studies of YFeO3. Phys. Rev. B 2014,
+89, 014420.
+[12] Park, K.; Sim, H.; Leiner, J.C.; Yoshida, Y.; Jeong, J.; Yano, S.; Gardner, J.; Bourges, P.; Klicpera,
+M.; Sechovsk´y, V.; Boehm, M.; Park, J.-G. Low-energy spin dynamics of orthoferrites AFeO3
+(A = Y, La, Bi). J. Phys. Condens. Matter 2018, 30, 235802.
+[13] Amelin, K.; Nagel, U.; Fishman, R.S.; Yoshida, Y.; Sim, H.; Park, K.; Park, J.-G.; R˜o˜om, T.
+Terahertz absorption spectroscopy study of spin waves in orthoferrite YFeO3 in a magnetic field.
+Phys. Rev. B 2018, 98, 174417.
+[14] Moskvin, A.S.; Bostrem, I.G. Cubic Anisotropy of Rare-Earth Orthoferrites. Sov. Phys. Solid St.
+1979, 21, 628.
+9
+

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@@ -0,0 +1,410 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf,len=409
+page_content='Simple Realistic Model of Spin Reorientation in 4f-3d  Compounds  Alexander Moskvin*, Evgenii Vasinovich, Anton Shadrin  Ural Federal University, Ekaterinburg, Russia  Abstract: Spin reorientation is an important phenomenon of rare-earth perovskites, orthoferrites  and orthochromites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' In this study, we consider a simple but realistic microscopic theory of the  spontaneous spin-reorientation transitions induced by the 4f-3d interaction, more specifically, the  interaction of the main Kramers doublet or non-Kramers quasi-doublet of the 4f ion with an effective  magnetic field induced by the 3d sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The obtained results indicate that the cause of both the  temperature and the character of the spin-reorientation transition is a competition between the second  and fourth order spin anisotropy of the 3d sublattice, the crystal field for 4f ions, and the 4f-3d  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Keywords: 4f-3d interaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' (quasi)doublets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' spin reorientation  1 Introduction  Rare-earth orthorhombic perovskites, orthoferrites RFeO3 and orthochromites RCrO3 (where  R is a rare-earth ion and yttrium), exhibit many important features such as weak ferro- and  antiferromagnetism, magnetization reversal, anomalous circular magnetooptics, and the phenomenon  of the spontaneous spin reorientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The spin reorientation (SR) is one of their unique properties  that have attracted a lot of attention back in the 70s of the last century [1, 2], though their exact  microscopic origin is still a challenge to theorists and experimentalists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The revival of interest in the mechanism of the spontaneous spin reorientation and  magnetic compensation in rare-earth perovskites in recent years is related with the discovery  of the magnetoelectric and the exchange bias effect, which can have a direct application in  magnetoelectronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Along with the emergence of new experimental studies (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' [3, 4]),  there also appeared theoretical works claiming to modify the mean-field theory of the spontaneous  spin-reorientation transitions [5] or to scrutinize the microscopic mechanism responsible for  spin reorientations and magnetization reversal [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' In fact, these results are not directly related  to the microscopic theory of the spontaneous spin reorientation in rare-earth orthoferrites and  orthochromites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' For instance, the authors of the most recent paper [6] did not take into account  alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='moskvin@urfu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='ru  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='12157v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='str-el]  28 Jan 2023  a number of interactions, such as the fourth-order anisotropy for the 3𝑑 sublattice of orthoferrites  and the crystal field for 𝑅-ions, which play a fundamental role in determining the spontaneous  spin reorientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The spin anisotropy of the second order in the 3𝑑 sublattice of orthorhombic  orthoferrites and orthochromites is generally not reduced to an effective uniaxial form as adopted in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Furthermore, the density functional theory does not allow in principle to give an adequate  description of such effects of higher orders of perturbation theory as spin anisotropy or antisymmetric  exchange [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  In this paper, we present the results of a simple but realistic microscopic model of the spontaneous  spin reorientation in rare-earth orthoferrites and orthochromites, which takes into account all the main  relevant interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' This model was developed back in the 80s of the last century [8], but has not  been published until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  2 Model formulation  The most popular examples of systems with the spontaneous SR transitions are magnets based  on 3𝑑 and 4𝑓 elements such as rare-earth orthoferrites RFeO3, orthochromites RCrO3, intermetallic  compounds RCo5, RFe2 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' In all cases, an important cause of the spontaneous SR is the 4𝑓 − 3𝑑  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Usually this interaction is taken into account by introducing an effective field of the  magnetically ordered 3𝑑 sublattice acting on the 4𝑓 ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  To consider the contribution of the rare-earth sublattice to the free energy at low temperatures, we  are developing a model which takes into account either the well isolated lower Kramers doublet of the  4𝑓 ions (with an odd number of the 4𝑓 electrons) or the well isolated two lower Stark sublevels with  close energies that form a quasi-doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Within the framework of such “single-doublet” approximation we consider the spontaneous SR  transition in orthorhombic weak ferromagnets RFeO3 and RCrO3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' where the free energy per ion can  be represented as follows  Φ(𝜃) = 𝐾1 cos 2𝜃 + 𝐾2 cos 4𝜃 − 𝑘𝑇 ln 2 cosh ∆(𝜃)  2𝑘𝑇 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  (1)  where 𝐾1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 𝐾2 are the first and second anisotropy constants of the 3𝑑 sublattice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' which are temperature  independent (at least in the SR region),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 𝜃 is the orientation angle of the antiferromagnetic,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' or N´eel  vector G of the 3𝑑 sublattice (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' in the 𝑎𝑐 plane), and ∆(𝜃) is the lower doublet (quasi-doublet)  splitting of the 4𝑓 ion in a magnetic field induced by the 3𝑑 sublattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Theoretical estimations [8–10] of the different contributions to the first constants of the magnetic  anisotropy for orthoferrites RFeO3 point to a competition of several main mechanisms with relatively  regular (Dzyaloshinskii-Moriya (DM) coupling, magnetodipole interaction) or irregular (single-ion  anisotropy, SIA) dependence on the type of R-ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' For instance, the microscopic theory predicts an  unexpectedly strong increase in values of the constant 𝐾1(𝑎𝑐) for LuFeO3 as compared with YFeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The SIA contribution to 𝐾1(𝑎𝑐) partially compensates for the large contribution of the DM interaction  in YFeO3, whereas in LuFeO3, they add up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' This result is confirmed by experimental data on the  2  measurement of the threshold field 𝐻𝑆𝑅 of spin reorientation Γ4 → Γ2 (𝐺𝑥 → 𝐺𝑧) in the orthoferrite  Lu0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='5Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='5FeO3, in which 𝐻𝑆𝑅 = 15 T as compared to 𝐻𝑆𝑅 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='5 T in YFeO3 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Thus, one can  estimate 𝐾1(𝑎𝑐) in LuFeO3 as around three times as much as 𝐾1(𝑎𝑐) in YFeO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Let us pay attention to recent works on the determination of the parameters of the spin Hamiltonian  in YFeO3 from measurements of the spin-wave spectrum by the inelastic neutron scattering [11,  12] and terahertz absorption spectroscopy [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' However, these authors started with a simplified  spin-Hamiltonian that took into account only Heisenberg exchange, DM interaction, and single-  ion anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Obviously, disregarding the magnetic dipole and exchange-relativistic anisotropy, the  “single-ion anisotropy” constants found by the authors are some effective quantities that are not directly  related to the SIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Unfortunately, despite numerous, including fairly recent, studies of the magnetic anisotropy of  orthoferrites, we do not have reliable experimental data on the magnitude of the contributions of  various anisotropy mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  As shown by theoretical calculations [8,9,14] the constants 𝐾2 of the fourth order spin anisotropy  rather smoothly decrease in absolute value, changing by no more than two times on going from La to  Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' But the most interesting was the conclusion about the different signs of these constants, positive  for the 𝑎𝑐 and 𝑏𝑐 planes and negative for the 𝑎𝑏 plane, thus indicating a different character of spin-  reorientation transitions in the corresponding planes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=', second-order transitions in the 𝑎𝑐 and 𝑏𝑐  planes and first-order transitions in the 𝑎𝑏 plane [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Indeed, all currently known spin-reorientation  transitions of the Γ4 −Γ2 (𝐺𝑥 −𝐺𝑧) type in orthoferrites RFeO3 (R = Pr, Nd, Sm, Tb, Ho, Er, Tm, Yb)  are smooth, with two characteristic temperatures of the second-order phase transitions to be a start  and finish of the spin reorientation, and the only known jump-like first order SR transition for these  crystals is the SR transition Γ4 − Γ1 (𝐺𝑥 − 𝐺𝑦) in the 𝑎𝑏 plane in DyFeO3 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' A unique example  that confirms the conclusions about the sign of the second anisotropy constant is a mixed orthoferrite  Ho0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='5Dy0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='5FeO3 [2] in which two spin-reorientation transitions 𝐺𝑥 − 𝐺𝑦 (𝑇 = 46 K) and 𝐺𝑦 − 𝐺𝑧  (18 ÷ 24 K) are realized through one phase transition of the first order in the 𝑎𝑏 plane and two phase  transitions of the second order in the 𝑏𝑐 plane, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The splitting value ∆(𝜃) for the Kramers doublet in a magnetic field H has the well-known form  ∆(𝜃) = 𝜇𝐵  [︀  (𝑔𝑥𝑥𝐻𝑥 + 𝑔𝑥𝑦𝐻𝑦)2 + (𝑔𝑥𝑦𝐻𝑥 + 𝑔𝑦𝑦𝐻𝑦)2 + 𝑔2  𝑧𝑧𝐻2  𝑧  ]︀1/2 ,  (2)  where it is taken into account that for the 4𝑓 ions in RFeO3 the ˆ𝑔-tensor (with the local symmetry 𝐶𝑠)  has the form  ˆ𝑔 =  ⎛  ⎜  ⎝  𝑔𝑥𝑥  𝑔𝑥𝑦  0  𝑔𝑥𝑦  𝑔𝑦𝑦  0  0  0  𝑔𝑧𝑧  ⎞  ⎟  ⎠ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  (3)  The effective field H for the SR transition 𝐺𝑥 → 𝐺𝑧 in the 𝑎𝑐 plane can be represented as follows  𝐻𝑥 = 𝐻(0)  𝑥 cos 𝜃, 𝐻𝑦 = 𝐻(0)  𝑦  cos 𝜃, 𝐻𝑧 = 𝐻(0)  𝑧  sin 𝜃,  (4)  3  so in the absence of an external magnetic field, for ∆(𝜃) we have the rather simple expression:  ∆(𝜃) =  (︂∆2  𝑎 − ∆2  𝑐  2  cos 2𝜃 + ∆2  𝑎 + ∆2  𝑐  2  )︂1/2  ,  (5)  where ∆𝑎,𝑐 are the doublet splitting for the cases of 𝜃 = 0 (𝐺𝑧-phase) and 𝜃 = 𝜋/2 (𝐺𝑥-phase)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The dependence ∆(𝜃) from (5) is also valid in the case of quasi-doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  A contribution of splitting ∆ to the free energy Φ(𝜃) for the rare-earth sublattice is usually  considered in the “high-temperature” approximation, when 𝑘𝑇 ≫ ∆ and the influence of the 4𝑓  sublattice are reduced only to renormalization of the first anisotropy constant 𝐾1:  𝐾*  1 = 𝐾1  (︂  1 − 1  𝜏  )︂  ,  (6)  where 𝜏 = 𝑇/𝑇𝑆𝑅 is the reduced temperature and 𝑇𝑆𝑅 = (∆2  𝑎 − ∆2  𝑐)/16𝑘𝐾1 is the characteristic  transition temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Below we will consider a specific situation when 𝐾1 > 0 and ∆𝑎 > ∆𝑐, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' when the configuration  𝐺𝑥 (𝜃 = 𝜋/2) is realized at high temperatures and a decrease in temperature can lead to the spin  reorientation 𝐺𝑥 → 𝐺𝑧 or 𝐺𝑥 → 𝐺𝑥𝑧 (transition to an angular spin structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The type of the phase  transition of the spin reorientation in the “high-temperature” approximation is determined by the sign  of the second constant 𝐾2: at 𝐾2 < 0 it will be realized by one first-order phase transition at 𝑇 = 𝑇𝑆𝑅,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 𝜏 = 1, or at 𝐾2 > 0 by two second-order phase transitions at 𝜏𝑠 = (1 + 𝛾)−1 and 𝜏𝑓 = (1 − 𝛾)−1,  where 𝜏𝑠 and 𝜏𝑓 are the reduced temperatures of the beginning and end of the SR phase transition and  𝛾 = 4𝐾2/𝐾1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  3 Analysis of the “single-doublet” model  A behavior of a system described by the free energy (1) can be analyzed rigorously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The condition  𝜕Φ/𝜕𝜃 = 0 reduces in this case to two equations:  sin 2𝜃 = 0,  (7)  𝛼𝜇 + 𝛽𝜇3 = tanh 𝜇  𝜏 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  (8)  where the following notations are introduced:  𝛼 = 1 − 𝛾 ∆2  𝑎 + ∆2  𝑐  ∆2  𝑎 − ∆2  𝑐  , 𝛽 =  2𝛾  𝜇2  𝑓 − 𝜇2  𝑠  , 𝜇 = ∆(𝜃)  2𝑘𝑇𝑆𝑅  , 𝜇𝑠 =  ∆𝑐  2𝑘𝑇𝑆𝑅  , 𝜇𝑓 =  ∆𝑎  2𝑘𝑇𝑆𝑅  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  (9)  This corresponds to three possible magnetic configurations:  The configuration 𝐺𝑥: 𝜃 = ±𝜋/2, stable at tanh 𝜇𝑠/𝜏 ≤ 𝛼𝜇𝑠 + 𝛽𝜇3  𝑠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The configuration 𝐺𝑧: 𝜃 = 0, 𝜋, stable at tanh 𝜇𝑓/𝜏 ≥ 𝛼𝜇𝑓 + 𝛽𝜇3  𝑓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  4  The angular configuration 𝐺𝑥𝑧: the temperature dependence of 𝜃(𝜏) is determined by solving  the equation (8) (see Figure 1), the state is stable at 𝜕𝜇/𝜕𝜏 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The peculiar 𝜇-𝜏 phase diagram which represents solutions of the master equation (8) given a fixed  value of the 𝛼 parameter and different value of the 𝛽 parameter is shown in Figure 1, where areas  with different character of the SR transition are highlighted in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' For the solutions in  the FO region, the SR goes through one first-order phase transition, in the SO region we arrive at one  or two second-order phase transitions, in the MO1,2 regions we arrive at a “mixture” of the first and  second-order phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' All the lines 𝜇(𝜏) on the right side converge to  √︀  |𝛼/𝛽| at 𝜏 → ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' on  the left side, when 𝜏 → 0 the branch point 𝜇 =  3  2𝛼 is obtained at 𝛽 = − 4  27𝛼3, and the point 𝜇 = 1/𝛼  at 𝛽 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' all the solutions, where 𝜇 can reach zero, converge to 𝜏 = 1/𝛼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  0  1/α  τ  1/α  3  2 α  μ  \uf603α / β1\uf604  \uf603α / β2\uf604  \uf603α / β3\uf604  FO  MO1  MO2  SO  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 1: (Color online) The peculiar 𝜇-𝜏 phase diagram which represents solutions of the master  equation (8) given a fixed value of the 𝛼 parameter and different value of the 𝛽 parameter (see text for  detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The character of the SR transition will be determined by the form of the solution of the equation  (8) in the region 𝜇𝑠 ≤ 𝜇 ≤ 𝜇𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Let us analyze this equation starting with the simplest case 𝐾2 = 0,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 𝛼 = 1, 𝛽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' In this case, the main equation transforms into the molecular field equation well  known in the basic theory of ferromagnetism:  𝜇 = tanh 𝜇  𝜏 = 𝐵 1  2  (︁𝜇  𝜏  )︁  ,  (10)  where 𝐵1/2(𝑥) is the Brillouin function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The equation has only one non-trivial solution at 0 ≤ 𝜏 ≤ 1,  0 ≤ 𝜇 ≤ 1, and the function 𝜇(𝜏) has the usual “Weiss” form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Thus, with the absence of the  cubic anisotropy (𝐾2 = 0) in the “single-doublet” model the SR will be realized either through two  second-order phase transitions at 𝜇𝑓 ≤ 1 (the complete spin-reorientation 𝐺𝑥 → 𝐺𝑧), or through one  second-order phase transition at 𝜇𝑓 > 1, but in this case the SR will be incomplete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' it will end  with a transition to the angular spin structure 𝐺𝑥𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The spin reorientation will begin at a temperature  5  𝑇𝑠 ≤ 𝑇𝑆𝑅 and 𝑇𝑠 is equal to 𝑇𝑆𝑅 only in the case 𝜇𝑠 = 0 (∆𝑐 = 0), which can be realized in the general  case only for Ising ions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Dy3+ in DyFeO3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' For this type of ions, the temperature dependence of  the “order parameter” 𝜇 (in fact the splitting ∆(𝜃) of the doublet) in a close range of 𝑇𝑆𝑅 will be very  sharp: 𝜇(𝑇) ∼ (𝑇 − 𝑇𝑆𝑅)−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Nevertheless, the SR will be continuous and the temperature range of  the SR ∆𝑇 = 𝑇𝑠 − 𝑇𝑓 at 𝜇 ≪ 1 can theoretically reach arbitrarily small values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Thus, the results of the rigorous analysis of the “single-doublet” model are fundamentally different  from the conclusions of the simplified model (the “high-temperature” approximation), according to  which for 𝐾2 = 0 the spin reorientation always occurs as the first-order phase transition at 𝑇 = 𝑇𝑆𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  For a positive second anisotropy constant (𝐾2 > 0, 𝛽 > 0), the main equation (8) has one non-  trivial solution in the region 0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 𝜇0 at 𝛼 > 0, and one in the region 0 ≤ 𝜏 ≤ ∞,  √︀  |𝛼/𝛽| ≤ 𝜇 ≤ 𝜇0 at 𝛼 ≤ 0, where 𝜇0 is determined from the solution of the equation 𝛼𝜇0 +𝛽𝜇3  0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The situation in this case is very similar to the previous one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' the beginning of the SR will always  be a second-order phase transition, and the reorientation will be complete (𝐺𝑥 → 𝐺𝑧) or incomplete  (𝐺𝑥 → 𝐺𝑥𝑧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Note that under the condition (𝜇2  𝑓 − 𝜇2  𝑠)/(𝜇2  𝑓 + 𝜇2  𝑠) ≥ 𝛾, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 𝛼 ≤ 0, the width of the  reorientation region becomes very large, even if 𝜇𝑠 differs slightly from 𝜇𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  For Ising ions at ∆𝑐 = 0, the SR beginning temperature is determined in exactly the same way as  in the “high-temperature” approximation 𝑇𝑠 = 𝑇𝑆𝑅/(1 + 𝛾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  For a negative second anisotropy constant (𝐾2 < 0, 𝛽 < 0), the several fundamentally different  solutions of the main equation (8) are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' For 𝐾*  2 ≥ 𝐾2, where 𝐾*  2 is determined from the  condition 𝛽 = − 1  3𝛼3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  2𝛾  𝜇2  𝑓 − 𝜇2  𝑠  = −1  3  (︃  1 − 𝛾 𝜇2  𝑓 + 𝜇2  𝑠  𝜇2  𝑓 − 𝜇2  𝑠  )︃3  ,  (11)  there is one non-trivial solution of the equation (8) in the region 1/𝛼 ≤ 𝜏 < ∞, 𝜇 ≤  √︀  𝛼/𝛽, but  here 𝜇(𝑇) decreases with decreasing temperature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 𝜕𝜇/𝜕𝜏 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' This solution is unstable and there  is no fundamental possibility for a smooth rotation of spins, the SR is always realized through the  first-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  In the intermediate range of values 𝐾2 (𝐾*  2 < 𝐾2 < 0 or − 1  3𝛼3 < 𝛽 < 0) the main equation  has two non-trivial solutions, and for one of them 𝜕𝜇/𝜕𝜏 > 0 (corresponding to bigger values of 𝜇),  and for the second 𝜕𝜇/𝜕𝜏 < 0 (corresponding to smaller values of 𝜇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' It is convenient to consider  separately three areas of variation 𝛽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' − 4  27𝛼3 < 𝛽 < 0:  a) the first solution: 0 ≤ 𝜏 < ∞, 𝜇> ≤ 𝜇 <  √︀  |𝛼/𝛽|,  b) the second solution: 0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 𝜇<,  where 𝜇>, 𝜇< are the bigger and smaller positive solution of the equation 𝛼𝜇 + 𝛽𝜇3 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 𝛽 = − 4  27𝛼3:  a) the first solution: 0 ≤ 𝜏 < ∞, 3/(2𝛼) ≤ 𝜇 <  √︀  |𝛼/𝛽|,  b) the second solution: 0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 3/(2𝛼),  moreover, in this case we have a branch point of the main equation solution at 𝜏 = 0, 𝜇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' − 1  3𝛼3 < 𝛽 < − 4  27𝛼3:  a) the first solution: 𝜏0 ≤ 𝜏 < ∞, 𝜇0 ≤ 𝜇 <  √︀  |𝛼/𝛽|,  6  b) the second solution: 𝜏0 ≤ 𝜏 ≤ 1/𝛼, 0 ≤ 𝜇 ≤ 𝜇0,  where the quantities 𝜇0, 𝜏0 correspond to the branch points of the main equation solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Illustrations of typical (a,b) and unconventional (c,d) SR transitions predicted by simple  (quasi)doublet model are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The Figure 2a, built with 𝐾1 = 1, 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='05, ∆𝑎 =  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='84, ∆𝑐 = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='82, which corresponds to 𝑇𝑆𝑅 = 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='73, 𝜇𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='162, 𝜇𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='337, 𝜏𝑠 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='04, 𝜏𝑓 =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='91, describes a typical smooth SR transition with two second-order phase transitions 𝐺𝑥 − 𝐺���𝑧 at  the beginning (𝜏𝑠) and 𝐺𝑥𝑧 − 𝐺𝑧 at the end (𝜏𝑓) of the spin reorientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The Figure 2b, built with 𝐾1 = 1, 𝛾 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='1, ∆𝑎 = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='19, ∆𝑐 = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='1, which corresponds to  𝑇𝑆𝑅 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='95, 𝜇𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='59, 𝜇𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='72, 𝜏𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='762, 𝜏𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='93, describes an abrupt first-order SR  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' For 𝜏 > 𝜏𝑓 there is the 𝐺𝑥-phase, which can remain stable up to 𝜏𝑠 when cooled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' For 𝜏 < 𝜏𝑠  there is the 𝐺𝑧-phase, which can remain stable up to 𝜏𝑓 when heated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The point 𝐴 marks a phase  transition point when the phases 𝐺𝑥 and 𝐺𝑧 have equal energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  The Figure 2c, built with 𝐾1 = 1, 𝛾 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='222, ∆𝑎 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='72, ∆𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='63, which corresponds  to 𝑇𝑆𝑅 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='65, 𝜇𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='307, 𝜇𝑓 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='266, 𝜏𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='778, 𝜏𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='523 and the Figure 2d, built with  𝐾1 = 1, 𝛾 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='25, ∆𝑎 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='71, ∆𝑐 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='02, which corresponds to 𝑇𝑆𝑅 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='56, 𝜇𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='396, 𝜇𝑓 =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='31, 𝜏𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='73, 𝜏𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='545 describe unconventional "mixed"SR transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' At 𝜏𝑠 there is the smooth  second-order phase transition 𝐺𝑥 − 𝐺𝑥𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' At 𝜏 ≤ 𝜏𝑓 we have two stable phases 𝐺𝑧 and 𝐺𝑥𝑧: at those  temperatures the sharp first-order phase transition 𝐺𝑥𝑧 − 𝐺𝑧 can happen, or the system could stay in  the angular 𝐺𝑥𝑧-phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  (a)  τ  τs  τf  μ  μs  μf  (c)  τ  τs  τf  μ  μs  μf  (d)  τ  τs  τf  μ  μs  μf  θ  Φ  θ  Φ  θ  Φ  τ > τf  τ < τs  A  (b)  τ  τs  τf  μ  μs  μf  A  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 2: Illustrations of typical (a,b) and unconventional (c,d) SR transitions predicted by simple  (quasi)doublet model (see text for detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The arrows indicate the direction of the antiferromagnetic  vector G in the 𝑎𝑐 plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' The insets in panel (b) show the 𝜃-dependence of the free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  7  Thus, there are not only the smooth and abrupt SR transitions, a characteristic feature of the range  of intermediate values 𝐾2 is the fundamental possibility of the existence of “mixed” SR transitions, in  which the spins first smoothly rotate through a certain angle and then jump to the position with 𝜃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  For this, it is sufficient that 𝜇𝑓 corresponds to a point on the upper branch of solutions, and 𝜇𝑠 to a point  on the lower branch of solutions at 𝜏𝑓 < 𝜏𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' In this case, the spin reorientation begins with the single  second-order transition 𝐺𝑥 → 𝐺𝑥𝑧 and then ends with the first-order phase transition 𝐺𝑥𝑧 → 𝐺𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  In contrast to the “high-temperature” approximation, the “single-doublet” model claims the nature  of the phase transition is determined not simply by the sign of the second anisotropy constant, but  also it depends on the ratio between 𝐾1, 𝐾2 and the doublet splitting in both phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Nevertheless,  if we apply the simplified model to describe the SR transition, we have to renormalize both the first  and the second anisotropy constant, giving the last one sometimes a rather complicated temperature  dependence, in particular with a change in sign when considering transitions of the “mixed” type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Of course, in this case Fe sublattice alone is not enough to provide the value of the effective second  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  4 Conclusion  The model of the spin-reorientation transitions induced by the 4𝑓 − 3𝑑 interaction in rare-earth  orthoferrites and orthochromites has been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' It is shown that both the temperature and  the character of the spin-reorientation transition following from the solution of the transcendental  equation (8) are the result of competition between the second and fourth order spin anisotropy of  the 3𝑑 sublattice, the crystal field for 4f ions, and the 4𝑓 − 3𝑑 interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' At variance with the  “high-temperature” approximation, the “single-doublet” model, along with typical smooth and abrupt  SR transitions, predicts the appearance of mixed-type SR transitions, with an initial second-order  transition and a final abrupt first-order transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Funding: The research was supported by the Ministry of Education and Science of the Russian  Federation, project № FEUZ-2020-0054, and by Russian Science Foundation, project № 22-22-00682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  References  [1] Belov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Usp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' 1976, 19, 574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  [2] Belov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Levitin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Orientational Transitions in Rare-  Earth Magnetics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Nauka: Moscow, Russia, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' (In Russian)  [3] Singh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Rajput, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Padmanabhan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Anas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Damay, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Kumar, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Eguchi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Jain,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Yusuf, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Maitra, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Malik V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Successive spin reorientations and rare earth ordering  in Nd0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='5Dy0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='5FeO3: Experimental and ab initio investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' B 2020, 102, 144432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  8  [4] Hoogeboom, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Kuschel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Bauer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Mostovoy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Kimel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' van Wees, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content='  Magnetic order of Dy3+ and Fe3+ moments in antiferromagnetic DyFeO3 probed by spin Hall  magnetoresistance and spin Seebeck effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Derkachenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Bousquet, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Magnetic phase diagram of rare-earth orthorhombic  perovskite oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Antisymmetric Exchange and Magnetic Anisotropy in Weak Ferromagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Thesis, Lomonosov Moscow State University, Moscow, Russia, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' (In Russian)  [9] Moskvin,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Structure–Property  Relationships  for  Weak  Ferromagnetic  Perovskites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Magnetochemistry 2021, 7, 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  [10] Kadomtseva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Agafonov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Moskvin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Podlesnyak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Granroth, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Fishman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Kolesnikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content='  Pomjakushina, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Conder, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Yoshida, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Jeong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Yano, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Bourges, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Klicpera,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Sechovsk´y, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
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+page_content=' Boehm, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Low-energy spin dynamics of orthoferrites AFeO3  (A = Y, La, Bi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Matter 2018, 30, 235802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  [13] Amelin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Nagel, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Fishman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Yoshida, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Sim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Park, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' R˜o˜om, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Terahertz absorption spectroscopy study of spin waves in orthoferrite YFeO3 in a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' B 2018, 98, 174417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  [14] Moskvin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Bostrem, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Cubic Anisotropy of Rare-Earth Orthoferrites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content=' Solid St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  1979, 21, 628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}
+page_content='  9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFLT4oBgHgl3EQfuy_h/content/2301.12157v1.pdf'}

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+Ion filling of a one-dimensional nanofluidic channel in the interaction
+confinement regime
+Paul Robin,1 Adrien Delahais,1 Lyd´eric Bocquet,1 and Nikita Kavokine2, 3, a)
+1)Laboratoire de Physique de l’´Ecole Normale Sup´erieure, ENS, Universit´e PSL, CNRS, Sorbonne Universit´e,
+Universit´e Paris Cit´e, Paris, France
+2)Department of Molecular Spectroscopy, Max Planck Institute for Polymer Research, Ackermannweg 10,
+55128 Mainz, Germany
+3)Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, NY 10010,
+USA
+(Dated: 12 January 2023)
+Ion transport measurements are widely used as an indirect probe for various properties of confined electrolytes.
+It is generally assumed that the ion concentration in a nanoscale channel is equal to the ion concentration
+in the macroscopic reservoirs it connects to, with deviations arising only in the presence of surface charges
+on the channel walls.
+Here, we show that this assumption may break down even in a neutral channel,
+due to electrostatic correlations between the ions arising in the regime of interaction confinement, where
+Coulomb interactions are reinforced due to the presence of the channel walls. We focus on a one-dimensional
+channel geometry, where an exact evaluation of the electrolyte’s partition function is possible with a transfer
+operator approach. Our exact solution reveals that in nanometre-scale channels, the ion concentration is
+generally lower than in the reservoirs, and depends continuously on the bulk salt concentration, in contrast to
+conventional mean-field theory that predicts an abrupt filling transition. We develop a modified mean-field
+theory taking into account the presence of ion pairs that agrees quantitatively with the exact solution and
+provides predictions for experimentally-relevant observables such as the ionic conductivity. Our results will
+guide the interpretation of nanoscale ion transport measurements.
+I.
+INTRODUCTION
+A channel connects two reservoirs filled with a salt so-
+lution at concentration cout. What is the salt concentra-
+tion cin inside the channel? The straightforward answer
+cin = cout is challenged as soon as the channel’s dimen-
+sions are at the nanometre scale1. A deviation typically
+occurs because of the presence of a surface charge density
+Σ on the channel walls. Indeed, a sufficiently long chan-
+nel must remain electrically neutral2, which results in an
+imbalance of the concentrations c±
+in of the positive and
+negative ions. In a cylindrical channel of radius R that
+is smaller than the electrolyte’s Debye length, the con-
+centrations are given by the famous Donnan equilibrium
+result3:
+c±
+in =
+�
+c2
+out + (2Σ/R)2 ± 2Σ/R.
+(1)
+Eq. (1) is widely used to infer a channel’s surface charge
+from measurements of its conductivity at different salt
+concentrations.
+For sufficiently small surface charges
+(2Σ/R ≪ cout), Eq. (1) predicts cin = cout even at ex-
+treme nanoscales.
+Importantly, this prediction under-
+lies the method for extracting confined ion mobilities
+from transport measurements, which has been applied
+down to 7-˚A-wide two-dimensional channels4. Yet, phys-
+ically, cin = cout stems from the assumption that the
+electrolyte solutions, both in the reservoirs and in the
+a)Electronic mail: [email protected]
+channel, behave as ideal gases of non-interacting ions.
+While such a description is valid in the bulk reservoirs
+at reasonable salt concentrations5, it must be challenged
+in the nanometre-scale channel which is subject to in-
+teraction confinement6 – a reinforcement of the effective
+Coulomb interactions between the ions due to the dielec-
+tric contrast between the solvent (water) and the channel
+wall3,6–14.
+Due to interaction confinement, ions face a self-energy
+barrier Es when entering the channel7,8.
+It was first
+noted by Parsegian7 that this should result in ion exclu-
+sion: the salt concentration within the channel is then
+given by an Arrhenius scaling cin = coute−Es/kBT under
+the assumption of non-interacting ions. However, the re-
+sult becomes more subtle as the confinement-reinforced
+ionic interactions are taken into account.
+Within a
+mean-field description of a spherical nanopore, Dres-
+ner15 predicted an abrupt filling transition, where cin
+was a discontinuous function of cout.
+Later, Palmeri
+and coworkers16,17 recovered a similar transition using a
+three-dimensional model of a cylindrical channel, treated
+within the variational field theory formalism of Netz and
+Orland18.
+While this approach could be applied to a
+realistic geometry, it took into account electrostatic cor-
+relations only approximately.
+An exact treatment of electrostatic correlations is pos-
+sible upon simplification of the geometry to a purely
+one-dimensional model, with the channel wall being
+taken into account by introducing an effective confined
+Coulomb interaction.
+The 1D electrolyte can then be
+mapped onto an Ising or 1D Coulomb-gas-type model;
+the transfer matrix solution of such models was used, for
+arXiv:2301.04622v1  [cond-mat.soft]  11 Jan 2023
+
+2
+Effective interaction
+Self-energy
+B
+A
+FIG. 1.
+Ion filling in the interaction confinement regime.
+A. Schematic of the ion filling problem: a cylindrical
+nanochannel (radius R ∼ 1 nm) is connected to macroscopic reservoirs of aqueous electrolyte. The salt concentration inside
+the channel, cin, may differ from that in the reservoirs, cout. B. Physics of interaction confinement. When a charged species
+enters a nanochannel, the dielectric contrast between water (ϵw ∼ 80) and walls (ϵm ∼ 2) constraints the electric field lines to
+remain within the channel. This process can be interpreted in terms of image charges inside the channel walls, and results in
+an electrostatic self-energy barrier for ions to enter the channel, and reinforced interactions between ions.
+example, to discuss the capacitance of nanoporous sys-
+tems19–21. The lattice models may be taken to the con-
+tinuum limit, and the resulting path integral solutions
+have been used to discuss various ion-exchange phase
+transitions that arise in the presence of fixed discrete
+charges inside the channel9,22,23 and the ionic Coulomb
+blockade phenomenon13.
+Such models are particularly
+rich theoretically, as they support a mapping to non-
+Hermitian quantum mechanics24.
+Nevertheless, to our
+knowledge, the fundamental problem of ion filling in
+an uncharged channel has not been tackled within this
+framework.
+In this paper, we treat the ion-filling problem in the
+interaction confinement regime using an exactly-solvable
+one-dimensional model.
+We find that the value of cin
+is strongly affected by the formation of Bjerrum pairs
+– pairs of oppositely charged ions – within the channel,
+which preclude the occurence of an abrupt filling transi-
+tion. This is in contrast to the prediction of Palmeri and
+coworkers16,17, and to the result of conventional mean-
+field theory. We then build on our exact results to pro-
+pose a modified mean-field model that accounts for the
+relevant physical ingredients, and, particularly, for the
+presence of ion pairs.
+The paper is organized as follows.
+In Section II,
+we present the one-dimensional model and its solution
+within a path-integral formalism. The reader interested
+only in the physical outcomes may skip directly to Sec-
+tion III, where we discuss the model’s prediction for the
+ion concentration within the channel, compare it to the
+mean-field solution, and interpret it in terms of tightly
+bound Bjerrum pairs. In Section IV, we establish a mod-
+ified mean-field theory, based on the notion of phantom
+pairs, that reproduces our exact solution. The mean-field
+theory allows us to determine the number of unpaired
+ions and produces experimentally relevant predictions for
+a nanochannel’s ionic conductance. Section V establishes
+our conclusions.
+II.
+1D COULOMB GAS MODEL
+A.
+Confined interaction
+We consider a cylindrical channel of radius R and
+length L, connected to macroscopic reservoirs (Fig. 1A).
+We first assume for simplicity that the channel is filled
+with water that has isotropic dielectric permittivity ϵw =
+80, and that it is embedded in an insulating medium
+with much lower permittivity ϵm (for a lipid membrane7,
+ϵm ∼ 2).
+The effective Coulomb interaction V (x) be-
+tween two monovalent ions separated by a distance x
+on the channel axis can be computed exactly by solving
+Poisson’s equation8,12,13. A simple approximate expres-
+sion can be obtained for x ∼ R (ref.3):
+V (x) ≈
+e2α
+2πϵ0ϵwRe−|x|/(αR),
+(2)
+where α is a numerical coefficient that depends on the
+ratio ϵw/ϵm (α = 6.3 for ϵw/ϵm = 40). The reinforce-
+ment of electrostatic interactions compared to the usual
+e2/4πϵ0ϵwr Coulomb interaction that ions experience in
+bulk water can be interpreted in terms of images charges
+within the channel walls (Fig. 1B). Two confined ions
+interact not only with each other, but also with their
+respective image charges.
+We introduce the parameters ξ ≡ αR and xT
+≡
+2πϵ0ϵwR2kBT/e2: both have the dimension of a length.
+
+3
+With these notations,
+V (x) = kBT ξ
+xT
+e−|x|/ξ.
+(3)
+The effects of ion valence and of anisotropic dielectric
+response of confined water can be taken into account by
+adjusting ξ and xT 13. Formally, the expression in Eq. (2)
+is valid for any channel radius.
+Yet, it is only physi-
+cally relevant if at x ∼ R the interaction is significant
+compared to kBT, which restricts in practice the appli-
+cability of Eq. (2) to R ≲ 2 nm. In such extreme 1D
+confinement, we may neglect the ions’ degrees of free-
+dom perpendicular to the channel axis and assume that
+they are constrained to move in one dimension. The par-
+tition function of such a 1D electrolyte may be computed
+exactly, as detailed in the next section.
+B.
+Path integral formalism
+Here, we detail the analytical solution for the partition
+function of a 1D Coulomb gas-like system that was first
+introduced in ref.13. We set kBT = 1 until the end of Sec.
+II. We start from a lattice model, in order to rigorously
+establish a path integral description in the continuum
+limit.
+Our computation is inspired by the original solution of the 1D Coulomb gas model by Lenard and Edwards25, and
+subsequent studies by Demery, Dean and coworkers19,21,26,27, as well as Shklovskii and coworkers22,23. We consider
+a one-dimensional lattice with sites 1, . . . , M as a model for the nanochannel of radius R and length L. Each lattice
+site i carries a spin Si, which takes the values {0, 1, −1}, corresponding respectively to no ion, a positive ion, or a
+negative ion occupying the site. We model the surface charge distribution as an extra fixed charge qi added at each
+lattice site. The spins interact with the Hamiltonian
+H({Si}) =
+ξ
+2xT
+M
+�
+i,j=1
+(Si + qi)(Sj + qj)e−|i−j|/ξ ≡
+1
+2xT
+(S + q)T C(S + q).
+(4)
+The system is in contact with a particle reservoir at concentration cout. Here the parameters ξ and xT are dimension-
+less, expressed in number of lattice sites.
+The grand partition function is given by
+Ξ =
+�
+S1,...,SM
+z
+�
+i |Si|e−
+1
+2xT (S+q)T C(S+q),
+(5)
+with z = coutπR2L/M the fugacity. The matrix C can be analytically inverted:
+C−1 =
+1
+2ξ sinh(1/ξ) ·
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+e1/ξ
+−1
+0
+0
+. . .
+0
+0
+−1
+2 cosh(1/ξ) −1
+0
+. . .
+0
+0
+...
+...
+... ...
+...
+...
+...
+... ... ...
+...
+...
+...
+... ...
+...
+...
+0
+0
+. . .
+0
+−1 2 cosh(1/ξ)
+−1
+0
+0
+. . . . . .
+0
+−1
+e1/ξ
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(6)
+Hence we can carry out a Hubbard-Stratonovich transformation, that is rewrite the partition function as a gaussian
+integral, introducing the integration variable ϕ:
+Ξ =
+�
+xM
+T
+(2π)Mdet(C) ·
+�
+S1,...,SM
+z
+�
+i |Si|
+�
+dϕe− xT
+2 ϕT C−1ϕ+i(S+q)T ϕ,
+(7)
+with det(C) =
+e1/ξ
+2 sinh(1/ξ) ·
+�
+ξ(1 − e−2/ξ)
+�M. After performing the sum over the spins, which is now decoupled, we
+obtain
+Ξ =
+�
+xM
+T
+(2π)Mdet(C) ·
+�
+dϕ1 . . . dϕM
+M
+�
+j=1
+(1 + 2z cos ϕj)
+M
+�
+j=1
+eiqjϕj . . .
+. . . exp
+�
+�−
+xT
+4ξ sinh(1/ξ)
+�
+�
+M−1
+�
+j=1
+(ϕj+1 − ϕj)2 + 2(cosh(1/ξ) − 1)
+M−1
+�
+j=2
+ϕ2
+j + (e1/ξ − 1)(ϕ2
+1 + ϕ2
+M)
+�
+�
+�
+� .
+(8)
+
+4
+We now take a continuum limit of the lattice model. We call a the physical lattice spacing and let ˜ξ = aξ, ˜xT = axT
+and ˜z = Mz/L. We then let a → 0 and M → ∞ while keeping the physical length of the system L = aM constant.
+We then drop the tilde sign to lighten the notation and obtain
+Ξ =
+�
+dϕ(0)e−xT ϕ(0)2/4ξ
+�
+[dϕ]e−S[ϕ]
+�
+dϕ(L)e−xT ϕ(L)2/4ξ
+(9)
+with
+S[ϕ] =
+� L
+0
+dx
+�
+xT
+4
+�dϕ
+dx
+�2
++ xT
+4ξ2 ϕ(x)2 − iq(x)ϕ(x) − 2z cos ϕ(x)
+�
+≡
+� L
+0
+L(ϕ, ˙ϕ).
+(10)
+q(x) is the one-dimensional density corresponding to the surface charge, and z ≡ πR2cout. At this point ξ and xT
+have the dimension of length. The path integral measure is defined as
+[dϕ] =
+lim
+a→0
+M→∞
+L=aM
+�
+�
+M
+�
+j=1
+� xT
+4πadϕj
+�
+� .
+(11)
+We now define the propagator P(ϕ, x|ϕ0, 0), or simply P(ϕ, x), as
+P(ϕ, x) =
+�
+dϕ(x)δ(ϕ(x) − ϕ)
+�
+[dϕ]e−
+� x
+0 L(ϕ, ˙ϕ)
+�
+dϕ(0)δ(ϕ(0) − ϕ0).
+(12)
+Considering an infinitesimal displacement ∆x,
+P(ϕ, x) =
+� xT
+4π∆x
+�
+d(∆ϕ)P(ϕ − ∆ϕ, x − ∆x) . . .
+. . . exp
+�
+−
+� x
+x−∆x
+dx′
+�
+xT
+4
+�∆ϕ
+∆x
+�2
++ xT
+4ξ2 ϕ2 − iq(x)ϕ − 2z cos ϕ
+��
+.
+(13)
+Expanding the propagator as P(ϕ − ∆ϕ, x − ∆x) = P(ϕ, x) − ∆x∂P/∂x − ∆ϕ∂P/∂ϕ + (1/2)(∆ϕ2)∂2P/∂ϕ2, and
+carrying out the gaussian integrals, we obtain
+P(ϕ, x) =
+�
+P(ϕ, x) − ∆x∂P
+∂x + O(∆x2)
+� �
+1 − ∆x
+� xT
+4ξ2 ϕ2 − iq(x)ϕ − 2z cos ϕ
+�
++ O(∆x2)
+�
++ ∆x
+xT
+∂2P
+∂x2 (1 + O(∆x)).
+(14)
+P(ϕ, x) thus solves the partial differential equation
+∂P
+∂x = 1
+xT
+∂2P
+∂ϕ2 +
+�
+iqϕ − xT
+4ξ2 ϕ2 + 2z cos ϕ
+�
+P,
+(15)
+with initial condition P(ϕ, 0) = δ(ϕ − ϕ0), which is the equivalent of a Schr¨odinger equation for the path integral
+representation (9). The partition function can thus be computed as
+Ξ =
+�
+dϕ(L)e−xT ϕ2/4ξP(ϕ, L|f0),
+(16)
+where P(ϕ, L|f0) is the solution of (15) with initial condition P(ϕ, 0) = f0(ϕ) ≡ e−xT ϕ2/4ξ.
+C.
+Transfer operator
+We introduce the Fourier transform of P with respect to ϕ:
+˜P(k, x) =
+1
+√
+2π
+�
+dϕe−ikϕP(ϕ, x).
+(17)
+
+5
+Then ˜P(k, x) satisfies
+∂ ˜P
+∂x = − k2
+xT
+˜P − q ∂ ˜P
+∂k + xT
+4ξ2
+∂2 ˜P
+∂k2 + z
+�
+˜P(k + 1, x) + ˜P(k − 1, x)
+�
+.
+(18)
+From now on, we restrict ourselves to an uncharged channel (q = 0). We then define the operator T such that
+[T ( ˜P)](k) = − k2
+xT
+˜P + xT
+4ξ2
+∂2 ˜P
+∂k2 + z
+�
+˜P(k + 1, x) + ˜P(k − 1, x)
+�
+,
+(19)
+which plays the role of a functional transfer matrix. Recalling eq. (16), the partition function then reads
+Ξ = ⟨f0|eLT |f0⟩
+(20)
+with f0(k) = e−ξk2/xT and ⟨f(k)|g(k)⟩ ≡
+�
+dkf ∗(k)g(k).
+Now, in the limit L → ∞, we may consider the largest eigenvalue λ of the operator T , and the associated eigen-
+function χ:
+[T (χ)](k) = λχ(k).
+(21)
+Then, up to an exponentially small correction,
+Ξ = |⟨f0|χ⟩|2⟨χ|χ⟩eλL.
+(22)
+D.
+Ion concentration
+Our aim is to compute the salt concentration cin in the nanoscale channel given a salt concentration cout in the
+reservoir. At the level of the lattice model, the probability to find, say, a positive ion at position k, can be computed
+by replacing a factor (1 + 2z cos ϕk) by zeiϕk in Eq. (8). In the continuum limit, we obtain the positive (negative) ion
+linear density at position x by inserting the operator zeiϕ (ze−iϕ) at position x:
+πR2⟨c±
+in(x)⟩ = 1
+Ξ
+�
+dϕ(0)dϕ(x)dϕ(L)e−xT ϕ(0)2/4ξP(ϕ(x), x|ϕ(0), 0)ze±iϕ(x)P(ϕ(L), L|ϕ(x), x)e−xT ϕ(L)2/4ξ,
+(23)
+Upon Fourier-transformation, the insertion of eiϕ amounts to a shift by unity. Introducing the operator,
+SQ : f �→ (g : k �→ f(k − Q)),
+(24)
+the concentrations are given by
+⟨c±
+in(x)⟩ =
+z
+πR2
+⟨f0|exT S±1e(L−x)T |f0⟩
+Ξ
+= cout
+⟨f0|exT S±1e(L−x)T |f0⟩
+Ξ
+,
+(25)
+since z = coutπR2. In the thermodynamic limit, and using Eq. (22) for the partition function, we obtain
+⟨c±
+in⟩ = cout
+⟨χ(k)|χ(k ∓ 1)⟩
+⟨χ(k)|χ(k)⟩
+.
+(26)
+Eq. (26) is the main result of our exact computation. In practice, the function χ(k) is determined numerically, by
+finite-difference integration of Eq. (18).
+III.
+PHYSICS OF ION FILLING
+A.
+Debye-H¨uckel solution
+We now go back to the ion filling problem (Fig. 1A)
+and present first a one-dimensional mean-field solution.
+Typically, the mean-field solution of an electrolyte prob-
+lem is obtained by solving the Poisson-Boltzmann equa-
+tion28,29. For the conventional Poisson-Boltzmann equa-
+tion to apply, we would need to consider the full three-
+dimensional geometry of our problem, and the effective
+interaction of Eq. (3) would be introduced implicitly
+through the boundary conditions at the channel walls15.
+
+6
+B
+A
+C
+Concentration
+Distance
+Anions
+Cations
+Debye cloud
+10-4
+10-2
+100
+Reservoir concentration (M)
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Channel conc./Res. conc.
+Exact solution
+Series expansion
+Poisson-Boltzmann
+Debye-Hückel
+10-4
+10-2
+100
+Reservoir concentration (M)
+10-3
+10-2
+10-1
+100
+Channel conc./Res. conc.
+Exact solution
+Series expansion
+Poisson-Boltzmann
+Debye-Hückel
+Bulk
+Self-energy
+barrier
+Bulk
+Self-energy
+barrier
+Ion pairs
+Weak interactions:
+Es = 0.5 kBT
+Strong interactions:
+Es = 6 kBT
+FIG. 2. Comparing mean-field approximations with the exact Coulomb gas solution. A. Schematic description
+of the mean-field approaches.
+The chemical potential of confined ions is determined by solving the (linear or nonlinear)
+Poisson-Boltzmann equation around a given ion, interacting with an oppositely charged Debye cloud.
+B. Dependence of
+the channel salt concentration cin on the reservoir salt concentration cout, in a weakly-interacting case (R = 1 nm, ξ = 7 nm,
+xT = 7 nm, Es = 0.5 kBT). We plot four different predictions for the ratio cin/cout: the exact field-theoretical solution (Eq. (26),
+blue circles), its low concentration expansion (Eq. (47), black line), the mean-field predictions from solving the full Poisson-
+Boltzmann equation (Eq. (40), orange curve) or from its Debye-H¨uckel linearization (Eq. (36), yellow line). The two mean-field
+predictions are indistinguishable.
+In all cases, the naive estimate cin = cout is recovered for high enough concentrations.
+In the dilute limit, the concentration inside the channel is well approximated by the Arrhenius scaling cin = coute−Es/kBT .
+C. Dependence of the channel salt concentration cin on the reservoir salt concentration cout, in a strongly-interacting case
+(R = 1 nm, ξ = 7 nm, xT = 0.6 nm, Es = 6 kBT). The color code is the same as in B. Here, the mean-field predictions strongly
+deviate from the exact solution, with the Debye-H¨uckel model predicting an abrupt filling transition. This discrepancy is due
+to the formation of Bjerrum pairs at intermediate concentrations, as evidenced by the scaling cin ∝ c2
+out in the exact solution.
+In order to obtain a mean-field solution directly in the
+1D geometry, we need to introduce a modified Poisson’s
+equation for the electrostatic potential Φ whose Green’s
+function coincides with Eq. (3):
+� d2
+dx2 − 1
+ξ2
+�
+φ = −2πR2 c+ − c−
+xT
+,
+(27)
+with φ ≡ eΦ/kBT the dimensionless potential.
+Im-
+posing that the ions follow a Boltzmann distribution
+(c± = cine∓φ, where cin is understood as the average con-
+centration inside the channel), we obtain the analogue of
+the Poisson-Boltzmann equation in our 1D geometry:
+� d2
+dx2 − 1
+ξ2
+�
+φ = 2πR2 cin
+xT
+sinh φ.
+(28)
+In order to proceed analytically, we make a Debye-
+H¨uckel-type approximation and linearize Eq. (28) with
+respect to φ. Then, the potential around an ion placed
+in the channel at x = 0 is given by
+φ(x) = ξeff
+xT
+e−|x|/ξeff,
+(29)
+with
+ξ2
+eff =
+ξ2
+1 + 4πR2cinξ2/xT
+.
+(30)
+The chemical potential inside the channel is the sum of
+an ideal gas entropic part and of an excess part due to
+interactions:
+µin = µent + µex,
+(31)
+with
+µent = kBT log coutΛ3,
+(32)
+Λ being the De Broglie thermal wavelength of the ions.
+µex can be obtained via a Debye charging process30:
+µex
+kBT =
+� 1
+0
+φλ(0)dλ, φλ(0) =
+λξ/xT
+�
+1 + 4λπR2cinξ2/xT
+.
+(33)
+We determine cin by imposing equality of the chemical
+potentials between the channel and the reservoir:
+µout = kBT log coutΛ3 = µin,
+(34)
+which yields
+cin = coute−µex/kBT .
+(35)
+Evaluating analytically the integral in Eq. (33), we obtain
+an implicit equation for cin.
+With the notation ˆcin ≡
+πR2cin,
+cin = cout exp
+�
+− ξ
+2xT
+×
+x2
+T
+6ξ2ˆc2
+inξ2
+�
+1 − 3
+2(1 + 4ˆcinξ2/xT )1/2
++1
+2(1 + 4ˆcinξ2/xT )3/2
+��
+.
+(36)
+
+7
+In Fig. 2B and C, we plot the ratio cin/cout as a func-
+tion of cout, as obtained by numerically solving Eq. (36).
+We fix ξ = 7 nm (which corresponds to a channel with
+R ≈ 1 nm and strong dielectric contrast), and vary
+xT to set the ionic interaction strength.
+The interac-
+tion strength may be quantified through the self-energy
+barrier, Es = kBT × ξ/(2xT ). The limiting behavior of
+cin/cout may be understood directly from Eq. (36). When
+cin is small, Eq. (36) reduces to the Arrhenius scaling
+cin = coute−Es/kBT : this results typically holds for bio-
+logical ion channels which may contain either 0 or 1 ion at
+any given time, and the effect of inter-ionic interactions
+is negligible. When cin is large, we recover cin = cout. In-
+deed, the excess term in the chemical potential vanishes
+at high concentrations, which is then dominated by the
+entropic term. The fact that µex → 0 as cin → ∞ is non-
+trivial: it can be seen, physically, as resulting from the
+Coulomb potential of each ion being perfectly screened
+by the other ions. At small values of Es, Eq. (36) has
+a single solution for all values of cout, which interpolates
+smoothly between the two limiting regimes.
+However,
+for Es ≳ 5kBT, it has three solutions in a certain range
+of cout, pointing to a pseudo-first-order phase transition
+between a low-concentration and a high-concentration
+phase, similar to the one predicted by Dresner15 and
+Palmeri et al.16. The transition occurs at ˆcin ∼ xT /ξ2: as
+per Eq. (30), this corresponds to the concentration where
+the effect of the screening cloud on an ion’s Coulomb po-
+tential becomes significant.
+B.
+Full Poisson-Boltzmann solution
+The physical content of the mean-field solution pre-
+sented above is similar to the one of Dresner, based on
+a linearized Poisson-Boltzmann equation15. The differ-
+ence in geometry, and the fact that he foregoes the use
+of the Debye charging process, do not seem to play a sig-
+nificant qualitative role. The solution of Palmeri et al.16
+takes ionic correlations into account to some extent, yet
+it still involves a Debye-H¨uckel-type linear equation for
+the mean-field interaction potential between the ions.
+One may ask whether the same phenomenology per-
+sists if one does not linearize the Poisson-Boltzmann
+equation. The full Poisson-Boltzmann equation cannot
+be solved analytically, but supports the following inte-
+gral form:
+�dφ
+dx
+�2
+− 1
+ξ2 φ2 = 4πR2 cin
+xT
+(cosh φ − 1) ,
+(37)
+where we have used the fact that φ should vanish at x →
+∞. For x → 0, the solution of Eq. (37) should reduce
+to the unscreened potential in Eq. (3) up to an additive
+constant, so that
+1
+x2
+T
+− 1
+ξ2 φ2(0) = 4πR2 cin
+xT
+(cosh φ(0) − 1) .
+(38)
+Once again, one may express the excess chemical po-
+tential of the confined ions through a Debye charging
+process:
+µex
+kBT =
+� 1
+0
+φλ(0)dλ,
+λ2
+x2
+T
+− 1
+ξ2 φ2
+λ(0) = 4πR2 λcin
+xT
+(cosh φλ(0) − 1) .
+(39)
+This result is the analogue of Eq. (33), with φλ(0) now
+being the solution of an implicit non-linear equation, so
+that µex must be determined numerically. As before, the
+concentration inside the channel is then given by:
+cin = coute−µex/kBT .
+(40)
+The prediction of the full Poisson-Boltzmann equation
+is shown in Fig. 2B and C: we find cin to be a smooth
+function of cout for all values of parameters, in contrast to
+the linearized solution. We may not, however, unambigu-
+ously conclude that the filling transition is an artifact of
+linearization, since the non-linear solution still involves a
+mean-field approximation and is not guaranteed to yield
+the correct result.
+Interestingly, the “physically-motivated” mean-field
+solution in Eq. (28) differs from the mean-field limit of
+our exact solution. It is obtained by taking the saddle-
+point approximation in the path-integral expression of
+the partition function (Eq. (9)).
+The Euler-Lagrange
+equation for the minimizer ϕ(x) of the action S[ϕ] in
+Eq. (10) is, upon identifying φ = −iϕ,
+� d2
+dx2 − 1
+ξ2
+�
+φ = 2πR2 cout
+xT
+sinh φ.
+(41)
+This is Eq. (28) with cin replaced with cout, and corre-
+sponds to a first order treatment of interactions. Indeed,
+if the ions are non-interacting, cin = cout. By solving the
+mean-field equation, we determine how the ions’ chemi-
+cal potential is affected by Debye screening, which then
+results in value of cin that is different from cout. Within
+a straightforward interaction expansion procedure, one
+should determine the effect of screening assuming the ze-
+roth order value for the ion concentration inside the chan-
+nel, which is cout: this corresponds to Eq. (41). Eq. (28)
+contains an additional self-consistency condition, as it
+assumes the actual value cin for the ion concentration,
+which is not known until Eq. (28) is solved. One may
+draw a loose condensed matter physics analogy, where
+Eq. (41) resembles the Born approximation for impu-
+rity scattering, while Eq. (28) is analogous to the self-
+consistent Born approximation.31
+C.
+Exact solution
+We now turn to the exact solution obtained in Sec.
+II to unambiguously solve the ion filling problem. We
+
+8
+determine cin according to Eq. (26):
+⟨c±
+in⟩ = cout
+⟨χ(k)|χ(k ∓ 1)⟩
+⟨χ(k)|χ(k)⟩
+,
+(42)
+where χ(k) is the highest eigenfunction of the trans-
+fer operator in Eq. (19), determined in practice by nu-
+merical integration. The exact results for cin, with the
+same parameter values as for the mean-field solution,
+are shown in Fig. 2 B and C. When interactions are
+weak (small values of Es, Fig. 2B), the exact and mean-
+field solutions are in good agreement. Notably, all so-
+lutions smoothly interpolate between the bulk scaling
+cin = cout at high concentration, and the Arrhenius scal-
+ing cin = coute−Es/kBT at low concentration. Conversely,
+in the strongly-interacting case (large Es, Fig. 2C), the
+exact result yields a much larger ion concentration that
+the mean-field solutions for intermediate values of cout.
+In this intermediate regime, cin remains a smooth func-
+tion of cout, and obeys the scaling cin ∝ c2
+out.
+Such a scaling is the signature of the formation of
+tightly bound Bjerrum pairs of positive and negative ions
+– strongly-correlated configurations that are not taken
+into account by mean-field solutions. Indeed, let us as-
+sume that the channel contains an ideal gas of ion pairs at
+concentration cin. We further assume that in a pair, the
+distance between the two ions is uniformly distributed
+in the interval [−xT /2, xT /2], and the binding energy of
+a pair is kBTξ/xT = 2Es.
+Then, the grand partition
+function reads
+Ξ =
+�
+N
+(ze−βEs)2N 1
+N!
+N
+�
+i=1
+L
+� xT /2
+−xT /2
+dx e2βEs
+(43)
+=
+�
+N
+(z2LxT )N
+N!
+= ez2LxT ,
+(44)
+where we recall that z = πR2cout and β ≡ 1/(kBT).
+Using that
+πR2cin = 1
+L
+∂ log Ξ
+∂(βµ) = z
+L
+∂ log Ξ
+∂z
+,
+(45)
+we obtain
+cin = 2z2xT
+πR2
+= 2πR2xT c2
+out.
+(46)
+We recover indeed the quadratic scaling.
+We may check that the prefactor in Eq. (46) is the
+correct one by evaluating analytically the expression in
+Eq. (26) in the low concentration limit zT ≡ zxT ≪ 1.
+An analytical expansion of the function χ(k) in powers
+of zT was derived in ref.13. Substituting it into Eq. (26),
+we obtain
+πR2cin = z(e−βEs + 2zT − 13
+2 z2
+T e−βEs
+−7z3
+T + O(z4
+T ) + O(e−2βEs)).
+(47)
+The first term in the expansion corresponds to cin =
+coute−βEs.
+At the lowest salt concentrations, forming
+Bjerrum pairs is too entropically unfavorable, and the
+concentration inside the channel is controlled by the self-
+energy barrier.
+However, as the salt concentration in-
+creases, there is no abrupt transition to a highly-screened
+concentrated phase inside the channel; instead, the chan-
+nel is progressively filled by Bjerrum pairs. This corre-
+sponds to the quadratic term in the expansion, with the
+prefactor agreeing indeed with Eq. (46).1 The expansion
+in Eq. (47) reproduces quite well the low-concentration
+behavior of the exact solution as shown in Fig. 2B and
+C. However, it fails at high concentrations, where it does
+not recover cin = cout.
+Our exact analysis of the ion statistics in a nanoscale
+channel has revealed that Bjerrum pairs are a crucial in-
+gredient of the filling process. We now develop a modified
+mean-field theory that accounts the presence of Bjerrum
+pairs and compare it to the exact solution.
+IV.
+PAIR-ENHANCED MEAN-FIELD THEORY
+A.
+Debye-H¨uckel-Bjerrum theory
+The traditional mean-field treatment of electrolytes is
+incapable of taking Bjerrum pairs into account, as it nat-
+urally neglects any strong ion-ion correlations – pairing
+being a fundamentally discrete phenomenon.
+An idea
+proposed by Bjerrum to amend the Debye-H¨uckel theory
+was to introduce ion pairs as a separate species encapsu-
+lating all “strong” ion-ion correlations32. More precisely,
+any two oppositely charged ions that are closer than some
+minimum distance can be considered as a single neutral
+entity – a Bjerrum pair. The remaining “free” ions should
+then only experience weak interactions with each other,
+and can be treated at the mean-field level. Importantly,
+this last remark justifies the Debye-H¨uckel linearization,
+as all non-linear effects are assumed to be hidden in the
+definition of ion pairs.
+As before, we consider that pairs behave like particles
+of an ideal gas, and that their maximum extension is
+given by xT . Defining cp
+in the concentration pairs inside
+the channel, the chemical potential of pairs is given by:
+µp
+in = kBT log
+cp
+inΛ6
+2πxT R2 ,
+(48)
+where the geometrical factor inside the logarithm ac-
+counts for the internal degrees of freedom of a pair. The
+chemical potential only has an entropic term, because
+the binding energy of the pair exactly compensates the
+self-energy of the two separate ions. The chemical equi-
+librium between free ions and pairs inside the channel
+1 This justifies a posteriori our choice of [−xT /2, xT /2] as the
+interval in which a paired-up ion is allowed to move.
+
+9
+Concentration
+Distance
+Anions
+Cations
+B
+A
+C
+Debye cloud
+Bjerrum pair
+Well-defined
+pair
+Phantom pair
+10-4
+10-2
+100
+Reservoir concentration (M)
+10-3
+10-2
+10-1
+100
+101
+102
+Channel conc./Res. conc.
+Exact solution
+Debye-Hückel-Bjerrum mean-field
+Phantom pair mean-field
+Strong interactions:
+Es = 6 kBT
+FIG. 3. Pair-enhanced mean-field theory. A. Treatment of ion pairing in mean-field approaches. Top panel: Mean-field
+theories inevitably underestimate ion-ion correlations. To circumvent this problem, two ions that are distant by less than xT
+are considered to form an ion pair, which is treated as a separate chemical species. Bottom panel: schematic representation
+of ion distribution around a fixed positive ion. The distribution is very peaked close to the central ion, due to the formation
+of an ion pair, and then relaxes smoothly to the mean value cin. B. Evolution of channel concentration cin as function of
+reservoir concentration cout, in a strongly-interacting cacse (R = 1 nm, ξ = 7 nm, xT = 0.6 nm, Es = 6 kBT). We plot the ratio
+cin/cout obtained from three different models taking Bjerrum pairs into account: the exact field-theoretical solution (Eq. (26),
+blue circles), the Debye-H¨uckel-Bjerrum mean-field theory (Eq. (51), red line) and our modified mean-field theory based on the
+notion of phantom pairs (Eq. (55), orange line), which reproduces the exact solution quantitatively for all values of parameters.
+At high concentration, the Debye-H¨uckel-Bjerrum prediction fails due to the uncontrolled proliferation of Bjerrum pairs. C.
+Formation of phantom pairs inside the nanochannel. At low concentration (top panel), pairs are well-separated and ions forming
+a pair are tightly bound to each other. At high concentration (bottom panel), ionic interactions are weakened as a result of
+Debye screening, and two quasi-non-interacting ions may find themselves within a distance xT of each other without actually
+binding: this is a phantom pair.
+can be written as:
+µ+
+in + µ−
+in = 2µin = µp
+in,
+(49)
+where µ+
+in and µ−
+in are the chemical potentials of cations
+and anions, respectively.
+We then obtain, using the
+Debye-H¨uckel solution for µin (equations (31) to (33)):
+cp
+in = 2πR2xT c2
+out,
+(50)
+which is the result obtained in the previous section. The
+average concentration in free ions cf
+in is not modified com-
+pared to the Debye-H¨uckel solution, and is therefore the
+solution of the self-consistent Eq. (36).
+One can then
+compute the total concentration inside the channel as
+cin = cf
+in + cp
+in, or, explicitly
+cin = coute−µex(cf
+in)/kBT + 2πR2xT c2
+out.
+(51)
+In other words, the only impact of pairs in Bjerrum’s
+computation is to add a quadratic term 2πR2xT c2
+out to
+the Debye-H¨uckel result, matching with the expansion
+(47) of the exact solution up order 2 in the bulk concen-
+tration. We compare the two predictions on Fig. 3B. The
+Debye-H¨uckel-Bjerrum solution is found to match the ex-
+act one quite well at low and intermediate concentrations.
+This result is, however, unphysical for cout ≳ 1/πR2xT :
+cin is found to grow much faster than the bulk concen-
+tration. One solution would be to consider higher-order
+terms in the mean-field treatment through the inclusion
+of triplets, quadruplets, etc. of ions, and all possible in-
+teractions between these entities.
+Truncating the sum
+at any finite order, however, would not yield a solution
+valid in the entire range of concentrations, nor is it guar-
+anteed to converge to the exact solution. This approach
+is also unsatisfactory as it would not yield a closed-form
+expression for cin and would not allow for qualitative un-
+derstanding of the underlying physics.
+Instead, we develop a different method that, through
+physics-driven arguments, prevents the divergence of cin
+at high bulk concentrations and reproduces quantita-
+tively the exact solution.
+B.
+Phantom pairs
+Eq. (51) overestimates the number of Bjerrum pairs in
+the channel because it fails to account for the presence
+of Bjerrum pairs in the reservoir. The electrolyte in the
+reservoir is treated as an ideal gas : the ions are non-
+interacting and they cannot form actual tightly-bound
+pairs. Nevertheless, we have defined any two oppositely
+charged ions that find themselves in a cylinder of radius
+R and length xT to be a separate chemical species. Such
+configurations may arise in the reservoir simply out of
+statistical chance: we dub them phantom pairs. For our
+
+10
+mean-field theory to be consistent, these phantom pairs
+need to be taken into account.
+Let cp
+out be the concentration of phantom pairs in the
+reservoir.
+The chemical equilibrium between phantom
+pairs and free ions imposes
+cp
+out = 2πR2xT (cf
+out)2.
+(52)
+In addition, one has cf
+out + cp
+out = cout, since an ion must
+either be free or part of a pair. Imposing this condition
+yields:
+cf
+out =
+√
+1 + 8πcoutxT R2 − 1
+4xT πR2
+.
+(53)
+We use this result to control the proliferation of pairs in
+the channel: we now equilibrate the free ions inside the
+nanochannel with only the free ions in the reservoir:
+cf
+in = cf
+oute−µex(cf
+in)/kBT ,
+(54)
+which corresponds to Eq. (35) with cout replaced by cf
+out.
+Eq. (54) is again a self-consistent equation, this time on
+the concentration of free ions cf
+in, that must be solved
+numerically. Lastly, equilibrating pairs with free ions in-
+side the channel (or, equivalently, pairs inside with pairs
+outside), we obtain:
+cin = cf
+in + 2πR2xT (cf
+out)2,
+(55)
+where the second term corresponds again to Bjerrum
+pairs. Eqs. (53) to (55) constitute the main result of our
+modified mean-field theory. Note that µex may be deter-
+mined at the Debye-H¨uckel level (Eq. (33)), or by solving
+the full Poisson-Boltzmann equation (Eq. (39)). In what
+follows, we will only discuss the latter, as it offers greater
+accuracy; however, the Debye-H¨uckel prediction provides
+reasonable results even in the case of strong interactions,
+and yields for a convenient analytical expression for µex
+as function of cf
+in.
+The
+prediction
+of
+our
+phantom
+pair
+Poisson-
+Boltzmann model is compared to the exact solution (26)
+in Fig. 3B. The two solutions are found to be in near
+perfect agreement for all values of parameters, even in
+strong coupling limit Es ≫ kBT.
+In the next two sections, we use our modified mean-
+field model to predict the conductance of a nanochannel,
+first in the case of a neutral channel, and then in presence
+of a surface charge.
+C.
+Conductance
+One strength of our modified mean-field model is that
+it offers insight into the physical properties of the con-
+fined system beyond the value of the ionic concentra-
+tion. In particular, the decomposition of the electrolyte
+into free ions and bound pairs allows us to estimate the
+channel’s conductance. Tightly bound Bjerrum pairs are
+electrically neutral, so that they do not contribute to the
+ionic current to first order in applied electric field: it
+would then be straightforward to assume that the chan-
+nel’s conductance is proportional to the concentration
+of free ions. However, the reasoning needs to be more
+subtle, since the channel, in the same way as the reser-
+voir, may contain non-interacting phantom pairs.
+In-
+deed, we have decomposed the confined electrolyte into
+tightly bound pairs, that have no ionic atmosphere, and
+free ions that are dressed by a Debye screening cloud.
+As the concentration increases, the interaction between
+dressed ions becomes weak, and two of them may find
+themselves within a distance xT without actually bind-
+ing. Such a phantom pair is expected to still contribute
+to the conductance. The concentration of phantom pairs
+in the channel is obtained by imposing their chemical
+equilibrium with the free ions treated as an ideal gas.
+Thus, we estimate the channel’s conductance as:
+G = 2 e2D
+kBT
+πR2
+L
+�
+cf
+in + 2xT πR2(cf
+in)2�
+,
+(56)
+where D is the diffusion coefficient of ions; the second
+term corresponds to the contribution of phantom pairs.
+In Fig. 4A, we compare this result to the Ohm’s law
+prediction where pairs are neglected and one assumes
+cin = cout. Ohm’s law is found to greatly overestimate
+the conductance at low concentration. In the dilute limit,
+we instead recover the Arrhenius scaling, where one as-
+sumes cin = coute−Es/kBT .
+Finally, we stress that Eq. (56) only accounts for the
+electrophoresis of free ions, and is therefore only valid
+in the limit of weak external electric fields.
+Stronger
+voltage drops will result in the breaking of ion pairs,
+causing a conductivity increase in a process known as
+the second Wien effect. This phenomenon is described in
+refs.13,14, and has been used to create solid-state voltage-
+gated nanochannels33.
+D.
+Effect of a surface charge
+Up till now, we have restricted ourselves to channels
+with uncharged walls.
+However, in most experimen-
+tally relevant situations, the channel walls bear a sur-
+face charge density Σ, which strongly impacts nanofluidic
+transport. While introducing a surface charge is tedious
+within the exact framework, we may readily assess the
+effect of surface charge in the interaction confinement
+regime using our pair-enhanced mean-field theory.
+In the limit where the channel’s radius is smaller than
+the Debye length, we assume that the presence of the
+surface charge amounts to a homogeneous Donnan po-
+tential drop VD inside the channel, which we do not need
+to determine explicitly. Then, the chemical potential of
+ions inside the channel reads:
+µ±
+in = µex ± eVD + kBT log c±
+inΛ3.
+(57)
+
+11
+B
+A
+10-3
+10-2
+10-1
+100
+101
+Reservoir concentration (M)
+10-6
+10-4
+10-2
+100
+Channel conductance (nS)
+Ohm law
+Phantom pair mean-field
+Arrhenius model
+Actual surface charge
+10-3 C/m2
+Apparent surface charge
+10-2 C/m2
+10-3
+10-2
+10-1
+100
+101
+Reservoir concentration (M)
+10-2
+10-1
+100
+101
+Conductance (nS)
+Donnan equilibrium
+Phantom pair mean-field
+FIG. 4. Channel conductance in the pair-enhanced mean-field model. A. Conductance of a nanochannel (R = 1 nm,
+ξ = 7 nm, xT = 0.7 nm, Es = 10 kBT) as function of the reservoir concentration. The red line corresponds to the prediction of
+the phantom pair mean-field model (Eq. (56)) for T = 300 K, D = 10−9 m2/s and L = 100 nm. The Ohm’s law bulk prediction
+(cin = cout, blue line) and the Arrhenius model (cin = coute−Es/kBT , yellow line) are also represented for comparison. B.
+Conductance of a nanochannel with a weak surface charge Σ = 10−3 C/m2. We represented the predictions of the conventional
+Donnan equilibrium (Eq. (1), blue line) and of the phantom pair mean-field theory (equations (56) and (59), red line). Because
+interaction confinement results in a lower ion concentration in the channel, the usual formula Σ ∼ Rc∗/2, where c∗ is the reservoir
+concentration for which conductance levels off overestimates the surface charge by one order of magnitude, as indicated on the
+plot.
+Note that the concentration in free anions c−
+in and cations
+c+
+in are now distinct, so that µex is defined as a function of
+the average free ion concentration cf
+in = (c+
+in+c−
+in)/2. In a
+channel that is sufficiently long for local electroneutrality
+to hold,
+c+
+in − c−
+in + 2Σ/R = 0.
+(58)
+Imposing chemical equilibrium with the reservoir, we ob-
+tain a modified version of the Donnan result (Eq. (1)):
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+cin = cf
+in + cp
+in
+cf
+in =
+��
+cf
+oute−βµex(cf
+in)�2
++
+� 2Σ
+R
+�2,
+cp
+in = 2πR2xT (cf
+out)2,
+(59)
+with cf
+out given by Eq. (53).
+One can again obtain the channel’s conductance
+through Eq. (56), which we compare to the Donnan /
+Ohm’s law result in Fig. 4B. Importantly, the Donnan
+result predicts that conductance becomes independent of
+concentration for cout ∼ 2Σ/R (see Eq. (1)). In practice,
+this result is commonly used to estimate experimentally
+the surface charge as Σ ∼ Rc∗/2, where c∗ is the reser-
+voir concentration for which conductance levels off. In
+contrast, in the interaction confinement regime, we pre-
+dict that the transition occurs instead at cf
+in ∼ 2Σ/R –
+corresponding to a higher reservoir concentration, due to
+the self-energy barrier. In this case, Donnan’s prediction
+overestimates the surface charge by typically one order
+of magnitude, as shown in Fig. 4B.
+Finally, let us stress that we considered here a charge
+homogeneously distributed along the channel’s surface.
+This assumption is relevant in the case of conducting
+wall materials, such as systems where the charge is im-
+posed via a gating electrode connected to the chan-
+nel walls.
+This situation, however, may be different
+in experimentally-available devices, where the surface
+charge generally consists in localized charged groups and
+defects on the channel walls. In this case, the physics be-
+come more involved as ions may form bound pairs with
+the fixed surface charges.
+Some of these physics have
+been revealed by the exact computations of Shklovskii
+and coworkers9,22; a technically simpler approach to
+these physics using our pair-enhanced mean-field theory
+would be possible, but extends beyond the scope of the
+present work.
+V.
+DISCUSSION AND PERSPECTIVES
+We have determined the salt concentration inside a
+nanometric channel connected to reservoirs filled with
+electrolyte.
+In the case of a fully 1D geometry, corre-
+sponding to a nanotube of radius R ∼ 1nm, we devel-
+oped an exact field-theoretical solution that allowed us
+to compute channel concentration cin as function of the
+reservoir concentration cout. This solution clears up the
+ambiguities of pre-existing mean-field theories, and con-
+tradicts the naive expectation cin = cout. In particular,
+the concentration inside the nanochannel is found to be
+always lower than in the bulk, as the confinement of elec-
+trostatic interactions creates an energy barrier for ions to
+
+12
+enter the channel.
+Yet, we found that cin is in fact higher than the predic-
+tion of the mean-field Debye-H¨uckel theory, as ion pairing
+is counterbalances to some extent the energy cost of in-
+teraction confinement. Such strong ion-ion correlations
+cannot be directly accounted for in a mean-field theory,
+and the filling transition that emerges in Debye-H¨uckel
+theory appears to be an artefact of linearization. To over-
+come this issue, one can add Bjerrum pairs as a separate
+chemical species within the Debye-H¨uckel model. Care-
+fully accounting for the statistical formation of unbound
+phantom pairs, we obtain a modified mean-field theory
+that reproduces the result of the exact computation with
+nearly-perfect accuracy, and that can be extended to ac-
+count for a non-zero surface charge on the channel wall.
+Despite the concurring results, the two original for-
+malisms developed in this work serve distinct purposes.
+The field-theoretical solution plays the role of a touch-
+stone model, owing to its exact treatment of all many-
+body interactions. Modeling electrolytes is a notoriously
+hard problem in statistical physics, and simplified models
+often lack a lack a reference solution for benchmarking
+their approximations. This difficulty is lifted in the 1D
+geometry: thanks to the existence of the exact solution,
+we have been able to build a quantitatively precise mean-
+field model, adding step-by-step the qualitative ingredi-
+ents necessary to reproduce the exact result.
+Moreover, the field theory formalism gives access to the
+entire statistics of the system, including finite-size effects
+which elude any mean-field treatment.
+The latter are
+expected to be relevant in many experimental situations,
+as a substantial amount of current works focuses on short
+pores, where the length of the channel is comparable to
+its radius. For instance, one can expect shorter channels
+to deviate from electroneutrality2 – something entirely
+impossible in the limit of infinitely long channels.
+On the other hand, our modified mean-field formalism
+has the advantage of mathematical simplicity, allowing
+for convenient physical interpretations. The simple dis-
+tinction between free ions and Bjerrum pairs can be used
+to straightforwardly estimate the channel’s conductance.
+The influence of ion-ion correlations on conductivity is
+of particular importance as conductance measurements
+underpin many nanofluidic experiments. In contrast, the
+exact solution does not provide any such insight on trans-
+port properties, as it is limited to thermal equilibrium.
+Furthermore, the mean-field model may easily be
+adapted to other geometries, whereas an exact treatment
+is only possible in the strictly 1D case. Extensions of our
+results to 2D nanochannels would be of significant in-
+terest. In particular, 2D nanochannels can be made out
+of various materials with different electronic properties,
+which directly impact the confined ionic interactions6.
+Therefore, 2D nanochannels could serve as a platform
+for exploring the impact of wall metallicity on the ion
+filling problem.
+Both our exact and mean-field solutions can be ex-
+pected to fail at very high concentrations. Indeed, our
+work relies on a simplified picture of electrolytes, where
+all steric effects are discarded. We considered point-like
+ions with no short-distance repulsion; therefore, no effect
+like saturation or layering can be accounted for.
+Sim-
+ilarly, we neglected any interaction with the solvent –
+for example, we did not consider the decrement in rela-
+tive permittivity at high salt concentrations34. However,
+since all electrostatic interactions are screened in the
+limit of high concentrations, such considerations should
+not impact the conclusions of the present work: partic-
+ularly, we would still expect that cin = cout at high con-
+centration.
+Lastly, let us briefly recall our results for the ion filling
+problem. In channels larger than a few nanometers, the
+conventional mean-field picture is valid, so that in ab-
+sence of any surface charge the salt concentration inside
+the channel equals that of the reservoirs: cin = cout. For
+nanometre-scale confinement and low concentrations, in-
+teraction confinement amounts to a finite energy barrier
+for ions to enter the channel: cin = coute−Es/kBT . As
+concentration increases, more ions are able to overcome
+the barrier by forming Bjerrum pairs, neutralizing the
+electrostatic cost of confinement, at the price of entropy:
+cin ∝ c2
+out. Only at high concentrations can one recover
+the intuitive estimate cin = cout, as intense screening can-
+cels out all electrostatic interactions. Overall, interaction
+confinement has a significant impact on the properties
+of nanofluidic systems, and the assumption cin = cout
+should be questioned any time the system’s size reaches
+the nanometre scale.
+ACKNOWLEDGMENTS
+N.K. acknowledges support from a Humboldt fellow-
+ship.
+L.B. acknowledges funding from the EU H2020
+Framework Programme/ERC Advanced Grant agree-
+ment number 785911-Shadoks.
+The Flatiron Institute
+is a division of the Simons Foundation.
+DATA AVAILABILITY STATEMENT
+The data that support the findings of this study are
+available from the corresponding author upon reasonable
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+(2013).
+

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+arXiv:2301.00262v1  [math.PR]  31 Dec 2022
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+KOHEI SUZUKI
+Abstract. In this article, we show 1-Bakry–Émery lower Ricci curvature
+bound BE1(0, ∞) of a Dirichlet form on the configuration space whose in-
+variant measure is sineβ ensemble for any β > 0.
+As a particular case of
+β = 2, our result proves BE1(0, ∞) for a Dirichlet form related to the unlablled
+Dyson Brownian motion. We prove furthermore several functional inequalities
+including the integral Bochner inequality, the local Poincaré and the local log-
+Sobolev inequalities as well as the log-Harnack and the dimension-free Harnack
+inequalities, the Lipschitz contraction property and the L∞-to-Lipschitz regu-
+larisation property of the semigroup with the L2-transportation-type extended
+distance. At the end of the article, we provide a sufficient condition for the
+synthetic lower Ricci curvature bound in the case of general invariant measures
+beyond sineβ.
+Contents
+1.
+Introduction
+1
+2.
+Notation and Preliminaries
+4
+3.
+Curvature bound for finite-particle systems
+11
+4.
+Curvature bound for infinite-particle systems
+15
+5.
+Dimension-free/log Harnack inequalities and Lipschitz regularisation
+25
+6.
+Generalisation
+27
+Appendix A.
+29
+References
+31
+1. Introduction
+The objective of this article is to reveal the structure of lower curvature bound be-
+hind an infinite particle system of diffusions with logarithmic interactions. Such an
+interacting particle system is realised as a continuous-time strong Markov process
+having continuous paths (called a diffusion process) taking values in the configu-
+ration space Υ = Υ(R) over R and having the sineβ (β > 0) ensemble µ as an
+Date: 31/12/2022.
+Key words and phrases.
+Dyson Brownian motion, sine beta ensemble, Ricci curvature bound.
+Department of Mathematical Science, Durham University
+E-mail: [email protected] .
+1
+
+2
+K. SUZUKI
+invariant measure. We study a corresponding Dirichlet form (EΥ,µ, D(EΥ,µ)) with
+square field ΓΥ (Prop. 4.15) whose invariant measure is sineβ ensemble µ on the
+configuration space Υ. The case of β = 2 is particularly related to the diffusion
+process called (unlabelled) Dyson Brownian motion (cf. [Spo87, KT10, Osa13]). The
+labelled interacting diffusions can be phrased formally as the following infinitely
+many stochastic differential equation with logarithmic interaction (see [Tsa16] for a
+rigorous construction):
+dXk
+t = β
+2 lim
+r→∞
+�
+i̸=k:|Xk
+t −Xi
+t|<r
+1
+Xk
+t − Xi
+t
+dt + dBk
+t ,
+k ∈ N ,
+whereby {Bk
+· }k∈N are infinitely many independent Brownian motions on R.
+The main result of this article is to show that the aforementioned Dirichlet
+form (EΥ,µ, D(EΥ,µ)) satisfies the non-negative lower Ricci curvature bound BE1(0, ∞)
+in the sense of Bakry–Émery.
+We prove, furthermore, several related functional
+inequalities including the integral Bochner inequality with respect to the integral
+Γ2-operator (ΓΥ,µ
+2
+, D(ΓΥ,µ
+2
+)) (see (4.22)), the local Poincaré, the local log-Sobolev
+inequalities as well as the dimension-free Harnack inequality, the log-Harnack in-
+equality, the Lipschitz contraction and the L∞-to-Lipschitz regularisation property
+with respect to the L2-transportation-type extended distance ¯dΥ on Υ (see (2.13)).
+The main results are summarised in the following.
+Theorem. Let β > 0 and µ be the sineβ ensemble. The form (EΥ,µ, D(EΥ,µ)) satis-
+fies the following:
+• (Thm. 4.17) 1-Bakry–Émery estimate BE1(0, ∞): for u ∈ D(EΥ,µ), t > 0,
+ΓΥ�
+T Υ,µ
+t
+u
+� 1
+2 ≤ T Υ,µ
+t
+�
+ΓΥ(u)
+1
+2�
+;
+• (Cor. 4.18) Integral Bochner inequality: for every (u, ϕ) ∈ D(ΓΥ,µ
+2
+)
+ΓΥ,µ
+2
+(u, ϕ) ≥ 0 ;
+• (Cor. 4.18) Local Poincaré inequality: for u ∈ D(EΥ,µ), t > 0,
+T Υ,µ
+t
+u2 − (T Υ,µ
+t
+u)2 ≤ 2tT Υ,µ
+t
+ΓΥ(u) ,
+T Υ,µ
+t
+u2 − (T Υ,µ
+t
+u)2 ≥ 2tΓΥ(T Υ,µ
+t
+u) ;
+• (Cor. 4.18) Local log-Sobolev inequality: for non-negative u ∈ D(EΥ,µ), t > 0,
+T Υ,µ
+t
+u log u − T Υ,µ
+t
+u log T Υ,µ
+t
+u ≤ tT Υ,µ
+t
+�ΓΥ(u)
+u
+�
+,
+T Υ,µ
+t
+u log u − T Υ,µ
+t
+u log T Υ,µ
+t
+u ≥ tΓΥ(T Υ,µ
+t
+u)
+T Υ,µ
+t
+u
+.
+• (Thm. 5.1) Log-Harnack inequality: for every non-negative u ∈ L∞(Υ, µ),
+t > 0, there exists Ω ⊂ Υ so that µ(Ω) = 1 and
+T Υ,µ
+t
+(log u)(γ) ≤ log(T Υ,µ
+t
+u)(η) + ¯dΥ(γ, η)2 ,
+∀γ, η ∈ Ω ;
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+3
+• (Thm. 5.1) Dimension-free Harnack inequality: for every non-negative u ∈
+L∞(Υ, µ), t > 0 and α > 1 there exists Ω ⊂ Υ so that µ(Ω) = 1 and
+(T Υ,µ
+t
+u)α(γ) ≤ T Υ,µ
+t
+uα(η) exp
+�
+α
+2(α − 1)
+¯dΥ(γ, η)2�
+,
+∀γ, η ∈ Ω ;
+• (Thm. 5.1) Lipschitz contraction: for every u ∈ Lip(¯dΥ, µ) and t > 0
+T Υ,µ
+t
+u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ
+t
+u
+and the following Lipschitz contraction holds:
+Lip¯dΥ( ˜T Υ,µ
+t
+u) ≤ Lip¯dΥ(u) ;
+• (Thm. 5.1) L∞(Υ, µ)-to-Lip(¯dΥ, µ) regularisation by semigroup: For u ∈
+L∞(µ) and t > 0,
+T Υ,µ
+t
+u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ
+t
+u
+and the following estimate holds:
+Lip¯dΥ( ˜T Υ,µ
+t
+u) ≤
+1
+√
+2t∥u∥L∞(µ)
+∀u ∈ Lipb(¯dΥ, µ) .
+At the end of this article, Theorem will be extended to general point processes
+beyond the sineβ ensemble, see Thm. 6.2.
+Comparison with Literature. To the best knowledge of the author, this is the
+first article addressing lower Ricci curvature bound in the setting of interacting and
+infinite particle systems of diffusion processes. In the case of non-interacting case
+where the invariant measure is the Poisson measure, it has been studied in [EH15]
+in the case of Riemannian manifolds and in [DS22] in the case of general diffusion
+spaces. In the case of finite particle systems, a variable Ricci curvature bound has
+been addressed in [VG20] in the case of Coulomb-type potentials where the curvature
+bound depends on the number of particles.
+Outline of the article. After preparing the notation and the preliminaries in
+Section 2, we discuss in Section 3 the synthetic lower Ricci curvature bound for
+Dirichlet forms (EΥ(Br),µη
+r, D(EΥ(Br),µη
+r)) on the configuration space Υ(Br) over the
+closed metric ball Br with radius r > 0 centred at 0, whose invariant measure is
+the projected regular conditional probability µη
+r on Υ(Br) conditioned at η on the
+compliment Bc
+r ⊂ R. The key point of the proof is that the logarithm of the Radon–
+Nikodým density Ψη
+r := − log(dµη
+r/ dπmr) with respect the Poisson measure πmr on
+Υ(Br) with the intensity mr being the Lebesgue measure restricted on Br is geodesi-
+cally convex in (Υ(Br), ¯dΥ) due to the following DLR (Dobrushin–Lanford–Ruelle)
+equation (⋆) proven in [DHLM20, Thm.1.1] with the number-rigidity ([Gho15] for
+sine2; [NR18] and [DHLM20] for sineβ): for µ-a.e. η, there exists a unique k =
+
+4
+K. SUZUKI
+k(η) ∈ N0 so that µη
+r(Υl(Br)) > 0 if and only if l = k where Υl(Br) := {γ ∈
+Υ(Br) : γ(Br) = l}, and for γ = �k
+i=1 δxi ∈ Υ(Br)
+dµη
+r = 1
+Zη
+r e−Ψk,η
+r
+dm⊙k
+r
+,
+(⋆)
+Ψk,η
+r (γ) := − log
+� k
+�
+i<j
+|xi − xj|β
+k
+�
+i=1
+lim
+R→∞
+�
+y∈ηBcr ,|y|≤R
+���1 − xi
+y
+���
+β
+�
+,
+where m⊙k
+r
+is the k-symmetric product measure of the Lebesgue measure mr re-
+stricted on Br ⊂ R and Zη
+r is the normalising constant.
+In Section 4, we prove BE1(0, ∞) for (EΥ,µ, D(EΥ,µ)) in the following steps: we
+first construct the truncated form (EΥ,µ
+r
+, D(EΥ,µ
+r
+)) on Υ whose gradient operator is
+truncated up to configurations on Br (Prop. 4.7). We then identify it with the super-
+position Dirichlet form ( ¯EΥ,µ
+r
+, D( ¯EΥ,µ
+r
+)) lifted from (EΥ(Br),µη
+r, D(EΥ(Br),µη
+r)) with re-
+spect to the conditioning η (Thm. 4.11). By this identification, we can lift BE(0, ∞)
+from the form (EΥ(Br),µη
+r, D(EΥ(Br),µη
+r)) onto the truncated form (EΥ,µ
+r
+, D(EΥ,µ
+r
+)).
+Showing the monotonicity of the form (EΥ,µ
+r
+, D(EΥ,µ
+r
+)) with respect to r and pass-
+ing to the limit r → ∞, we prove BE1(0, ∞) for the limit form (EΥ,µ, D(EΥ,µ))
+(Thm. 4.17). As a consequence of BE1(0, ∞), we obtain local Poincaré and local
+log-Sobolev inequalities (Cor. 4.18).
+In Section 5, we prove the log-Harnack inequality, the dimension-free Harnack
+inequality, the Lipschitz contraction and L∞(µ)-to-Lip(¯dΥ) properties (Thm. 5.1).
+Our proof strategy is to lift the corresponding functional inequalities from the space
+of finitely many configurations.
+In Section 6, we extend Theorem to the case of general point processes beyond
+sineβ (Thm. 6.2).
+Acknowledgement. A large part of the current work has been completed while
+the author was at Bielefeld University. He gratefully acknowledges funding by the
+Alexander von Humboldt Stiftung to support his stay.
+Data Availability Statement. Data sharing not applicable to this article as no
+datasets were generated or analysed during the current study.
+2. Notation and Preliminaries
+2.1. Numbers, Tensors, Function Spaces. We write N := {1, 2, 3, . . .}, N0 =
+{0, 1, 2, . . .}, N := N∪{+∞} and N0 := N0 ∪{+∞}. The uppercase letter N is used
+for N ∈ N0, while the lowercase letter n is used for n ∈ N0. We shall adhere to the
+following conventions:
+• the superscript □×N (the subscript □×N) denotes (N-fold) product objects;
+• the superscript □⊗N (the subscript □⊗N) denotes (N-fold) tensor objects;
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+5
+• the superscript □⊙N (the subscript □⊙N) denotes (N-fold) symmetric tensor
+objects;
+Let (X, τ) be a topological space with σ-finite Borel measure ν. We use the following
+symbols:
+(a) Lp(ν) (1 ≤ p ≤ ∞) for the space of ν-equivalence classes of functions u
+with |u|p ν-integrable when 1 ≤ p < ∞, and with u ν-essentially bounded
+when p = ∞. The Lp(ν)-norm is denoted by ∥u∥p
+Lp(ν) := ∥u∥p
+p :=
+�
+X |u|p dν
+for 1 ≤ p < ∞, and ∥u∥L∞(ν) := ∥u∥∞ = esssupX u. In the case of p = 2, the
+inner-product is denoted by (u, v)L2(ν) :=
+�
+X uv dν;
+(b) Lp
+s(ν⊗n) := {u ∈ Lp(ν⊗n) : u is symmetric} where u is said to be symmetric
+if and only if u(x1, . . . , xk) = u(xσ(1), . . . , xσ(k)) for any element σ ∈ S(k) in
+the k-symmetric group.
+(c) Cb(X) for the space of τ-continuous bounded functions on X; if X is locally
+compact, C0(X) denotes the space of τ-continuous and compactly supported
+functions on X; C∞
+0 (R) for the space of compactly supported smooth func-
+tions on R;
+(d) We write 1A for the indicator function on A, i.e., 1A(x) = 1 if x ∈ A, and
+1A(x) = 0 otherwise.
+2.2. Dirichlet forms. We refer the reader to [MR90, BH91] for this subsection.
+Throughout this paper, a Hilbert space always means a Hilbert space with inner
+product (·, ·)H taking value in R.
+Dirichlet forms.
+Given a bilinear form (Q, D(Q)) on a Hilbert space H, we write
+Q(u) := Q(u, u) ,
+Qα(u, v) := Q(u, v) + α(u, v)H , α > 0 .
+Let (X, Σ, ν) be a σ-finite measure space. A symmetric Dirichlet form on L2(ν) is
+a non-negative definite densely defined closed symmetric bilinear form (Q, D(Q))
+on L2(ν) satisfying the Markov property
+u0 := 0 ∨ u ∧ 1 ∈ D(Q)
+and
+Q(u0) ≤ Q(u) ,
+u ∈ D(Q) .
+Throughout this article, Dirichlet form always means symmetric Dirichlet form. If
+not otherwise stated, D(Q) is always regarded as a Hilbert space with norm
+∥ · ∥D(Q) := Q1( · )1/2 :=
+�
+Q( · ) + ∥ · ∥2
+L2(ν) .
+In order to distinguish Dirichlet forms defined in different base spaces with different
+reference measures, we often use the notation QX,ν to specify the base space X and
+the reference measure ν.
+Square field.
+A Dirichlet form (Q, D(Q)) admits square field Γ if there exists a
+dense subspace H ⊂ D(Q) ∩ L∞(ν) having the following property: for any u ∈ H,
+
+6
+K. SUZUKI
+there exists v ∈ L1(ν) so that
+2Q(uh, u) − Q(h, u2) =
+�
+X
+hv dν
+∀h ∈ D(Q) ∩ L∞(ν) .
+Such v is denoted by Γ(u). The square field Γ can be uniquely extended as an
+operator on D(Q) × D(Q) → L1(ν) ([BH91, Thm. I.4.1.3]).
+Semigroups and generators.
+We refer the reader to [MR90, Chap. I, Sec. 2] for
+the following contents.
+Let (Q, D(Q)) be a symmetric closed form on a Hilbert
+space H. The infinitesimal generator (A, D(A)) corresponding to (Q, D(Q)) is the
+unique densely defined closed operator on H satisfying the following integration-by-
+parts formula:
+−(u, Av)H = Q(u, v)
+∀u ∈ D(Q), v ∈ D(A) .
+The resolvent operator {Rα}α≥0 is the unique bounded linear operator on H satis-
+fying
+Qα(Rαu, v) = (u, v)H
+∀u ∈ H
+v ∈ D(Q) .
+The semigroup {Tt}t≥0 is the unique bounded linear operator on H satisfying
+Gαu =
+� ∞
+0
+e−αtTtu dt
+u ∈ H .
+Locality.
+Let (Q, D(Q)) be a Dirihclet form on L2(ν). It is called local ([BH91,
+Def. 5.1.2]) if for any F, G ∈ C∞
+c (R) and any u ∈ D(Q),
+supp[F] ∩ supp[G] = ∅ =⇒ Q(F0 ◦ u, G0 ◦ u) = 0 ,
+where F0(x) := F(x) − F(0) and G0(x) := G(x) − G(0).
+2.3. Metric space. Let X be any non-empty set. A function d: X ×2 → [0, ∞] is
+an extended distance if it is symmetric and satisfying the triangle inequality, and it
+does not vanish outside the diagonal in X ×2, i.e. d(x, y) = 0 iff x = y; a distance
+if it is finite. Let x0 ∈ X and r ∈ [0, ∞). We write Br(x0) := {dx0 ≤ r}, where
+dx0 := d(x0, ·).
+Lipschitz algebras.
+A function f : X → R is d-Lipschitz if there exists a con-
+stant L > 0 so that
+��u(x) − u(y)
+�� ≤ L d(x, y) ,
+x, y ∈ X .
+(2.1)
+The smallest constant L so that (2.1) holds is the (global) Lipschitz constant of u,
+denoted by Lipd(u). For any non-empty A ⊂ X we write Lip(A, d), resp. Lipb(A, d)
+for the family of all finite, resp. bounded, d-Lipschitz functions on A. For simplic-
+ity of notation, further let Lip(d) := Lip(X, d), resp. Lipb(d) := Lipb(X, d). Set also
+Lip1(d) := {u ∈ Lip(d) : Lipd(u) ≤ 1} and Lip1
+b(d) := Lip1(d) ∩ Lipb(d). For a given
+measure ν, we set
+Lip(d, ν) := {u ∈ Lip(d) : u is ν-measurable} ,
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+7
+as well as Lipb(d, ν) and Lip1
+b(d, ν) denoting the corresponding subspaces of ν-
+measurable functions respectively.
+Geodesical convexity.
+A metric space (X, d) is called a geodesic space if for any
+x0, x1 ∈ X there exists a constant speed geodesic ω : [0, 1] → X connecting x0 and x1:
+ω0 = x0 ,
+ω1 = x1 ,
+d(ωt, ωs) = |t − s|d(ω0, ω1)
+∀t, s ∈ [0, 1] .
+For a function U : X → R ∪ {+∞}, define D(U) := {x ∈ X : U(x) < ∞}. We say
+that U is K-geodesically convex for K ∈ R if for any x0, x1 ∈ D(U) there exists a
+constant speed geodesic ω : [0, 1] → X with ω0 = x0 and ω1 = x1 and
+U(ωt) ≤ (1 − t)U(ω0) + tU(ω1) − K
+2 t(1 − t)d2(ω0, ω1)
+∀t ∈ [0, 1] .
+When K = 0, we say that U is geodesically convex.
+2.4. Cheeger energies. A complete separable geodesic metric space (X, d) equipped
+with fully supported Radon measure ν with finite total mass ν(X) < ∞ is called a
+metric measure space in this article. Let (X, d, ν) be a metric measure space. For
+u ∈ Lip(d), the slope |Ddu|(x) is defined as
+|Ddu|(x) :=
+
+
+
+
+
+lim sup
+y→x
+|u(x) − u(y)|
+d(x, y)
+if x is not isolated;
+0
+otherwise .
+The slope is universally measurable, see [AGS14a, Lem. 2.6]. The Cheeger energy
+Chd,ν : L2(ν) → R ∪ {+∞} is defined as the L2(ν)-lower semi-continuous envelope
+of
+�
+X |Ddu|2 dν:
+Chd,ν(u) := inf
+�
+lim inf
+n→∞
+�
+X
+|Ddun|2 dν : un ∈ Lip(d) ∩ L2(ν)
+L2
+−→ u
+�
+.
+The domain is denoted by W 1,2(X, d, ν) := {u ∈ L2(ν) : Chd,ν(u) < ∞}.
+The
+Cheeger energy Chd,ν can be expressed by the following integration, see [AGS14a,
+Thm. 4.5] : there exists a measurable function |∇u|∗ ∈ L2(ν) so that |∇u|∗ ≤ |Ddu|
+ν-a.e. for every u ∈ Lip(d) and
+Chd,ν(u) =
+�
+X
+|∇u|2
+∗ dν
+∀u ∈ W 1,2(X, d, ν) ,
+where |∇u|∗ is called minimal relaxed slope.
+2.5. Riemannian Curvature-dimension condition. Let (X, d, ν) be a metric
+measure space. The following definition is an equivalent characterisation of RCD(K, ∞)
+by [AGS15, Cor. 4.18]. We say that (X, d, ν) satisfies the Riemannian Curvature-
+Dimension Condition RCD(K, ∞) for K ∈ R if
+(i) Chd,ν is quadratic, i.e., Chd,ν(u + v) + Chd,ν(u − v) = 2Chd,ν(u) + 2Chd,ν(v);
+(ii) Sobolev-to-Lipschitz property holds, i.e., every u ∈ W 1,2(X, d, ν) with |∇u|∗ ≤
+1 has a d-Lipschitz ν-representative ˜u satisfying Lip(˜u) ≤ 1;
+
+8
+K. SUZUKI
+(iii) Chd,ν satisfies BE2(K, ∞), i.e., |∇Ttu|2
+∗ ≤ e−2KtTt|∇u|2
+∗ for every u ∈ W 1,2(X, d, ν)
+and t > 0.
+In this case, the Cheeger energy Chd,ν is a local Dirichlet form ([AGS14b, §4.3]). We
+note that, while [AGS15, Cor. 4.18] is stated in terms of the minimal weak upper
+gradient denoted by |∇ · |w, it is identical to the minimal relaxed slope |∇ · |∗ due to
+[AGS14a, Thm. 6.2].
+2.6. Configuration spaces. A configuration on a locally compact Polish space X
+is any N0-valued Radon measure γ on X, which can be expressed by γ = �N
+ii δxi
+for N ∈ ¯N, where δx denotes the Dirac measure at x, i.e., δx(A) = 1 if and only if
+x ∈ A. The configuration space Υ = Υ(X) is the space of all configurations over X.
+The space Υ is equipped with the vague topology, i.e., the topology generated by
+the duality of the space C0(X) of continuous functions with compact support. We
+write the restriction γA := γ ⇂A for a Polish subspace A ⊂ X and the corresponding
+restriction map is denoted by
+prA : Υ −→ Υ(A): γ �−→ γA .
+(2.2)
+The N-particle configuration space is denoted by
+ΥN := {γ ∈ Υ : γ(X) = N} ,
+N ∈ N0 .
+Let Sk be the k-symmetric group. It can be readily seen that the k-particle config-
+uration space Υk is isomorphic to the quotient space X×k/Sk:
+Υk ∼= X⊙k := X×k/Sk ,
+k ∈ N0 .
+(2.3)
+The associated projection map from X×k to the quotient space X×k/Sk is denoted
+by Pr. For η ∈ Υ and r > 0, we set
+Υη
+r := {γ ∈ Υ : γBcr = ηBcr} .
+(2.4)
+Conditional probability.
+Let µ be a Borel probability measure on Υ. Let
+µ(· | prBcr(·) = ηBcr)
+denote the regular conditional probability of µ conditioned at η ∈ Υ with respect
+to the σ-field generated by the projection map γ ∈ Υ �→ prr(γ) = γBr ∈ Υ(Br) (see
+e.g., [DS21a, Def. 3.32] for the precise definition). Let µη
+r be the probability measure
+on Υ(Br) defined as
+µη
+r := (prr)#µ(· | prBcr(·) = ηBcr) ,
+(2.5)
+and its restriction on Υk(Br) is denoted by µk,η
+r
+:= µη
+r|Υk(Br).
+Note: The conditional probability µ(· | prBcr(·) = ηBcr) is a probability measure on
+the whole space Υ whose support is Υη
+r = {γ ∈ Υ : γBcr = ηBcr}. We may project the
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+9
+conditional probability to the probability measure µη
+r on Υ(Br) as in (2.5) without
+loss of information in the sense that
+prr : Υη
+r → Υ(Br) is a bi-measure-preserving bijection .
+(2.6)
+Namely, the projection map prr is bijective with the inverse map pr−1
+r
+defined
+as pr−1
+r (γ) := γ + η, and both prr and pr−1
+r
+are measure-preserving between the
+two measures µ(· | prBcr(·) = ηBcr) and µη
+r.
+For a measurable function u: Υ → R, r > 0 and for η ∈ Υ, we set
+uη
+r(γ) := u(γ + ηBcr)
+γ ∈ Υ(Br) .
+(2.7)
+By the property of the conditional probability, it is straightforward to see that for
+any u ∈ L1(µ),
+�
+Υ
+u dµ =
+�
+Υ
+��
+Υ(Br)
+uη
+r dµη
+r
+�
+dµ(η) .
+(2.8)
+See, e.g., [DS21a, Prop. 3.44]. For a measurable set Ω ⊂ Υ, define a section Ωη
+r ⊂
+Υ(Br) at η ∈ Υ on Bc
+r by
+Ωη
+r := {γ ∈ Υ(Br) : γ + ηBcr ∈ Ω} .
+(2.9)
+By applying the disintegration formula (2.8) to u = 1Ω, we obtain
+µ(Ω) =
+�
+Υ
+µη
+r(Ωη
+r) dµ(η) .
+(2.10)
+Poisson measure.
+Let (X, τ, ν) be a locally compact Polish space with Radon
+measure ν satisfying ν(X) < ∞. The Poisson measure πν on Υ(X) with intensity ν
+is defined in terms of the symmetric tensor measure ν⊙ as follows:
+πν(·) := e−ν(X)
+∞
+�
+k=1
+ν⊙k�
+· ∩ Υk(X)
+�
+= e−ν(X)
+∞
+�
+k=1
+1
+k!(Pr)#ν⊗k�
+· ∩ Υk(X)
+�
+.
+(2.11)
+L2-transportation distance.
+Let (X, d) be a locally compact complete separable
+metric space.
+For i = 1, 2 let proji : X×2 → X denote the projection to the ith
+coordinate for i = 1, 2. For γ, η ∈ Υ, let Cpl(γ, η) be the set of all couplings of γ
+and η, i.e.,
+Cpl(γ, η) := {q ∈ M (X
+×2): (proj1)♯q = γ , (proj2)♯q = η} .
+Here M (X ×2) denotes the space of all Radon measures on X ×2. The L2-transportation
+extended distance on Υ(X) is
+dΥ(γ, η) :=
+inf
+q∈Cpl(γ,η)
+��
+X×2 d2(x, y) dq(x, y)
+�1/2
+,
+inf ∅ = +∞ .
+(2.12)
+We refer the readers to [DS21a, §4.2, p.52] for details regarding the L2-transportation
+extended distance dΥ.
+It is important to note that dΥ is an extended distance,
+
+10
+K. SUZUKI
+attaining the value +∞ and dΥ is lower semi-continuous with respect to the product
+vague topology τ ×2
+v
+but never τ ×2
+v -continuous.
+We introduce a variant of the L2-transportation extended distance, called L2-
+transportation-type extended distance ¯dΥ defined as
+¯dΥ(γ, η) :=
+
+
+
+dΥ(γ, η)
+if γBcr = ηBcr for some r > 0 ,
++∞
+otherwise .
+(2.13)
+By definition, dΥ ≤ ¯dΥ on Υ, and dΥ = ¯dΥ on Υ(Br) for any r > 0. In particular,
+we have
+Lip(Υ, dΥ) ⊂ Lip(Υ, ¯dΥ) ,
+Lip¯dΥ(u) ≤ LipdΥ(u) ,
+u ∈ Lip(Υ, dΥ) .
+(2.14)
+It can be readily seen that
+¯dΥ(γ, η) < ∞
+⇐⇒
+γBcr = ηBcr , γ(Br) = η(Br)
+for some r > 0 .
+(2.15)
+When we work with the configuration space Υ(Rn) over the n-dimensional Eu-
+clidean space Rn or over any Polish subset in Rn, we always choose the Euclidean
+distance d(x, y) = |x − y| and the L2-transportation distance dΥ and ¯dΥ associated
+with d.
+2.7. sineβ ensemble. Let β > 0 and CβEk be the circular β ensemble on the k-
+particle configuration space, i.e., it is the probability measure Pk,β on the space Υk(S1)
+over the unit circle S1 ⊂ C defined as
+dPk,β :=
+1
+Zk,β
+�
+1≤j<l≤k
+��eiθj − eiθl��β dθ1
+2π · · · dθk
+2π ,
+where the normalisation constant Zk,β is given in terms of Gamma function Γ:
+Zk,β := Γ( 1
+2βk + 1)
+Γ( 1
+2βk + 1)k .
+According to [KS09, Def. 1.6], the circular β ensemble CβE is defined as the limit
+probability measure Pβ whose Laplace transform is determined as
+�
+exp
+�
+−
+�
+x∈γ
+f(x)
+�
+dPβ(γ) = lim
+k→∞
+�
+exp
+�
+−
+k
+�
+i=1
+f(kθi)
+�
+dPk,β(θ1, . . . , θk) ,
+for all f ∈ C0(R). In [VV09] a probability measure µβ on Υ(R) called sine β ensem-
+ble has been constructed by a limit of Gaussian β-ensemble. These two measures Pβ
+and µβ turned out to be identical each other by the work of [Nak14]. Throughout the
+rest of the article, we use the symbol µ = µβ to denote sineβ ensemble (equivalently,
+circular β ensemble) and we do not specify the inverse temperature β as there is no
+particular role played by a special β.
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+11
+Number-rigidity.
+A Borel probability µ on Υ = Υ(Rn) is said to be number
+rigid (in short: (R)) if for any bounded domain E ⊂ Rn, there exists Ω ⊂ Υ so that
+µ(Ω) = 1 and, for any γ, η ∈ Ω
+γEc = ηEc implies γE = ηE .
+(R)
+Namely, the configuration outside E determines the number of particle inside E. The
+number-rigidity has been proven in [Gho15] for the sine2 ensemble and in [NR18],
+[DHLM20] for the sineβ ensemble for general β > 0.
+3. Curvature bound for finite-particle systems
+In this section, we study Dirichlet forms on the configuration space Υ(Br) over
+metric balls Br ⊂ R. We denoted by m and mr the Lebesgue measure on R and
+its restriction on the metric ball Br := [−r, r] respectively, and take the Euclidean
+distance d(x, y) := |x − y| for x, y ∈ Br.
+3.1. Construction of Dirichlet forms on Υk(Br). Let W 1,2
+s (m⊗k
+r ) be the space
+of m⊗k
+r -classes of (1, 2)-Sobolev and symmetric functions on the product space B×k
+r ,
+i.e.,
+W 1,2
+s (m⊗k
+r ) :=
+�
+u ∈ L2
+s(m⊗k
+r ) :
+�
+B×k
+r
+|∇⊗ku|2 dm⊗k
+r
+< ∞
+�
+,
+where ∇⊗k denotes the weak derivative on R×k: ∇⊗ku := (∂1u, . . . , ∂ku). The space
+W 1,2
+s (m⊗k
+r ) consisting of symmetric functions, the projection Pr : B×k
+r
+→ Υk(Br) ∼=
+B×k
+r /Sk naturally acts on W 1,2
+s (m⊗k
+r ) and the resulting quotient space is denoted by
+W 1,2(m⊙k
+r ), which is the (1, 2)-Sobolev space on Υk(Br):
+W 1,2(m⊙k
+r ) :=
+�
+u ∈ L2(m⊙k
+r ) :
+�
+Υk(Br)
+|∇⊙ku|2 dm⊙k
+r
+< ∞
+�
+,
+where ∇⊙k is the quotient operator of the weak gradient operator ∇⊗k through the
+projection Pr and m⊙k
+r
+is the symmetric product measure defined as
+m⊙k
+r
+:= 1
+k!(Pr)#m⊗k
+r
+.
+For 0 < r < R < ∞, k ∈ N0 and η ∈ Υ(Bc
+r), we introduce the following finite
+Borel measure on Υk(Br): for γ = �k
+i=1 δxi
+dµk,η
+r,R(γ) := e−Ψk,η
+r,R(γ) dm⊙k
+r (γ) ,
+(3.1)
+Ψk,η
+r,R(γ) := − log
+� k
+�
+i<j
+|xi − xj|β
+k
+�
+i=1
+�
+y∈ηBcr ,|y|≤R
+���1 − xi
+y
+���
+β
+�
+.
+The corresponding weighted Sobolev norm is denoted by
+EΥ(Br),µk,η
+r,R(u) :=
+�
+Υk(Br)
+|∇⊙ku|2 dµk,η
+r,R ,
+u ∈ Lipb(Υk(Br), dΥ) ,
+(3.2)
+
+12
+K. SUZUKI
+where we note that as Lip(Υk, dΥ) ⊂ W 1,2(m⊙k
+r ) and |∇⊙ku| ≤ LipdΥ(u) due to the
+Rademacher theorem descendent from the one in the product Sobolev space W 1,2(m⊗k
+r )
+through the quotient, the expression |∇⊙ku| and its integral against the probability
+measure µk,η
+r,R make sense for u ∈ Lipb(Υk(Br), dΥ).
+Proposition 3.1. The form (3.2) is well-defined and closable. The closure is a local
+Dirichlet form on L2(µk,η
+r,R) and its domain is denoted by D
+�
+EΥ(Br),µk,η
+r,R).
+Proof. The well-definedness follows from the following inequality:
+�
+Υk(Br)
+|∇⊙ku|2 dµk,η
+r,R ≤
+���e−Ψk,η
+r,R
+���
+L∞(Υk(Br),µk,η
+r,R)
+�
+Υk(Br)
+|∇⊙ku|2 dm⊙k < ∞ .
+(3.3)
+The closability of EΥ(Br),µk,η
+r,R descends from the closability of the corresponding
+Dirichlet form on the product space B×k
+r
+defined on the space of symmetric d×k-
+Lipschitz functions:
+EB×k
+r
+,µk,η
+r,R :=
+�
+B×k
+r
+|∇⊗ku|2e−Ψk,η
+r,R dm⊗k
+r
+,
+where the closability of EB×k
+r
+,µk,η
+r,R is a consequence of the continuity of the den-
+sity e−Ψk,η
+r,R on B×k
+r
+and the standard Hamza-type argument by [MR85, Fuk97], see
+for an accessible reference, e.g., [MR90, pp. 44-45]. The locality of the form is an
+immediate consequence of the locality of the gradient operator ∇⊙k.
+■
+Let µ be the sineβ ensemble. Due to [DHLM20, Thm. 1.1], the following limit
+exists for µ-a.e. η, all x ∈ Br and r > 0:
+lim
+R→∞
+�
+y∈ηBcr ,|y|≤R
+���1 − x
+y
+���
+β
+.
+Recall that µη
+r has been defined in (2.5). By [DHLM20, Thm. 1.1] and the number-
+rigidity (R) of µ, for µ-a.e. η there exists k = k(η) so that
+µη
+r(Υl(Br)) > 0 if and only if l = k(η) ,
+(3.4)
+and for γ = �k
+i=1 δxi,
+dµη
+r = dµk,η
+r
+= e−Ψk,η
+r
+Zη
+r
+dm⊙k
+r
+,
+(3.5)
+Ψk,η
+r (γ) := − log
+� k
+�
+i<j
+|xi − xj|β
+k
+�
+i=1
+lim
+R→∞
+�
+y∈ηBcr ,|y|≤R
+���1 − xi
+y
+���
+β
+�
+,
+where Zη
+r is the normalising constant. The corresponding weighted Sobolev norm is
+defined as
+EΥ(Br),µk,η
+r (u) :=
+�
+Υk(Br)
+|∇⊙ku|2 dµk,η
+r
+,
+u ∈ Lipb(Υk(Br), dΥ) .
+(3.6)
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+13
+Proposition 3.2. Let µ be the sineβ ensemble for β > 0. The form (3.6) is well-
+defined and closable for µ-a.e. η. The closure is a local Dirichlet form on L2(µk,η
+r )
+and its domain is denoted by D(EΥ(Br),µk,η
+r ).
+Proof. As e−Ψk,η
+r,R
+R→∞
+−−−→ e−Ψk,η
+r
+uniformly on Υk(Br) for µ-a.e. η by [DHLM20,
+Lem. 2.3 and Proof of Thm. 2.1 in p. 183], the density e−Ψk,η
+r
+is continuous on
+B⊙k
+r , hence the same proof as Prop. 3.1 applies to conclude the statement.
+■
+3.2. Curvature bound for finite-particle systems. We show that the poten-
+tial Ψk,η
+r,R defined in (3.1) is geodesically convex in (Υ(Br), dΥ).
+Proposition 3.3. Ψk,η
+r,R is geodesically convex in (Υk(Br), dΥ) for any 0 < r < R <
+∞, k ∈ N and η ∈ Υ(Bc
+r),
+Proof. Note that if u1, . . . , uk are convex and α1, . . . , αk ≥ 0, then �k
+i=1 αiui is
+again convex. Note also that for any 0 < r < R, any y ∈ [−R, −r] ∪ [r, R] and any
+i, j ∈ {1, 2, . . . , k} with i ̸= j, the functions − log |xi − xj| and − log |1 − xi
+y | are
+convex in the following areas for any σ ∈ Sk:
+�
+(x1, . . . , xk) ∈ B×k
+r
+: xσ(1) < xσ(2) < · · · < xσ(k)
+�
+.
+The following expression, therefore, concludes that Ψk,η
+r,R is geodesically convex as a
+function on Υk(Br): for any γ = �k
+i=1 δxi
+Ψk,η
+r,R(γ) = −β
+k
+�
+i<j
+log(|xi − xj|) − β
+k
+�
+i=1
+�
+y∈ηBcr ,|y|≤R
+log
+���1 − xi
+y
+��� .
+(3.7)
+The proof is complete.
+■
+Thanks to the geodesical convexity of the potential Ψk,η
+r,R shown in Prop. 3.3,
+the Dirichlet form (EΥ(Br),µk,η
+r,R, D(EΥ(Br),µk,η
+r,R)) satisfies the Riemannian Curvature-
+Dimension condition RCD(0, ∞).
+Proposition 3.4. The space (Υk(Br), dΥ, µk,η
+r,R) satisfies RCD(0, ∞) for every k ∈
+N0, 0 < r < R < ∞ and η ∈ Υ, and it holds that
+�
+EΥ(Br),µk,η
+r,R, D(EΥ(Br),µk,η
+r,R)
+�
+=
+�
+ChdΥ,µk,η
+r,R, W 1,2(Υk(Br), dΥ, µk,η
+r,R)
+�
+.
+Proof. Noting that B×k
+r
+is a convex subset in Rk, the space (B×k
+r , d×k, m⊗k
+r ) is a
+geodesic subspace of Rk and, therefore, satisfies RCD(0, ∞) by the Global-to-Local
+property of RCD(0, ∞), see [AGS14b, Thm. 6.20]. Noting that the k-particle con-
+figuration space (Υk(Br), dΥ, µk,η
+r,R) is the quotient space of (B×k
+r , d×k, m⊗k) with
+respect to the symmetric group Sk and that the property RCD(0, ∞) is preserved
+under the quotient operation with respect to Sk thanks to [GKMS18], we obtain
+that (Υk(Br), dΥ, m⊙k) satisfies RCD(0, ∞) as well. By the geodesical convexity of
+the potential Ψk,η
+r,R shown in Prop. 3.3 and the continuity of the density e−Ψk,η
+r,R, the
+weighted space (Υk(Br), dΥ, µk,η
+r,R) satisfies RCD(0, ∞) by [AGS14b, Prop. 6.21].
+
+14
+K. SUZUKI
+To conclude the statement, it suffices to check the identity
+EΥ(Br),µk,η
+r,R = ChdΥ,µk,η
+r,R .
+By the Rademacher theorem on Υk(Br) descendent from the Rademacher theorem
+on B×k
+r , the slope |DdΥu| coincides with |∇⊙ku| for u ∈ Lip(Υk(Br), dΥ). Thus,
+EΥ(Br),µk,η
+r,R(u) =
+�
+Υk(Br)
+|DdΥu|2 dµk,η
+r,R
+u ∈ Lipb(Υk(Br), dΥ) .
+(3.8)
+Since ChdΥ,µk,η
+r,R is the L2-lower semi-continuous envelope, the functional ChdΥ,µk,η
+r,R is
+the maximal L2-lower semi-continuous functional satisfying
+ChdΥ,µk,η
+r,R(u) ≤
+�
+Υk(Br)
+|DdΥu|2 dµk,η
+r,R .
+As EΥ(Br),µk,η
+r,R is closed by Prop. 3.1, in particular, EΥ(Br),µk,η
+r,R is L2-lower semi-
+continuous. Therefore, combining the maximality of ChdΥ,µk,η
+r,R with (3.8), it holds
+that
+EΥ(Br),µk,η
+r,R ≤ ChdΥ,µk,η
+r,R and W 1,2(Υk(Br), dΥ, µk,η
+r,R) ⊂ D(EΥ(Br),µk,η
+r,R)
+and
+ChdΥ,µk,η
+r,R(u) = EΥ(Br),µk,η
+r,R(u)
+u ∈ Lipb(Υk(Br), dΥ) .
+As Lipb(Υk(Br), dΥ) is dense both in D(EΥ(Br),µk,η
+r,R) and W 1,2(Υk(Br), dΥ, µk,η
+r,R) by
+construction, the proof is completed.
+■
+In view of Prop. 3.4 and the approximation Ψk,η
+r,R to Ψk,η
+r
+as R → ∞, we prove
+that (Υk(Br), dΥ, µk,η
+r ) satisfies RCD(0, ∞) as well.
+Proposition 3.5. Let µ be the sineβ ensemble for β > 0. For any 0 < r < ∞ and
+µ-a.e. η ∈ Υ, the space (Υk(Br), dΥ, µk,η
+r ) satisfies RCD(0, ∞), where k = k(η) as
+in (3.4). Furthermore,
+�
+EΥ(Br),µk,η
+r , D(EΥ(Br),µk,η
+r )
+�
+=
+�
+ChdΥ,µk,η
+r , W 1,2(Υk(Br), dΥ, µk,η
+r )
+�
+.
+Proof. Since the potential Ψk,η
+r,R is geodesically convex for any R and it converges
+pointwise to Ψk,η
+r
+as R → ∞ for µ-a.e. η by [DHLM20, Lem. 2.3 and Proof of Thm.
+2.1 in p. 183], the potential Ψk,η
+r
+is again geodesically convex on (Υk(Br), dΥ). Fur-
+thermore, as the density e−Ψk,η
+r,R converges uniformly to e−Ψk,η
+r
+on Υk(Br) as R → ∞
+for µ-a.e. η by [DHLM20, Lem. 2.3 and Proof of Thm. 2.1 in p. 183], the den-
+sity e−Ψk,η
+r
+is continuous on Υ(Br). Noting the fact that the constant multiplication
+(by the normalisation constant Zη
+r ) does not change the lower Ricci curvature bound
+(see e.g., [Stu06, Prop. 4.13]), the same proof as Prop. 3.4 applies to conclude the
+statement.
+■
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+15
+4. Curvature bound for infinite-particle systems
+In this section, we construct a local Dirichlet form on Υ = Υ(R) associated with
+sineβ ensemble µ and show the BE1(0, ∞) property by the following steps: we first
+construct truncated Dirichlet forms on Υ whose gradient operators are truncated up
+to configurations inside Br. We then identify them with the superposition Dirichlet
+forms lifted from Υ(Br), thanks to which we can show BE1(0, ∞) for the truncated
+forms. We take the monotone limit of the truncated forms to construct a Dirichlet
+form with invariant measure sineβ ensembles µ and BE1(0, ∞) extends to the limit
+form. In the end of this section, we discuss several applications of the BE1(0, ∞)
+property.
+4.1. Superposition of Dirichlet forms from Υ(Br) onto Υ. In this subsection,
+we construct the truncated Dirichlet forms on Υ. We first construct square field
+operators on Υ and Υ(Br) respectively.
+For so doing, we introduce a map Uγ,x
+transferring functions on the configuration space Υ to functions on the base space R.
+For u : Υ → R, define Uγ,x(u) : R → R by
+Uγ,x(u)(y) := u
+�
+1X\{x} ·γ + δy
+�
+− u
+�
+1X\{x} ·γ
+�
+,
+γ ∈ Υ,
+x ∈ γ .
+(4.1)
+In the context of configuration spaces, the operation Uγ,x has been firstly discussed
+in [MR00, Lem. 1.2], see also [DS21a, Lem. 2.16]. We introduce the localisation of
+the operator Uγ,x on Br. Recall that for a measurable function u: Υ → R, r > 0
+and for η ∈ Υ, we set in (2.7)
+uη
+r(γ) := u(γ + ηBcr) for γ ∈ Υ(Br).
+Lemma 4.1. For u : Υ(Br) → R, define Ur
+γ,x(u) : Br → R by
+Ur
+γ,x(u)(y) := u(1X\{x} · γ + δy) − u(1X\{x} · γ)
+γ ∈ Υ(Br), x ∈ γ .
+The operation Ur
+γ,x maps from Lip(Υ(Br), dΥ) to Lip(Br) and Lipschitz constants
+are contracted by Ur
+γ,x for any r > 0:
+Lip(Ur
+γ,x(u)) ≤ LipdΥ(u)
+∀γ ∈ Υ(Br)
+∀x ∈ γ .
+Furthermore, for any u : Υ → R,
+Ur
+γBr ,x(uγ
+r)(y) = Uγ,x(u)(y)
+for every γ ∈ Υ, x ∈ γBr and y ∈ Br .
+Proof. Let u ∈ Lip(Υ(Br), dΥ). Then
+|Ur
+γ,x(u)(y) − Ur
+γ,x(u)(z)| = |u(1X\{x} ·γ + δy) − u(1X\{x} ·γ + δz)|
+≤ LipdΥ(u)dΥ(1X\{x} ·γ + δy, 1X\{x} ·γ + δz)
+= LipdΥ(u)|y − z| ,
+which concludes the first assertion.
+
+16
+K. SUZUKI
+We verify the second assertion. For every x ∈ γBr and y ∈ Br,
+Uγ,x(u)(y) = u(1X\{x} · γ + δy) − u(1X\{x} · γ)
+= u(1X\{x} · γBr + γBcr + δy) − u(1X\{x} · γBr + γBcr)
+= ur,γ(1X\{x} · γBr + δy) − ur,γ(1X\{x} · γBr)
+= Ur
+γBr ,x(uγ
+r)(y) .
+The proof is complete.
+■
+We now define a square field operator on Υ truncated up to particles inside Br.
+Definition 4.2 (Truncated square field on Υ). Let u : Υ → R be a measurable
+function so that Uγ,x(u)|Br ∈ W 1,2(mr) for µ-a.e. γ and every x ∈ γBr. The following
+operator is called the truncated square field ΓΥ
+r ,
+(4.2)
+ΓΥ
+r (u)(γ) :=
+�
+x∈γBr
+|∇Uγ,x(u)|2(x) .
+Thanks to Lem. A.1, Formula (4.2) is well-defined for µ-a.e. γ. Indeed, as Uγ,x(u)|Br ∈
+W 1,2
+loc (mr), the weak gradient ∇Uγ,x(u) is well-defined pointwise on a measurable set
+Σ ⊂ Br with mr(Σc) = 0. By applying Lem. A.1, Formula (4.2) is well-defined on
+the set Ω(r) of µ-full measure.
+Based on the truncated square field ΓΥ
+r , we introduce the truncated form on Υ
+defined on a certain core.
+Definition 4.3 (Core). For r > 0, let Cr be defined as the space of µ-classes of
+measurable functions u so that
+(a) u ∈ L∞(µ);
+(b) uη
+r ∈ Lipb(Υ(Br), dΥ) for µ-a.e. η and r > 0;
+(c) The following integral is finite:
+EΥ,µ
+r
+(u) :=
+�
+Υ
+ΓΥ
+r (u) dµ < ∞ .
+(4.3)
+Note that, thanks to Lem. 4.1, if a measurable function u : Υ → R satisfies (b),
+then Uγ,x(u)|Br ∈ Lip(Br, d) ⊂ W 1,2(mr). Thus, the expression ΓΥ
+r (u) in (4.3) is
+well-posed.
+Definition 4.4 (Square field on Υ(Br)). Fix r > 0 and η ∈ Υ. For a µ-measurable
+function u : Υ(Br) → R satisfying u|Υk(Br) ∈ D(EΥ(Br),µk,η
+r ) for any k ∈ N0, we
+define the following square field operator on Υ(Br):
+ΓΥ(Br)(u) :=
+∞
+�
+k=0
+���∇⊙k�
+u|Υk(Br)
+����
+2
+,
+(4.4)
+and define the following form:
+EΥ(Br),µη
+r(u) :=
+�
+Υ(Br)
+ΓΥ(Br)(u) dµη
+r ,
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+17
+D(EΥ(Br),µη
+r) := {u : Υ(Br) → R, EΥ(Br),µη
+r(u) < ∞} .
+Due to the number-rigidity (R), the Dirichlet form EΥ(Br),µη
+r is equal to EΥ(Br),µk,η
+r
+up to the normalising constant multiplication, therefore, it is a Dirichlet form as
+well. The corresponding semigroup is denoted by {T Υ(Br),µη
+r
+t
+}t≥0.
+Remark 4.5. The number-rigidity (R) is not necessary to conclude that EΥ(Br),µη
+r
+is a Dirichlet form since any countable sum of Dirichlet forms is a Dirichlet form
+(see e.g., [MR90, Exercise 3.9]).
+Before discussing properties of truncated forms, we prepare a lemma, which states
+that the operation (·)η
+r defined in (2.7) maps from Lip(Υ, ¯dΥ) to Lip(Υ(Br), dΥ) and
+contracts Lipschitz constants.
+Lemma 4.6. Let u ∈ Lip(Υ, ¯dΥ). Then, uη
+r ∈ Lip(Υ(Br), dΥ) and
+LipdΥ(uη
+r) ≤ Lip¯dΥ(u) ,
+∀η ∈ Υ ,
+r > 0 .
+(4.5)
+Proof. Let γ, ζ ∈ Υ(Br) and η ∈ Υ. Then,
+|uη(γ) − uη(ζ)| = |u(γ + ηBcr) − u(ζ + ηBcr)| ≤ Lip¯dΥ(u)¯dΥ(γ + ηBcr, ζ + ηBcr)
+= Lip¯dΥ(u)dΥ(γ, ζ) .
+The proof is completed.
+■
+The following proposition relates the two square fields ΓΥ
+r and ΓΥ(Br).
+Proposition 4.7 (Truncated form). The following relations hold on Cr:
+ΓΥ
+r (u)(γ + ηBcr) = ΓΥ(Br)(uη
+r)(γ) ,
+µ-a.e. η, µη
+r-a.e. γ ∈ Υ(Br) ,
+(4.6)
+EΥ,µ
+r
+(u) =
+�
+Υ
+EΥ(Br),µη
+r(uη
+r) dµ(η) ,
+u ∈ Cr .
+Furthermore, the Rademacher-type property holds: Lipb(¯dΥ, µ) ⊂ Cr and
+ΓΥ
+r (u) ≤ Lip¯dΥ(u)2
+∀u ∈ Lipb(¯dΥ, µ) .
+(4.7)
+As a consequence, the form (EΥ,µ
+r
+, Cr) in (4.3) is a densely defined closable Markovian
+form and the closure (EΥ,µ
+r
+, D(EΥ,µ
+r
+)) is a local Dirichlet form on L2(µ). The L2-
+semigroups corresponding to (EΥ,µ
+r
+, D(EΥ,µ
+r
+)) is denoted by {T Υ,µ
+r,t }t≥0.
+Proof. We first prove (4.6). Let u ∈ Cr. Thanks to (b) in Def. 4.3 and Lem. 4.1,
+Uγ,x(u) ∈ Lip(Br, d) ,
+µ-a.e. γ
+∀r > 0 .
+Thus, noting Lip(Br, d) ⊂ W 1,2(mr), the LHS of (4.6) is well-defined. The RHS
+of (4.6) is also well-defined by (b) in Def. 4.3 and the fact that Lipb(Υ(Br), dΥ) ⊂
+D(EΥ(Br),µη
+r) by construction. Thus, for µ-a.e. η, we can take a measurable set Ω =
+Ω(η) ⊂ Υ(Br) with µη
+r(Ω) = 1 so that (4.6) is well-defined for every γ ∈ Ω. As µη
+r is
+absolutely continuous with respect to the Poisson measure πmr, we may assume that
+
+18
+K. SUZUKI
+every γ ∈ Ω does not have multiple points, i.e., γ({x}) ∈ {0, 1} for every x ∈ Br.
+Let γ ∈ Ω ∩ Υk(Br). Then, according to (4.4),
+ΓΥ(Br)(uη
+r)(γ) =
+���∇⊙k�
+uη
+r
+����
+2
+(γ)
+=
+�
+x∈γ
+��∇uη
+r
+�
+1Br\{x} ·γ + δ•
+���2(x)
+=
+�
+x∈γ
+���∇
+�
+uη
+r
+�
+1Br\{x} ·γ + δ•
+�
+− uη
+r
+�
+1Br\{x} ·γ
+�����
+2
+(x)
+=
+�
+x∈γBr
+���∇
+�
+u
+�
+1X\{x} ·(γ + ηBcr) + δ•
+�
+− u
+�
+1X\{x} ·(γ + ηBcr)
+�����
+2
+(x)
+= ΓΥ
+r (u)(γ + ηBcr)
+where the second equality followed from the definition of the symmetric gradient
+operator ∇⊙k, for which we used the fact that γ ∈ Ω does not have multiple points;
+the third equality follows simply as uη
+r
+�
+1Br\{x} ·γ
+�
+does not depend on the variable
+denoted as •, on which the weak gradient ∇ operates; the fifth equality followed
+from the definition of the square field ΓΥ
+r . As this argument holds for arbitrary
+k ∈ N0, (4.6) has been shown. The Markov property and the locality of EΥ,µ
+r
+follow
+immediately from (4.6) since EΥ(Br),µη
+r possesses the corresponding properties by
+construction.
+We now show the Rademacher-type property: Lipb(¯dΥ, µ) ⊂ Cr and
+ΓΥ
+r (u) ≤ Lip¯dΥ(u)2
+∀u ∈ Lipb(¯dΥ, µ)
+∀r > 0 .
+(4.8)
+We first show Lipb(¯dΥ, µ) ⊂ Cr. The verification of (a) in Def. 4.3 is obvious. The
+verification of (b) in Def. 4.3 follows from the Lipschitz contraction (4.5) of the
+operator (·)η
+r. The verification of (c) in Def. 4.3 follows by showing (4.8) as µ is a
+probability measure.
+We now prove (4.8). As the Cheeger energy ChdΥ,µk,η
+r
+coincided with EΥ(Br),µk,η
+r
+by
+Prop. 3.5, the Rademacher-type property for EΥ(Br),µk,η
+r
+follows from that for ChdΥ,µk,η
+r ,
+the latter of which is an immediate consequence by the definition of the Cheeger
+energy. Therefore, we have that
+ΓΥ(Br)(u) ≤ LipdΥ(u)2
+∀u ∈ Lip(Υ(Br), dΥ)
+∀r > 0 .
+(4.9)
+In view of the relation between ΓΥ
+r and ΓΥ(Br) in (4.6) and the Lipschitz contrac-
+tion (4.5) of the operator (·)η
+r, we concluded (4.8).
+Noting that Lipb(dΥ, µ) ⊂ L2(µ) is dense (e.g., [AGS14a, Prop. 4.1]) and the
+fact that Lipb(dΥ, µ) ⊂ Lipb(¯dΥ, µ) ⊂ Cr by (2.14) and (4.8), we obtain that the
+form (EΥ,µ
+r
+, Cr) is densely defined.
+We now show the closability. Noting that EΥ(Br),µη
+r is closable for µ-a.e. η by
+Prop. 3.5, the superposition form ( ¯EΥ,µ
+r
+, D( ¯EΥ,µ
+r
+)) (defined below in Def. 4.8) is
+closable (indeed it is closed) by [BH91, Prop. V.3.1.1]. As the two forms (EΥ,µ
+r
+, Cr)
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+19
+and ( ¯EΥ,µ
+r
+, D( ¯EΥ,µ
+r
+)) coincide on Cr and Cr ⊂ D( ¯EΥ,µ
+r
+) by construction, the closability
+of (EΥ,µ
+r
+, Cr) is inherited from the closedness of the superposition form ( ¯EΥ,µ
+r
+, D( ¯EΥ,µ
+r
+)).
+The proof is complete.
+■
+The superposition of the Dirichlet form EΥ(Br),µη
+r onto Υ is now defined below.
+Definition 4.8 (Superposition Dirichlet form, e.g., [BH91, Prop. V.3.1.1]).
+D( ¯EΥ,µ
+r
+) :=
+�
+u ∈ L2(µ) :
+�
+Υ
+EΥ(Br),µη
+r(uη
+r) dµ(η) < ∞
+�
+,
+(4.10)
+¯EΥ,µ
+r
+(u) :=
+�
+Υ
+EΥ(Br),µη
+r(uη
+r) dµ(η) .
+It is known that ( ¯EΥ,µ
+r
+, D( ¯EΥ,µ
+r
+)) is a Dirichlet form on L2(µ) [BH91, Prop. V.3.1.1].
+The L2-semigroup and the infinitesimal generator corresponding to ( ¯EΥ,µ
+r
+, D( ¯EΥ,µ
+r
+))
+are denoted by { ¯T Υ,µ
+r,t }t≥0 and ( ¯AΥ,µ
+r
+, D( ¯AΥ,µ
+r
+)) respectively.
+The semigroup { ¯T Υ,µ
+r,t }t≥0 corresponding to the superposition form ¯EΥ,µ
+r
+can be
+obtained as the superposition of the semigroup {T Υ(Br),µη
+r
+t
+}t≥0 associated with the
+form EΥ(Br),µη
+r. For the following proposition, we refer the reader to [Del21, (iii)
+Prop. 2.13].
+Proposition 4.9 ([Del21, (iii) Prop. 2.13]). The following holds:
+¯T Υ,µ
+r,t u(γ) = T Υ(Br),µγ
+r
+t
+uγ
+r(γBr) ,
+(4.11)
+for µ-a.e. γ ∈ Υ, any t > 0.
+Remark 4.10. The proof of [Del21, (iii) Prop. 2.13] has been given in terms of direct
+integral in a general setting. As the measure µη
+r can be identified to the conditional
+probability µ(· | ·Bcr = ηBcr) by a bi-measure-preserving isomorphism as remarked
+in (2.6), our setting is a particular case of direct integrals discussed in [Del21].
+We now discuss the relation between EΥ,µ
+r
+and ¯EΥ,µ
+r
+. As the former form is con-
+structed as the smallest closed extension of (EΥ,µ
+r
+, Cr), it is clear by definition that
+EΥ,µ
+r
+= ¯EΥ,µ
+r
+on
+Cr ,
+D(EΥ,µ
+r
+) ⊂ D( ¯EΥ,µ
+r
+) .
+The following theorem proves that the opposite inclusion holds as well.
+Theorem 4.11. (EΥ,µ
+r
+, D(EΥ,µ
+r
+)) = ( ¯EΥ,µ
+r
+, D( ¯EΥ,µ
+r
+)).
+Proof. The inclusion D(EΥ,µ
+r
+) ⊂ D( ¯EΥ,µ
+r
+) with the inequality ¯EΥ,µ
+r
+≤ EΥ,µ
+r
+is straight-
+forward by definition. Noting ¯EΥ,µ
+r
+= EΥ,µ
+r
+on CR and D(EΥ,µ
+r
+) is the closure of Cr,
+it suffices to show that Cr ⊂ D( ¯EΥ,µ
+r
+) is dense. Thanks to Lem. A.4, we only need
+to show that ¯T Υ,µ
+r,t Cr ⊂ Cr.
+As ¯T Υ,µ
+r,t
+is an L∞-contraction semigroup by the sub-Markovian property of the
+semigroup (see, e.g., [MR90, Def. I.4.1]), we obtain ¯T Υ,µ
+r,t Cr ⊂ L∞(µ), which verifies
+(a) in Def. 4.3
+
+20
+K. SUZUKI
+Verification of (b) in Def. 4.3.
+Let u ∈ Cr and we show that ¯T Υ,µ
+r,t u satisfies (b)
+in Def. 4.3. By Prop. 4.9, we can identify the following two operators:
+¯T Υ,µ
+r,t u = T Υ(Br),µ·
+r
+t
+u·
+r(·Br) .
+This implies that
+�
+¯T Υ,µ
+r,t u
+�η
+r(·) = ¯T Υ,µ
+r,t u(· + ηBcr) = T Υ(Br),µη
+r
+t
+uη
+r(·) .
+Take k = k(η) as in (3.4). As the conditional probability µη
+r is supported only on
+Υk(Br), we only need to show
+T Υ(Br),µk,η
+r
+t
+uη
+r ∈ Lipb(Υk(Br), dΥ) .
+(4.12)
+As (Υk(Br), dΥ, µk,γ
+r ) is RCD(0, ∞) for k = k(η) for µ-a.e. η by Prop. 3.5, the corre-
+sponding semigroup satisfies L∞(µk,η
+r )-to-Lipb(Υk(Br), dΥ)-regularisation property
+([AGS14a, Thm. 6.5]), which shows that for µ-a.e. η
+T Υ(Br),µk,η
+r
+t
+v ∈ Lipb(Υk(Br), dΥ)
+∀v ∈ L∞(µk,η
+r ) ,
+and its Lipchitz constant is bounded as
+LipdΥ(T Υ(Br),µk,η
+r
+t
+v) ≤ c(t, K)∥v∥L∞(µk,η
+r
+) ,
+with constant c(t, K) depending only on t and the curvature bound K = 0 (to be
+more precise, c(t, 0) =
+1
+√
+2t). This proves (4.12), which completes the verification
+of (b).
+Verificaiton of (c) in Def. 4.3.
+Let u ∈ Cr. Thanks to the verification of (b), the
+square field ΓΥ
+r ( ¯T Υ,µ
+r,t u) is well-defined, and by (4.6) it holds that for µ-a.e. η
+ΓΥ
+r ( ¯T Υ,µ
+r,t u)(γ + ηBcr) = ΓΥ(Br)�
+( ¯T Υ,µ
+r,t u)η
+r
+�
+(γ)
+µη
+r-a.e. γ ∈ Υ(Br) .
+(4.13)
+In view of the contraction property of the semigroup with respect to the form by
+general theory (see, e.g., [FOT11, p.23, Lem. 1.3.3]), viz.
+EΥ(Br),µη
+r(T Υ(Br),µη
+r
+t
+uη
+r) ≤ EΥ(Br),µη
+r(uη
+r)
+as well as Prop. 4.9 and (4.13), we obtain
+�
+Υ
+ΓΥ
+r ( ¯T Υ,µ
+r,t u) dµ =
+�
+Υ
+EΥ(Br),µη
+r�
+( ¯T Υ,µ
+r,t u)η
+r
+�
+dµ(η)
+=
+�
+Υ
+EΥ(Br),µη
+r(T Υ(Br),µη
+r
+t
+uη
+r) dµ(η)
+≤
+�
+Υ
+EΥ(Br),µη
+r(uη
+r) dµ(η)
+= EΥ,µ
+r
+(u) < ∞ .
+The verification of (c) is completed. Therefore, we confirmed ¯T Υ,µ
+r,t Cr ⊂ Cr, which
+concludes the statement.
+■
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+21
+As a consequence of Thm. 4.11 and Prop. 4.9, we obtain the superposition formula
+for the semigroup {T Υ,µ
+r,t }t≥0 in terms of the semigroup {T Υ(Br),µη
+r
+t
+}t≥0.
+Corollary 4.12 (Coincidence of semigroups). The following three operators coin-
+cide:
+T Υ,µ
+r,t u(γ) = ¯T Υ,µ
+r,t u(γ) = T Υ(Br),µγ
+r
+t
+uγ
+r(γBr) ,
+(4.14)
+for µ-a.e. γ ∈ Υ, any t > 0.
+4.2. Monotone limit form. We now construct a Dirichlet form on Υ with sineβ-
+invariant measure µ as the monotone limit of (EΥ,µ
+r
+, D(EΥ,µ
+r
+) as r → ∞. The follow-
+ing proposition follows immediately from the definitions of the square field ΓΥ
+r and
+the core Cr.
+Proposition 4.13 (Monotonicity). The form (EΥ,µ
+r
+, D(EΥ,µ
+r
+) and the square field
+ΓΥ
+r are monotone increasing as r ↑ ∞, viz.,
+ΓΥ
+r (u) ≤ ΓΥ
+s (u) ,
+EΥ,µ
+r
+(u) ≤ EΥ,µ
+s
+(u) ,
+D(EΥ,µ
+s
+) ⊂ D(EΥ,µ
+r
+)
+r ≤ s .
+Proof. As Cr is a core of the form (EΥ,µ
+r
+, D(EΥ,µ
+r
+)), it suffices to check Cs ⊂ Cr
+and ΓΥ
+r (u) ≤ ΓΥ
+s (u) on Cs. Let u ∈ Cs and we show u ∈ Cr. By a simple reasoning
+similar to the proof of Lem. 4.6, we can see
+uη
+r ∈ Lipb(Υ(Br), dΥ)
+µ-a.e. η
+if uη
+s ∈ Lipb(Υ(Bs), dΥ)
+µ-a.e. η .
+By Def. 4.2, it is straightforward to see ΓΥ
+r (u) ≤ ΓΥ
+s (u). Thus,
+EΥ,µ
+r
+(u) =
+�
+Υ
+ΓΥ
+r (u) dµ ≤
+�
+Υ
+ΓΥ
+s (u) dµ =
+�
+Υ
+ΓΥ
+s (u) dµ = EΥ,µ
+s
+(u) < ∞ .
+Therefore, we conclude u ∈ Cr. The proof is completed.
+■
+We now define a Dirichlet form on Υ whose invariant measure is the sineβ mea-
+sure µ by the monotone limit of (EΥ,µ
+r
+, D(EΥ,µ
+r
+)).
+Definition 4.14 (Monotone limit form). The form (EΥ,µ, D(EΥ,µ)) is defined as the
+monotone limit:
+D(EΥ,µ) := {u ∈ ∩r>0D(EΥ,µ
+r
+) : EΥ,µ(u) = lim
+r→∞ EΥ,µ
+r
+(u) < ∞} ,
+(4.15)
+EΥ,µ(u) := lim
+r→∞ EΥ,µ
+r
+(u) .
+The form (EΥ,µ, D(EΥ,µ)) is a Dirichlet form on L2(µ) as it is the monotone limit
+of Dirichlet forms (e.g., by [MR90, Exercise 3.9]). The square field ΓΥ is defined as
+the monotone limit of ΓΥ
+r as well:
+ΓΥ(u) := lim
+r→∞ ΓΥ
+r (u)
+u ∈ D(EΥ,µ) .
+(4.16)
+The corresponding L2(µ)-semigroup is denoted by {T Υ,µ
+t
+}t≥0.
+
+22
+K. SUZUKI
+We now show that the form (EΥ,µ, D(EΥ,µ)) is a local Dirichlet form on L2(µ) and
+satisfies the Rademacher-type property with respect to the L2-transportation-type
+distance ¯dΥ.
+Proposition 4.15. The form (EΥ,µ, D(EΥ,µ)) is a local Dirichlet form on L2(µ).
+Furthermore, (EΥ,µ, D(EΥ,µ)) satisfies Rademacher-type property:
+Lip(¯dΥ, µ) ⊂ D(EΥ,µ) ,
+ΓΥ(u) ≤ Lip¯dΥ(u)2
+∀u ∈ Lip(¯dΥ, µ) .
+(4.17)
+Proof. The local property of (EΥ,µ, D(EΥ,µ)) follows from the fact that (EΥ,µ, D(EΥ,µ))
+is the monotone limit of the local Dirichlet form (EΥ,µ
+r
+, D(EΥ,µ
+r
+)).
+We show the
+Rademacher-type property. Since ΓΥ is the limit square field of ΓΥ
+r as in (4.16), it
+suffices to show
+Lip(¯dΥ, µ) ⊂ Cr
+and
+ΓΥ
+r (u) ≤ Lip¯dΥ(u)2
+∀u ∈ Lip(¯dΥ, µ)
+∀r > 0 ,
+which has been already proven in Prop. 4.7.
+We verified (4.17).
+The proof is
+complete.
+■
+Proposition 4.16. The semigroup {T Υ,µ
+t
+}t≥0 is the L2(µ)-strong operator limit of
+the semigroups {T Υ,µ
+r,t }t≥0, viz.,
+L2(µ)– lim
+r→∞ T Υ,µ
+r,t u = T Υ,µ
+t
+u
+∀u ∈ L2(µ) ,
+t > 0 .
+Proof. The statement follows from the monotonicity of (EΥ,µ
+r
+, D(EΥ,µ
+r
+) as r ↑ ∞
+proven in Prop. 4.13 and [RS80, S.14, p.373].
+■
+4.3. Bakry–Émery Curvature bound for (EΥ,µ, D(EΥ,µ)). In this subsection,
+we prove the Bakry–Émery curvature bound for the form (EΥ,µ, D(EΥ,µ)).
+Theorem 4.17. Let β > 0 and µ be the sineβ ensemble. The form (EΥ,µ, D(EΥ,µ))
+satisfies the 1-Bakry–Émery curvature dimension condition BE1(0, ∞):
+ΓΥ�
+T Υ,µ
+t
+u
+� 1
+2 ≤ T Υ,µ
+t
+�
+ΓΥ(u)
+1
+2�
+∀u ∈ D(EΥ,µ)
+∀t > 0 .
+(BE1(0, ∞))
+Proof. We first prove BE1(0, ∞) for the form (EΥ,µ
+r
+, D(EΥ,µ
+r
+)). Let u ∈ D(EΥ,µ
+r
+). By
+Prop. 3.5, [Han18, Thm. 1.1], by the expression (3.4) of µη
+r in terms of µk,η
+r
+and
+by the definition (4.4) of ΓΥ(Br), there exists Ξ1
+r ⊂ Υ with µ(Ξ1
+r) = 1 so that for
+every η ∈ Ξ1
+r there exists a measurable set Ω1,η
+r
+⊂ Υ(Br) with µη
+r(Ω1,η
+r ) = 1 satisfying
+that for every γ ∈ Ω1,η
+r , the following 1-Bakry–Émery gradient estimate holds:
+ΓΥ(Br)(T Υ(Br),µη
+r
+t
+uη
+r)
+1
+2(γ) ≤ T Υ(Br),µη
+r
+t
+�
+ΓΥ(Br)(uη
+r)
+� 1
+2(γ) .
+(4.18)
+By Prop. 4.7, there exists Ξ2
+r ⊂ Υ with µ(Ξ2
+r) = 1 so that for every η ∈ Ξ2
+r there
+exists a measurable set Ω2,η
+r
+⊂ Υ(Br) with µη
+r(Ω2,η
+r ) = 1 satisfying that for every
+γ ∈ Ω2,η
+r
+ΓΥ
+r (T Υ,µ
+r,t u)(γ + ηBcr) = ΓΥ(Br)��
+T Υ,µ
+r,t u
+�η
+r
+�
+(γ) ;
+(4.19)
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+23
+ΓΥ
+r (u)(γ + ηBcr) = ΓΥ(Br)(uη
+r)(γ) .
+By Cor. 4.12, there exists Λ3
+r ⊂ Υ with µ(Λ3
+r) = 1 so that for every γ ∈ Λ3
+r
+T Υ,µ
+r,t u(γ) = T Υ(Br),µγ
+r
+t
+uγ
+r(γ) .
+(4.20)
+By the standard disintegration argument, we can write
+Λ3
+r =
+�
+η∈Ξ3r
+pr−1
+r (Ω3,η
+r ) ∩ Υη
+r ,
+where Ω3,η
+r
+= (Λ3
+r)η
+r := {γ ∈ Υ(Br) : γ + ηBcr ∈ Λ3
+r} and Ξ3
+r = prBcr(Λ3
+r), and Υη
+r
+has been defined in (2.4).
+By the disintegration formula (2.10), µ(Ξ3
+r) = 1 and
+µη
+r(Ω3,η
+r ) = 1 for every η ∈ Ξ3
+r.
+Let Ξr := Ξ1
+r ∩ Ξ2
+r ∩ Ξ3
+r and Ωη
+r := Ω1,η
+r
+∩ Ω2,η
+r
+∩ Ω3,η
+r
+for η ∈ Ξr. Set
+Kr :=
+�
+η∈Ξr
+pr−1
+r (Ωη
+r) ∩ Υη
+r .
+By construction, µ(Ξr) = 1 and µη
+r(Ωη
+r) = 1 for every η ∈ Ξr. By (4.18), (4.19) and
+(4.20), the following inequalities hold for every γ ∈ Kr:
+ΓΥ
+r (T Υ,µ
+r,t u)
+1
+2(γ) = ΓΥ
+r (T Υ,µ
+r,t u)
+1
+2(γBr + γBcr)
+(4.21)
+= ΓΥ(Br)((T Υ,µ
+r,t u)γ
+r)
+1
+2(γBr)
+≤ T Υ(Br),µγ
+r
+t
+ΓΥ(Br)(uγ
+r)
+1
+2(γBr)
+= T Υ(Br),µγ
+r
+t
+�
+(ΓΥ
+r (u)γ
+r)
+1
+2�
+(γBr)
+= T Υ,µ
+r,t ΓΥ
+r (u)
+1
+2(γ) .
+Let Θ := {γ ∈ Υ : ΓΥ
+r (T Υ,µ
+r,t u)
+1
+2(γ) ≤ T Υ,µ
+r,t ΓΥ
+r (u)
+1
+2(γ)}. Then Θ is µ-measurable by
+construction, and thanks to (4.21), it holds that Kr ⊂ Θ. By applying Lem. A.2, we
+obtain µ(Θ) = 1, which concludes BE1(0, ∞) for the truncated form (EΥ,µ
+r
+, D(EΥ,µ
+r
+))
+for any r > 0.
+We now prove BE1(0, ∞) of the form (EΥ,µ, D(EΥ,µ)).
+Let u ∈ D(EΥ,µ) ⊂
+∩r>0D(EΥ,µ
+r
+). By the L2(µ)-lower semi-continuity of the square field ΓΥ, the mono-
+tonicity ΓΥ
+r ≤ ΓΥ
+r′ for r ≤ r′ (we will use it in the following displayed formulas in
+the first equality and in the second inequality), the convergence of T Υ,µ
+r′,t
+to T Υ,µ
+t
+as r′ → ∞ in the L2-strong operator sense by Prop. 4.16, and BE1(0, ∞) for the
+truncated form (EΥ,µ
+r
+, D(EΥ,µ
+r
+)) for any r > 0, the following inequalities hold true:
+ΓΥ(T Υ,µ
+t
+u)1/2 = lim
+r→∞ ΓΥ
+r (T Υ,µ
+t
+u)1/2 ≤ lim
+r→∞ lim inf
+r′→∞ ΓΥ
+r (T Υ,µ
+r′,t u)1/2
+≤ lim inf
+r′→∞ ΓΥ
+r′(T Υ,µ
+r′,t u)1/2
+≤ lim inf
+r′→∞ T Υ,µ
+r′,t ΓΥ
+r′(u)1/2
+= T Υ,µ
+t
+ΓΥ(u)1/2 .
+
+24
+K. SUZUKI
+The last equality in the above displayed formulas followed from the following argu-
+ment: by the L2(µ)-contraction property of T Υ,µ
+r′,t , the monotonicity of ΓΥ
+r′ as r′ ↑ ∞,
+and the convergence of the semigroups T Υ,µ
+r′,t
+to T Υ,µ
+t
+as r′ → ∞ in the L2-strong
+operator sense by Prop. 4.16,
+��T Υ,µ
+r′,t ΓΥ
+r′(u)1/2 − T Υ,µ
+t
+ΓΥ(u)1/2��
+L2(µ)
+=
+��T Υ,µ
+r′,t ΓΥ
+r′(u)1/2 − T Υ,µ
+r′,t ΓΥ(u)1/2��
+L2(µ) +
+��T Υ,µ
+r′,t ΓΥ(u)1/2 − T Υ,µ
+t
+ΓΥ(u)1/2��
+L2(µ)
+≤
+��ΓΥ
+r′ (u)1/2 − ΓΥ(u)1/2��
+L2(µ) +
+��T Υ,µ
+r′,t ΓΥ(u)1/2 − T Υ,µ
+t
+ΓΥ(u)1/2��
+L2(µ)
+r′→∞
+−−−→ 0 .
+The proof is completed.
+■
+4.4. Integral Bochner, local Poicaré and local log-Sobolev inequalities.
+As an application of BE(0, ∞) proven in Thm. 4.17, we show several functional
+inequalities. We define the integral Γ2-operator as follows:
+ΓΥ,µ
+2
+(u, ϕ) :=
+�
+Υ
+�1
+2ΓΥ(u)AΥ,µϕ − ΓΥ(u, AΥ,µu)ϕ
+�
+dµ ,
+(4.22)
+D(ΓΥ,µ
+2
+) :=
+�
+(u, ϕ) : D(AΥ,µ)×2 : AΥ,µu ∈ D(EΥ,µ), ϕ, AΥ,µu ∈ L∞(µ)
+�
+,
+where AΥ,µ denotes the L2(µ)-infinitesimal generator associated with (EΥ,µ, D(EΥ,µ)).
+Corollary 4.18. Let µ be the sineβ ensemble with β > 0. The following hold:
+(a) (lntegral Bochner inequality) for every (u, ϕ) ∈ D(ΓΥ,µ
+2
+)
+ΓΥ,µ
+2
+(u, ϕ) ≥ 0 ;
+(b) (local Poincaré inequality) for u ∈ D(EΥ,µ) and t > 0,
+T Υ,µ
+t
+u2 − (T Υ,µ
+t
+u)2 ≤ 2tT Υ,µ
+t
+ΓΥ(u) ,
+T Υ,µ
+t
+u2 − (T Υ,µ
+t
+u)2 ≥ 2tΓΥ(T Υ,µ
+t
+u) ;
+(c) (local logarithmic Sobolev inequality) for non-negative u ∈ D(EΥ,µ)
+and t > 0,
+T Υ,µ
+t
+u log u − T Υ,µ
+t
+u log T Υ,µ
+t
+u ≤ tT Υ,µ
+t
+�ΓΥ(u)
+u
+�
+,
+T Υ,µ
+t
+u log u − T Υ,µ
+t
+u log T Υ,µ
+t
+u ≥ tΓΥ(T Υ,µ
+t
+u)
+T Υ,µ
+t
+u
+.
+Proof. The statement (a) follows from BE(0, ∞) proven in Thm. 4.17 and [AGS15,
+Cor. 2.3]. The statement (b) and (c) are consequences of BE(0, ∞), see e.g., [BGL14,
+Thm.s 4.7.2, 5.5.2.].
+■
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+25
+5. Dimension-free/log Harnack inequalities and Lipschitz
+regularisation
+In this section, we prove functional inequalities involving the Bakry–Émery cur-
+vature bound BE(0, ∞) and the L2-transportation-type extended distance ¯dΥ.
+Theorem 5.1. Let µ be the sineβ ensemble with β > 0. Then the following inequal-
+ities hold:
+(a) (log-Harnack inequality) for every non-negative u ∈ L∞(Υ, µ), t > 0,
+there exists Ω ⊂ Υ so that µ(Ω) = 1 and
+T Υ,µ
+t
+(log u)(γ) ≤ log(T Υ,µ
+t
+u)(η) + ¯dΥ(γ, η)2 ,
+every γ, η ∈ Ω ;
+(b) (dimension-free Harnack inequality) for every non-negative u ∈ L∞(Υ, µ),
+t > 0 and α > 1 there exists Ω ⊂ Υ so that µ(Ω) = 1 and
+(T Υ,µ
+t
+u)α(γ) ≤ T Υ,µ
+t
+uα(η) exp
+�
+α
+2(α − 1)
+¯dΥ(γ, η)2�
+,
+for every γ, η ∈ Ω ;
+(c) (Lipschitz contraction) For u ∈ Lipb(¯dΥ, µ) and t > 0,
+T Υ,µ
+t
+u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ
+t
+u
+and the following estimate holds:
+Lip¯dΥ( ˜T Υ,µ
+t
+u) ≤ Lip¯dΥ(u) ;
+(d) (L∞-to-Lip regularisation) For u ∈ L∞(µ) and any t > 0,
+T Υ,µ
+t
+u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ
+t
+u
+and the following estimate holds:
+Lip¯dΥ( ˜T Υ,µ
+t
+u) ≤
+1
+√
+2t∥u∥L∞(µ) .
+Proof. We prove (a). By the relation between T Υ,µ
+r,t
+and T Υ(Br),µ·
+r
+t
+(·Br) in Prop. 4.12,
+there exists a measurable set Ωr
+sem ⊂ Υ with µ(Ωr
+sem) = 1 so that for every η ∈ Ωr
+sem
+T Υ,µ
+r,t (η) = T Υ(Br),µη
+r
+t
+(ηBr) .
+(5.1)
+Let u ∈ L∞(µ). Thanks to Lem. A.3, there exists Ωr
+∞ ⊂ Υ so that µ(Ωr
+∞) = 1
+and
+uη
+r ∈ L∞(µη
+r),
+∀η ∈ Ωr
+∞,
+∀r ∈ N .
+By Prop. 3.5, there exists a measurable set Ωr
+rcd ⊂ Υ so that µ(Ωr
+rcd) = 1 and
+(Υk, dΥ, µη
+r) is RCD(0, ∞) with k = k(η) as in (3.4) for every η ∈ Ωr
+rcd.
+Let Ωr := Ωr
+sem ∩ Ωr
+∞ ∩ Ωr
+rcd. As the log-Harnack inequality holds in RCD spaces
+(see, [AGS15, Lem. 4.6]), the following holds for every η ∈ Ωr and k = k(η)
+T Υk(Br),µk,η
+r
+t
+(log uη
+r)(γ) ≤ log(T Υk(Br),µk,η
+r
+t
+uη
+r)(ζ) + dΥ(γ, ζ)2 ,
+∀ γ, ζ ∈ Υk(Br) .
+(5.2)
+
+26
+K. SUZUKI
+Noting the convergence of the semigroups {T Υ,µ
+r,t }t≥0 to {T Υ,µ
+t
+}t≥0 in the L2(µ)-
+operator sense by Prop. 4.16, there exist Ωcon ⊂ Υ with µ(Ωcon) = 1 and a (non-
+relabelled) subsequence of {r} so that for every γ ∈ Ωcon
+T Υ,µ
+r,t (log u)(γ)
+r→∞
+−−−→ T Υ,µ
+t
+(log u)(γ) ,
+log(T Υ,µ
+r,t u)(γ)
+r→∞
+−−−→ log(T Υ,µ
+t
+u)(γ) .
+(5.3)
+Let Ω′ = Ωcon ∩r∈N Ωr, which by construction satisfies µ(Ω′) = 1. Our goal is now
+to prove that there exists Ω ⊂ Ω′ with µ(Ω) = 1 so that
+T Υ,µ
+t
+(log u)(γ) ≤ log(T Υ,µ
+t
+u)(η) + ¯dΥ(γ, η)2 ,
+every γ, η ∈ Ω .
+(5.4)
+Thanks to (5.3), Formula (5.4) comes down to the corresponding inequality for the
+semigroup {T Υ,µ
+r,t }t≥0 for any r > 0:
+T Υ,µ
+r,t (log u)(γ) ≤ log(T Υ,µ
+r,t u)(η) + ¯dΥ(γ, η)2 ,
+every γ, η ∈ Ω .
+(5.5)
+We prove (5.5) by contradiction. Suppose that for any Ω ⊂ Ω′ with µ(Ω) = 1,
+there exists γ, η ∈ Ω so that
+T Υ,µ
+r,t (log u)(γ) ≥ log(T Υ,µ
+r,t u)(η) + ¯dΥ(γ, η)2 .
+(5.6)
+We may assume that ¯dΥ(γ, η) < ∞, otherwise, we have nothing to prove. Thus,
+by (2.15), there exists r > 0 so that
+γBcr = ηBcr ,
+γ(Br) = η(Br) .
+(5.7)
+By making use of (5.1), (5.2), (5.7), we obtain
+T Υ,µ
+r,t (log u)(γ) = T Υ,µ
+r,t (log u)(γBr + γBcr)
+(5.8)
+= T Υ(Br),µγ
+r
+t
+(log uγ
+r)(γBr)
+≤ log(T Υ(Br),µγ
+r
+t
+u)(ηBr) + dΥ(γBr, ηBr)2
+= log(T Υ,µ
+r,t u)(η) + ¯dΥ(γ, η)2 ,
+which contradicts (5.6), therefore, the proof of (a) is completed.
+The proof of (b) follows precisely in the same strategy as above by replacing
+T Υ,µ
+t
+(log u), log(T Υ,µ
+t
+u) and ¯dΥ(γ, η)2 by (T Υ,µ
+t
+u)α, T Υ,µ
+t
+uα and
+α
+2(α−1)¯dΥ(γ, η)2 re-
+spectively, and noting that the dimension-free Harnack inequality holds on RCD(K, ∞)
+spaces ([Li15, Thm. 3.1]).
+The proof of (c): Note that uη
+r ∈ Lip(Υ(Br), dΥ) whenever u ∈ Lip(Υ, ¯dΥ) and
+LipdΥ(uη
+r) ≤ Lip¯dΥ(u) by Lem. 4.6. Note also that the sought conclusion of (c) can
+be rephrased as
+˜T Υ,µ
+t
+u(γ) − ˜T Υ,µ
+t
+u(η) ≤ Lip¯dΥ(u)¯dΥ(γ, η)
+∀γ, η ∈ Υ .
+Thus, by the same proof strategy as in (a) replacing T Υ,µ
+t
+(log u)(γ) and log(T Υ,µ
+t
+u)(η)
+with T Υ,µ
+t
+u(γ) and T Υ,µ
+t
+u(η), and noting that the Lipschitz contraction property
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+27
+holds on RCD spaces ([AGS14b, (iv) in Thm. 6.1]), we conclude that there exists
+Ω ⊂ Υ with µ(Ω) = 1 so that
+T Υ,µ
+t
+(γ) − T Υ,µ
+t
+(η) ≤ Lip¯dΥ(u)¯dΥ(γ, η)
+∀γ, η ∈ Ω .
+The conclusion now follows from the McShane extension Theorem [DS21b, Lem. 2.1].
+The proof of (d) is the same as that of (c) but using the L∞-to-Lip property
+([AGS14b, Thm. 6.5]) in RCD(K, ∞) spaces instead of [AGS14b, (iv) in Thm. 6.1]).
+The proof is complete.
+■
+6. Generalisation
+We have been so far working in the case of sineβ ensemble. In this section, we
+seek to generalise the aforementioned statements to general probability measures
+on Υ = Υ(Rn) for n ∈ N. In this section, we denote by m and mr the Lebesgue
+measure on Rn and its restriction on Br(0) respectively, and we take the Euclidean
+distance d(x, y) := |x − y| for x, y ∈ Rn. Let µ be a Borel probability on Υ and
+assume that it is fully supported on Υ with respect to the vague topology τv. Let
+K(µη
+r) ⊂ N0 be defined as
+K(µη
+r) := {k ∈ N0 : µk,η
+r (Υk(Br)) > 0} .
+Assumption 6.1. Let K ∈ R and µ be a fully supported Borel probability with
+respect to the vague topolgoy τv. Assume the following conditions:
+(a) the measure µη
+r is absolutely continuous with respect to the Poisson mea-
+sure πmr, and µk,η
+r
+is equivalent to πmr|Υk(Br) for any k ∈ K(µη
+r), µ-a.e. η and
+any r > 0;
+(b) the density
+dµk,η
+r
+dπmr|Υk(Br)
+is τv-continuous on Υk(Br), and the logarithmic density
+Ψk,η
+r
+= − log
+�
+dµk,η
+r
+dπmr|Υk(Br)
+�
+is K-geodesically convex with respect to dΥ on Υk(Br) for any k ∈ K(µη
+r),
+µ-a.e. η and any r > 0.
+Under (a) in Assumption, the local Dirichlet form (EΥ,µ, D(EΥ,µ)) is constructed
+in the same proof as in the case of sineβ ensemble as we have not use any partic-
+ular property of K = 0. We further show the synthetic curvature bound for the
+form (EΥ,µ, D(EΥ,µ)) and related functional inequalities.
+Theorem 6.2. Suppose that µ satisfies Assumption 6.1. Then the form (EΥ,µ, D(EΥ,µ))
+satisfies
+
+28
+K. SUZUKI
+(a) (Bakry–Émery inequality BE1(K, ∞))
+ΓΥ�
+T Υ,µ
+t
+u
+� 1
+2 ≤ e−KtT Υ,µ
+t
+�
+ΓΥ(u)
+1
+2�
+∀u ∈ D(EΥ,µ) ;
+(b) (lntegral Bochner inequality) for every (u, ϕ) ∈ D(ΓΥ,µ
+2
+)
+ΓΥ,µ
+2
+(u, ϕ) ≥ K
+�
+Υ
+ΓΥ(u)ϕ dµ ;
+(c) (local Poincaré inequality) for u ∈ D(EΥ,µ) and t > 0,
+T Υ,µ
+t
+u2 − (T Υ,µ
+t
+u)2 ≤ 1 − e−2Kt
+K
+T Υ,µ
+t
+ΓΥ(u) ,
+T Υ,µ
+t
+u2 − (T Υ,µ
+t
+u)2 ≥ e−2Kt − 1
+K
+ΓΥ(T Υ,µ
+t
+u) ;
+(d) (local logarithmic Sobolev inequality) for non-negative u ∈ D(EΥ,µ)
+and t > 0,
+T Υ,µ
+t
+u log u − T Υ,µ
+t
+u log T Υ,µ
+t
+u ≤ 1 − e−2Kt
+2K
+T Υ,µ
+t
+�ΓΥ(u)
+u
+�
+,
+T Υ,µ
+t
+u log u − T Υ,µ
+t
+u log T Υ,µ
+t
+u ≥ e−2Kt − 1
+2K
+ΓΥ(T Υ,µ
+t
+u)
+T Υ,µ
+t
+u
+.
+(e) (log Harnack inequality) for every non-negative u ∈ L∞(Υ, µ), t > 0,
+there exists Ω ⊂ Υ so that µ(Ω) = 1 and
+T Υ,µ
+t
+(log u)(γ) ≤ log(T Υ,µ
+t
+u)(η) +
+K
+2(1 − e−2Kt)
+¯dΥ(γ, η)2 ,
+∀γ, η ∈ Ω ;
+(f) (dimension-free Harnack inequality) for every non-negative u ∈ L∞(Υ, µ),
+t > 0 and α > 1 there exists Ω ⊂ Υ so that µ(Ω) = 1 and
+(T Υ,µ
+t
+u)α(γ) ≤ T Υ,µ
+t
+uα(η) exp
+�
+αK
+2(α − 1)(1 − e−2Kt)
+¯dΥ(γ, η)2�
+,
+∀γ, η ∈ Ω ;
+(g) (Lipschitz contraction) For u ∈ Lip(¯dΥ, µ) and t > 0,
+T Υ,µ
+t
+u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ
+t
+u
+and the following estimate holds:
+Lip¯dΥ( ˜T Υ,µ
+t
+u) ≤ e−KtLip¯dΥ(u) ;
+(h) (L∞-to-Lip regularisation) For u ∈ L∞(µ) and t > 0,
+T Υ,µ
+t
+u has a ¯dΥ-Lipschitz µ-modification ˜T Υ,µ
+t
+u
+and the following estimate holds:
+Lip¯dΥ( ˜T Υ,µ
+t
+u) ≤
+1
+�
+2I2K(t)
+∥u∥L∞(µ)
+∀t > 0 ,
+where IK(t) :=
+� t
+0 eKr dr.
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+29
+Proof. Thanks to Assumption 6.1, the space (Υk(Br), dΥ, µk,η
+r ) satisfies RCD(K, ∞)
+for every k ∈ K(µη
+r) as in the same proof of Prop. 3.4. As we have not used any
+particular properties of K = 0 for the proofs in the case of sineβ, the completely
+same proofs (up to constant multiplication depending only on K) work in the case of
+general K ∈ R and general µ satisfying Assumption 6.1, which therefore concludes
+the statements of Thm. 6.2.
+■
+Appendix A.
+Let m and mr be the Lebesgue measure on Rn and its restriction on Br respectively.
+Set Υ = Υ(Rn).
+Lemma A.1. Let µ be a Borel probability on Υ satisfying that µη
+r is absolutely
+continuous with respect to the Poisson measure πmr for any r > 0 and µ-a.e. η. Let
+Σ ⊂ Br so that mr(Σc) = 0. Let Ω(r) := {γ ∈ Υ : γΣ = γBr}. Then,
+µ
+�
+Ω(r)
+�
+= 1
+∀r > 0 .
+Proof. We fix r > 0 and write simply Ω = Ω(r). By the disintegration formula (2.10),
+µ(Ω) =
+�
+Υ
+µη
+r(Ωη
+r) dµ(η) .
+Thus, it suffices to show µη
+r(Ωη
+r) = 1 for µ-a.e. η. This is equivalent to show
+µη
+r(Ωη
+r) =
+�
+k∈N0
+µk,η
+r (Ωη
+r) = 1 .
+(A.1)
+As µk,η
+r
+is absolutely continuous with respect to πmr|Υk(Br), it suffices to prove
+πmr|Υk(Br)((Ωη
+r)c) = 0 for every k ∈ N0 and η ∈ Υ.
+We show that (recall the definition of symmetric product Σ⊙k in (2.3))
+Σ⊙k ⊂ Ωη
+r ∩ Υk(Br)
+∀η ∈ Υ .
+(A.2)
+Let γ ∈ Σ⊙k. Then by the definition of Ω, it holds that γ + ηBcr ∈ Ω for any η ∈ Υ.
+Thus, by recalling the definition (2.9) of Ωη
+r, we obtain γ ∈ Ωη
+r ∩ Υk(Br). Thus,
+(A.2) holds true.
+By using (A.2), πmr|Υk(Br) = e−mr(Br)m⊙k
+r
+by (2.11) and m⊙k
+r
+�
+(Σ⊙k)c�
+= 0 by
+hypothesis, we conclude that for every η ∈ Υ
+πmr|Υk(Br)((Ωη
+r)c) = e−mr(Br)m⊙k
+r
+��
+Ωη
+r ∩ Υk(Br)
+�c�
+≤ e−mr(Br)m⊙k
+r
+�
+(Σ⊙k)c�
+= 0 .
+The proof is complete.
+■
+We recall that for η ∈ Υ, we set Υη
+r := {γ ∈ Υ : γBcr = ηBcr}.
+Lemma A.2 (disintegration lemma). Assume that there exists a measurable set
+Ξ ⊂ Υ with µ(Ξ) = 1 so that for every η ∈ Ξ, there exists a family of measurable
+
+30
+K. SUZUKI
+sets Ωη ⊂ Υ(Br) so that µη
+r(Ωη) = 1 for every η ∈ Ξ. Let Ω ⊂ Υ be the (not
+necessarily measurable) subset defined by
+Ω :=
+�
+η∈Ξ
+pr−1
+r (Ωη) ∩ Υη
+r .
+Assume further that there exists a measurable set Θ ⊂ Υ so that Ω ⊂ Θ. Then,
+µ(Θ) = 1.
+Caveat.
+As the set Ω is defined as uncountable union of measurable sets, the mea-
+surability of Ω is not necessarily true in general. The disintegration formula (2.10)
+is, therefore, not necessarily applicable directly to Ω, which motivates the aforemen-
+tioned lemma.
+Proof of Lem. A.2. Let Θη
+r = {γ ∈ Υ(Br) : γ + ηBcr ∈ Θ} be a section of Θ at ηBcr
+as in (2.9). Then, Ωη ⊂ Θη
+r by assumption. Thus, µη
+r(Θη
+r) ≥ µη
+r(Ωη) ≥ 1. By the
+disintegration formula in (2.10), we have that
+µ(Θ) =
+�
+Υ
+µη
+r(Θη
+r) dµ(η) ≥ 1 .
+The proof is completed.
+■
+Lemma A.3. Let µ be a Borel probability on Υ. Let Ω ⊂ Υ satisfy µ(Ω) = 1.
+Then, there exists Ω′ ⊂ Ω with µ(Ω′) = 1 and
+µη
+r(Ωη
+r) = 1 ,
+∀η ∈ Ω′ .
+(A.3)
+Proof. By the disintegration formula (2.10),
+1 = µ(Ω) =
+�
+Υ
+µη
+r(Ωη
+r) dµ(η) =
+�
+Ω
+µη
+r(Ωη
+r) dµ(η) ,
+by which the statement is readily concluded.
+■
+Lemma A.4. Let (Q, D(Q)) be a closed form on a complete separable Hilbert
+space H. Let {Tt} and (A, D(A)) be the corresponding semigroup and infinitesi-
+mal generator respectively. Suppose that there exists an algebra C ⊂ D(Q) so that
+C ⊂ H is dense and TtC ⊂ C for any t > 0. Then C is dense in D(Q).
+Proof. It holds that TtD(A) ⊂ D(A) by the general property of semigroups associ-
+ated with closed forms. Thus, combining it with the hypothesis TtC ⊂ C,
+Tt(C ∩ D(A)) ⊂ C ∩ D(A) .
+Thus, by [RS75, Thm. X.49], C ∩ D(A) is dense in the graph norm in the space
+(A, D(A)). Namely, we obtained
+(A, C ∩ D(A)) is essentially self-adjoint .
+
+CURVATURE BOUND OF DYSON BROWNIAN MOTION
+31
+The density C ⊂ D(Q) now follows by the density of C ∩ D(A) in the graph norm,
+by the density of D(A) ⊂ D(Q) due to the general property of closed forms, by the
+density of C ⊂ H and by a simple integration-by-parts argument
+−Q(u, u) = (Au, u)H ≤ ∥Au∥H∥u∥H .
+The proof is complete.
+■
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+

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+Hybrid Quantum-Classical Autoencoders for
+End-to-End Radio Communication
+Zsolt Tabi∗, Bence Bak´o∗†, D´aniel T. R. Nagy∗‡, P´eter Vaderna‡, Zs´ofia Kallus‡,
+P´eter H´aga‡, Zolt´an Zimbor´as∗†
+∗E¨otv¨os Lor´and University, Budapest, Hungary
+Email: [email protected]
+†Wigner Research Centre for Physics, Budapest, Hungary
+Email: [email protected], [email protected]
+‡Ericsson Research, Budapest, Hungary
+Email: {daniel.a.nagy, peter.vaderna, zsofia.kallus, peter.haga}@ericsson.com
+Abstract—Quantum neural networks are emerging as poten-
+tial candidates to leverage noisy quantum processing units for
+applications. Here we introduce hybrid quantum-classical au-
+toencoders for end-to-end radio communication. In the physical
+layer of classical wireless systems, we study the performance
+of simulated architectures for standard encoded radio signals
+over a noisy channel. We implement a hybrid model, where
+a quantum decoder in the receiver works with a classical
+encoder in the transmitter part. Besides learning a latent space
+representation of the input symbols with good robustness against
+signal degradation, a generalized data re-uploading scheme for
+the qubit-based circuits allows to meet inference-time constraints
+of the application.
+Index Terms—variational quantum algorithms, quantum ma-
+chine learning, quantum autoencoder, radio communication
+I. INTRODUCTION
+One of the most popular Quantum Machine Learning
+(QML) methods are Quantum Neural Networks (QNNs) [1],
+[2]. These are special variational quantum circuits, designed as
+the quantum analogues of classical neural networks. QNNs can
+be optimized with gradient-based or gradient-free optimization
+algorithms forming hybrid quantum-classical training loops
+[3], [4]. Various QNN architectures have been proposed such
+as quantum convolutional neural networks [5], generative mod-
+els [6], long short-term memories [7] and autoencoders [8]–
+[10]. Beside the high activity in algorithmic research within
+QML, their novel benchmarking and requirement setting ap-
+plications are also motivating a wide variety of works [11].
+Although QML is still in a phase of basic research with
+many open questions, its early implementations in wireless
+communication systems spark both scientific curiosity and
+commercial interest [12]. However, high-performance, near
+real-time applications might impose a new set of requirements
+on these solutions.
+Wireless communication has undergone tremendous evolu-
+tion during the last decades. The increasing adoption of AI
+and ML methods is opening up new development possibilities
+in various parts of the radio stack. In the design of the sixth
+generation (6G) wireless networks, AI and ML technologies
+are considered to be tightly integrated into the system and
+smart algorithms can be applied to all aspects of network
+operations and procedures [13]. Considering the improvement
+of quantum computers it is envisioned that quantum algorithms
+and especially QML will play a significant role in future
+networks [12].
+The structure of this paper is as follows. In Sec. II, we
+give a high-level overview of wireless communication systems
+together with an autoencoder solution used in the radio phys-
+ical layer. Our novel hybrid classical-quantum autoencoder
+prototype is presented in Sec. III. We discuss our result in
+Sec. IV. Finally, we conclude with an outlook in Sec. V.
+II. AUTOENCODER ARCHITECTURE IN END-TO-END
+COMMUNICATION
+I
+Q
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+(a)
+I
+Q
+(b)
+Fig. 1: 16-QAM constellation diagrams. (a) Transmitted
+symbols; (b) Received noisy signal.
+s
+Transmitter
+Channel
+Receiver
+ˆs
+x
+y
+Fig. 2: High-level representation of a communication sys-
+tem. A message is transmitted through a noisy communication
+channel to be recovered by the receiver.
+The components of communication networks are organized
+in a layered architecture where each layer is responsible for
+different communication aspects [14]. The physical layer is the
+arXiv:2301.02609v1  [quant-ph]  6 Jan 2023
+
+Fig. 3: The proposed hybrid quantum-classical autoencoder embedded into the end-to-end communication architecture.
+The transmitter maps each message s to a symbol, then sends it through the channel. The channel is represented by an
+additive noise acting on the signal x. The receiver, realized by a quantum decoder, consists of a multi-layer QNN adapting
+data re-uploading. It decodes the noisy signal and gives an estimate of the original message.
+lowest layer. It provides the means of transmitting a stream of
+raw bits over a data channel connecting the network elements.
+The physical layer on the transmitter side converts data in
+the form of bits to electromagnetic waves to be transmitted
+wirelessly, while the receiver converts the electromagnetic
+waves received by an antenna to binary data. The main
+challenge in wireless data transmission is to overcome the
+channel impairments so that the messages can be recovered
+with small error rate.
+The role of modulation is to convert digital data into radio
+waves. It can be achieved in different ways, the informa-
+tion can be encoded by varying (shifting) either amplitude,
+frequency or phase of the electromagnetic wave. The more
+states the modulation has, the more bits are transferred in one
+symbol, resulting in higher data rate. However, with higher
+order modulation the signal is more sensitive to channel errors,
+so the applied modulation usually depends on the channel
+quality. Fig. 1 shows a constellation diagram of the symbol
+representation of 16-QAM modulation, where 4-bit strings
+can be represented as complex numbers in a scheme resistant
+to general noise patterns while achieving high data rate with
+minimal channel uses.
+Based on [15], we model a simple communications system,
+shown in Fig. 2, consisting of a transmitter that modulates
+message s ∈ M = {0, . . . , M − 1} into a signal x and
+sends it over a noisy channel to the receiver that tries to
+decode the received signal y resulting in the received message
+ˆs. The transmitted signal x suffers degradation due to the
+noise present on the channel. In case of transmission over a
+complex channel with n discrete channel uses, the transmitter
+can be represented as the transformation f : M �→ R2n,
+mapping the message s to x ∈ R2n signal with certain
+constraints imposed by the transmitting hardware (e.g., energy
+constraint or average power constraint). The channel can be
+modeled as a conditional probability density function p(y|x)
+that produces the output signal y ∈ R2n given the input signal,
+usually via some noise model (e.g., additive white Gaussian
+noise (AWGN)). The receiver is represented as the mapping
+g : R2n �→ M that recovers some estimate ˆs of the original
+message from the received signal. In this work, we focus on
+the case of single channel use (n = 1), however, this model
+can be easily adopted to cases of n > 1. Also, the number of
+transmit-receive pairs can be increased to get a Multiple-Input
+and Multiple-Output (MIMO) system [15], [16].
+Both of f and g transformations can be created in various
+ways. In case of simple noise models applied in the channel
+the transformations can be designed as explicit mathematical
+formulas. In the case of complex noise scenarios that are
+difficult to describe with mathematical models, a possible way
+is to train deep neural networks, especially autoencoders, to
+solve the encoding and decoding tasks [15].
+An autoencoder is a special type of deep neural network,
+with the aim to compress or denoise data [17]–[19]. Autoen-
+coders consist of an encoder function f : RD �→ RL, and
+a decoder function g : RL �→ RD. The encoder transforms
+its input χ into a latent space representation f(χ) ∈ RL,
+whereas the decoder tries to reconstruct it: ˆχ = g(f(χ)).
+Usually we have L < D, i.e., the encoder produces a compact
+representation of the data. f and g are typically deep neural
+networks trained jointly to minimize a loss function of the
+form L(χ, g(f(χ)).
+In telecommunication, opposite to the general compressing
+and denoising interpretation, autoencoders can be effectively
+used in the presented communication system to learn how to
+represent input messages as signals [15]. This model differs
+from the “typical” autoencoder concept in the sense that it does
+not try to remove noise from the input, instead it learns how
+to represent the input in a way that is robust against a given
+noisy channel acting in the latent space of the autoencoder.
+As a result of the training process, the latent space (or hidden
+layer) of the autoencoder contains the learned constellation of
+symbols (or codebook). The learned constellation is optimized
+for best mapping of the input messages to signals that can be
+accurately decoded with the largest success probability for the
+
+Quantum Decoder
+Encoder
+cod
+Noise model p(y|x)
+argmax
+Message vector
+Qubit encoding
+Qubit encoding
+Normalization
+ayer
+0.1
+abit readou
+La
+·
+0.8
+3
+0.05
+Transmitter
+Receiver
+hannelEncoding
+First Layer
+· · ·
+· · ·
+· · ·
+· · ·
+|0⟩
+Rx(y1)
+R(α1, β1, γ1)
+|0⟩
+Rx(y2)
+R(α2, β2, γ2)
+|0⟩
+R(α3, β3, γ3)
+|0⟩
+R(α4, β4, γ4)
+(a)
+First Layer with Encoding
+· · ·
+· · ·
+· · ·
+· · ·
+|0⟩
+Rx(y1)
+R(α1, β1, γ1)
+|0⟩
+Rx(y2)
+R(α2, β2, γ2)
+|0⟩
+R(α3, β3, γ3)
+|0⟩
+R(α4, β4, γ4)
+(b)
+First Layer with Double Encoding
+· · ·
+· · ·
+· · ·
+· · ·
+|0⟩
+Rx(y1)
+R(α1, β1, γ1)
+|0⟩
+Rx(y2)
+R(α2, β2, γ2)
+|0⟩
+Rx(y1)
+R(α3, β3, γ3)
+|0⟩
+Rx(y2)
+R(α4, β4, γ4)
+(c)
+First Layer with Weighted Double Encoding
+· · ·
+· · ·
+· · ·
+· · ·
+|0⟩
+Rx(w1 · y1)
+R(α1, β1, γ1)
+|0⟩
+Rx(w2 · y2)
+R(α2, β2, γ2)
+|0⟩
+Rx(w3 · y1)
+R(α3, β3, γ3)
+|0⟩
+Rx(w4 · y2)
+R(α4, β4, γ4)
+(d)
+Fig. 4: Quantum decoder implementations with encoding schemes. Ansatz circuits with (a) simple data encoding; (b) simple
+data re-uploading; (c) double data re-uploading; (d) weighted double data re-uploading.
+specific channel model. Whereas the encoder learns how to
+produce optimal symbols, the receiver learns how to decode
+these symbols after they have been corrupted by the channel,
+i.e., how to recover x after sampling from p(y|x).
+III. HYBRID QUANTUM AUTOENCODER FOR RADIO
+PHYSICAL LAYER
+A. Hybrid quantum autoencoder overview
+Quantum autoencoder architectures have previously been
+proposed to compress as well as denoise quantum data [8],
+[10]. Hybrid quantum-classical autoencoders enable many
+variations for quantum or classical encoding/decoding or the
+use of classical data. In this work, a hybrid quantum-classical
+autoencoder is applied for processing classical information.
+Building on the physical layer autoencoder presented in
+Sec. II, we propose a hybrid quantum-classical autoencoder
+with classical encoder on the transmitter side and a quantum
+decoder on the receiver side – trained in an end-to-end
+solution. The encoder projects the original message to a lower
+dimensional representation, robust to the channel degradation
+effect. Once the signal is passed to the quantum decoder, the
+compressed information is mapped to a higher dimensional
+Hilbert space of the qubits by a QNN that has been previously
+shown to be efficient for classification tasks [20], [21].
+In our model, the classical encoder consists of an embedding
+followed by a normalization. A simple linear embedding is
+used to produce the constellation, satisfying the average power
+constraint by normalization. The decoder is realized by a
+general strongly connected quantum neural network which
+we refer to as a quantum decoder. By simulating increasing
+levels of noise in the channel, we can present a performance
+evaluation of the various neural network architectures.
+B. Quantum decoder architectures
+A general QNN architecture has three main components
+as shown in Fig. 3: qubit encoding for embedding the input
+data, the parameterized QNN layers, and the qubit readout
+given as a probability distribution over the possible con-
+stellation symbols obtained from suitable measurements with
+high enough number of shots. To encode the output of the
+channel, we choose angle embedding with parameterized Rx
+rotations [22]. With this embedding, there are multiple ways
+to encode two-dimensional feature vectors into four qubits. As
+for the variational ansatz, we use strongly entangling layers
+introduced in quantum classifiers as they are known to be
+expressive reaching ‘wide corners of the Hilbert space’ [23].
+The measurements are performed in the computational basis
+and the obtained probability distribution over the 16 basis
+states is the output of the decoder.
+The simplest single-layer realization of such a QNN struc-
+ture is presented in Fig. 4a. To improve this ansatz, we can
+apply the data re-uploading trick recently introduced in [24].
+This technique, as shown in Fig. 4b, repeats the input encoding
+block before each layer of the QNN circuit. The intuition be-
+hind the effectiveness of this method is that by re-introducing
+the input before each layer, one can mimic the computational
+structure of typical classical deep neural networks, where the
+copying of the classical information is readily available, which
+
+would be, without this trick, prohibited by the no-cloning
+theorem in quantum machine learning. The expressivity of a
+model can be further increased by applying the encoding on
+different subsystems in parallel [25]. With this in mind, we
+further enhance the ansatz by encoding the first feature into
+both qubit no. 1 and no. 3 and the second input feature into
+both qubit no. 2 and no. 4. This double data re-uploading
+ansatz is presented in Fig. 4c. As a final improvement, we
+considered the role of the number of trainable parameters. As
+the expressive power of the ansatz is highly dependent on the
+number of trainable parameters, one should try to include as
+many parameters as possible. One way to increase the number
+of parameters while keeping the circuit as shallow as possible
+– to respect the limited hardware capabilities and the inference
+time constraints of the application – is to introduce trainable
+weights in the data re-uploading blocks, as shown in Fig. 4d.
+This modification keeps the depth constant.
+C. Training and fine-tuning
+For our hybrid autoencoder to achieve low estimation errors,
+the training of the end-to-end system requires to be further
+improved via hyper-parameter tuning.
+First, the training of the hybrid model is done on batches
+uniformly sampled from the set of messages {0, . . . , 15}.
+These are sent as two dimensional encoded symbols through
+the AWGN channel with SNR of 15 dB and i.i.d. noise.
+The accuracy of the model is measured by evaluating
+the Symbol Error Rate (SER), a key performance indica-
+tor commonly used in radio communication. The network
+weight updates are calculated with the sparse categorical cross-
+entropy of the distribution generated by the decoder and the
+ground truth symbols. This loss function is used to calculate
+gradients in a mini-batch gradient descent with batch size of 64
+and Adam optimizer [26]. We simulate the hybrid autoencoder
+using PennyLane [27], a quantum machine learning framework
+with its TensorFlow [28] backend. Second, we evaluate the
+reached model accuracy at various hyper-parameter settings.
+The search is conducted by KerasTuner [29] after partitioning
+the space as the simulator compute times are prohibitive of a
+full grid search.
+We start by first evaluating the learning rate parameter set
+η ∈ {0.1, 0.01, 0.001} using the simple ansatz presented on
+Fig. 4a with L = 8 layers with 1000-shot measurements and
+1000 training steps. Based on these results, the only viable
+value of η = 0.1 is set for the rest of this study.
+We continue with evaluating modifications to the basic
+ansatz but keeping the number of layers L = 8 and 1000
+training steps fixed, to minimize the overall computation time.
+The results are shown in Fig. 5. For the basic circuit, the SER
+fluctuates around its initial value without showing convergence
+to a desirable level. A significant accuracy improvement of
+roughly 40% is achieved by implementing single data re-
+uploading (Fig. 4b with ansatz of 1×DR). Introducing the
+double data re-uploading layer (2×DR with ansatz of Fig. 4c)
+leads to another 15% improvement. Finally, we can even fur-
+ther increase the performance by another 20% when using the
+0
+200
+400
+600
+800
+1000
+Number of training steps
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+Symbol Error Rate
+basic
+1x DR
+2x DR
+2x wDR
+Fig. 5: Learning curves of circuit architectures. The number
+of data re-uploading (none, single, double) and the weighted
+data encoding have high impact on the convergence properties
+of the quantum autoencoder.
+weighted double data re-uploading technique (2×wDR with
+ansatz Fig. 4d). Based on these results, the weighted double
+data re-uploading ansatz is chosen for further experiments.
+As a last step, we optimize the number of layers. The hybrid
+autoencoder using the best performing ansatz is trained with
+8 to 24 layers. Increasing the number of layers clearly shows
+the improvement in SER as well as in convergence time as
+seen in Fig. 6.
+IV. PERFORMANCE EVALUATION
+A. Validation
+Comparing our hybrid architecture to the classical method is
+crucial to validate the solution. Based on the learning curves
+presented, the shallowest network reaching accuracy similar
+to the classical solution contains L = 16 layers. Further
+increasing the number of layers leads to small improvements
+in accuracy but it is suboptimal in terms of circuit depth.
+Although the hybrid quantum autoencoder models are
+trained at SNR of 15 dB we further validate the results at
+different values. The evaluation is shown in Fig. 7. We see
+that the trained networks generalize well on previously unseen
+SNR values, and reach performance similar to the classical
+baseline.
+In Fig. 8, the constellation diagrams produced by autoen-
+coders having different numbers of layers are shown. If the
+trained autoencoder has good performance, it is expected that
+the symbols are uniformly distributed in the diagram, similarly
+to Fig 1. We see that increasing the number of layers leads to
+a more balanced distribution of symbols in the Q − I space,
+which implies that the symbols can be well separated in case
+of noisy channels.
+B. Time characteristics
+In radio telecommunication, the latency of the data trans-
+mission is also an important performance metric. In some use
+
+0
+200
+400
+600
+800
+1000
+Number of training steps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Symbol Error Rate
+L=8
+L=12
+L=16
+L=24
+classical
+Fig. 6: Learning curves of classical and hybrid autoen-
+coders for a set of layer numbers. We find that that
+the minimal number of layers necessary to achieve results
+comparable to the classical baseline is 16. Throughout these
+tests, we used ansatz according to Fig. 4d.
+cases it is even critical that the end-to-end delay falls below a
+certain threshold. In 5G networks, it is possible to achieve
+ms level latency. Hence, in addition to the accuracy it is
+inevitable to investigate the time characteristics of the autoen-
+coder model. After transpiling [30] the circuit ansatz to IBM
+QPU backend ibmq_belem and ibmq_santiago [31] and
+constructing the pulse-level scheduling, we can calculate the
+theoretical execution times on both QPUs. The transpiled
+circuits are deeper than the original ansatz, because we need
+SWAP gates due to limited qubit connectivity and the basis
+gate-set of the device can differ from the one used in Fig 4.
+In Table I, we present the circuit depth and the approximate
+per shot execution times of quantum decoders depending on
+the number of layers. The time values in the table suggest
+the following feasibility considerations for running QNN in a
+real-time system. The number of shots highly determines the
+reliability of the result of the inference. When the quantum
+decoder is executed with 1000 shots (a level already acceptable
+in current systems for this problem size), the inference time
+is the order of magnitude of 100ms which is higher than the
+accepted level in real-time radio systems. However, this can
+be reduced to the accepted level of below 10ms because the
+probability distribution is expected to be highly peaked for
+well-trained autoencoders.
+V. CONCLUSION AND OUTLOOK
+We presented a novel hybrid implementation of a quantum-
+classical autoencoder for end-to-end radio communication.
+The decoder was implemented as a variational quantum circuit.
+We showed that the use of advanced double re-uploading
+encoding schemes allows for the inference-time constraints of
+the application to be met without losing accuracy required
+from the autoencoder.
+By implementing a combination of parallel encodings and
+weighted data re-uploading, we showed how these schemes
+5
+0
+5
+10
+15
+20
+Signal-to-noise Ratio [dB]
+10
+2
+10
+1
+10
+0
+Symbol Error Rate
+L=8
+L=12
+L=16
+L=24
+classical
+Fig. 7: Validation of inference accuracy of the trained
+classical and hybrid autoencoders. With increasing SNR
+values, hybrid models generalize to validation data on par with
+the classical.
+I
+Q
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+(a)
+I
+Q
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+(b)
+Fig. 8: Constellations (latent space representations) learned
+by the hybrid autoencoder trained with SNR=15. (a) L = 8
+layers (b) L = 24 layers.
+TABLE I: Estimated execution times of the quantum
+decoder. The circuit was run on the ibmq_belem and
+ibmq_santiago depending on the number of layers, cal-
+culated with Qiskit’s transpiler.
+ibmq_belem
+ibmq_santiago
+# layers
+depth
+time [µs/shot]
+depth
+time [µs/shot]
+8
+125
+54.3
+145
+30.4
+12
+187
+78.4
+221
+43.6
+16
+260
+111.8
+297
+56.9
+20
+311
+124.2
+373
+70.12
+24
+379
+149.8
+449
+83.4
+can improve not just the QNN expressivity but also the
+performance of the whole autoencoder model. We expect these
+quantum-enhanced models to outperform classical ones in
+more complex channel noise scenarios, a direction for future
+study.
+
+ACKNOWLEDGMENT
+Zsolt Tabi and Zimbor´as Zolt´an would like to thank the
+support of the Hungarian National Research, Development and
+Innovation Office (NKFIH) through the Quantum Information
+National Laboratory of Hungary and through the Grants No.
+FK 135220, K124351 and TKP2021-NVA-29.
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+MORPHOLOGY-BASED NON-RIGID REGISTRATION OF
+CORONARY COMPUTED TOMOGRAPHY AND INTRAVASCULAR
+IMAGES THROUGH VIRTUAL CATHETER PATH OPTIMIZATION
+Karim Kadry∗
+Institute of Medical Engineering and Science
+Massachusetts Institute of Technology
+Cambridge, MA 02139
+[email protected]
+Abhishek Karmakar
+Meinig School of Biomedical Engineering
+Cornell University
+Ithaca, NY 14850
+[email protected]
+Andreas Schuh
+Biomedical Image Analysis Group
+Imperial College London
+HeartFlow, Inc., USA
+London, UK
+[email protected]
+Kersten Peterson
+HeartFlow, Inc., USA
+Redwood City, CA, 94063, USA
+[email protected]
+Michiel Schaap
+HeartFlow, Inc., USA
+Redwood City, CA, 94063, USA
+[email protected]
+David Marlevi
+Department of Molecular Medicine and Surgery
+Karolinska Institute
+Stockholm, Sweden
+[email protected]
+Charles Taylor
+Department of Electrical Engineering
+HeartFlow, Inc., USA
+Redwood City, CA, 94063, USA
+[email protected]
+Elazer Edelman
+Institute of Medical Engineering and Science
+Massachusetts Institute of Technology
+Cambridge, MA 02139
+[email protected]
+Farhad Nezami
+Department of Surgery
+Brigham and Women’s Hospital Harvard Medical School
+Boston, MA 02115
+[email protected]
+ABSTRACT
+Coronary Computed Tomography Angiography (CCTA) provides information on the presence, extent,
+and severity of obstructive coronary artery disease. Large-scale clinical studies analyzing CCTA-
+derived metrics typically require ground-truth validation in the form of high-fidelity 3D intravascular
+imaging. However, manual rigid alignment of intravascular images to corresponding CCTA images is
+both time consuming and user-dependent. Moreover, intravascular modalities suffer from several
+non-rigid motion-induced distortions arising from distortions in the imaging catheter path. To address
+these issues, we here present a semi-automatic segmentation-based framework for both rigid and
+non-rigid matching of intravascular images to CCTA images. We formulate the problem in terms
+of finding the optimal virtual catheter path that samples the CCTA data to recapitulate the coronary
+artery morphology found in the intravascular image. We validate our co-registration framework on a
+cohort of n = 40 patients using bifurcation landmarks as ground truth for longitudinal and rotational
+registration. Our results indicate that our non-rigid registration significantly outperforms other co-
+registration approaches for luminal bifurcation alignment in both longitudinal (mean mismatch: 3.3
+frames) and rotational directions (mean mismatch: 28.6 degrees). By providing a differentiable
+framework for automatic multi-modal intravascular data fusion, our developed co-registration modules
+∗Corresponding Author
+arXiv:2301.00060v1  [cs.CV]  30 Dec 2022
+
+arXiv Template
+A PREPRINT
+significantly reduces the manual effort required to conduct large-scale multi-modal clinical studies
+while also providing a solid foundation for the development of machine learning-based co-registration
+approaches.
+1
+Introduction
+Coronary computed tomography angiography (CCTA) is a three dimensional image modality that provides information
+on the presence, extent and severity of obstructive coronary artery disease (CAD) (Tzimas et al. [2022]). As such, CCTA
+allows for the detection of stenotic atherosclerotic sections and assists clinicians in diagnosing CAD and planning
+treatment. CCTA Images can also be used to create computational models of coronary blood flow, allowing for
+the non-invasive estimation of fractional flow reserve (FFR-CT); a key diagnostic parameter in assessing functional
+impairment (Uzu et al. [2019]).
+Albeit widespread in use, CCTA provides primary information on luminal anatomy, with limited capacity in assessing
+soft-tissue intraplaque tissue components. CCTA also suffers from blooming artifacts in the presence of highly
+attenuating calcium deposits (Kim et al. [2015], Budoff et al. [2008]), which, combined with comparably low image
+resolution, creates difficulties in resolving highly calcified arteries. Multiple studies have also been conducted to
+quantify the degree to which CCTA can accurately assess CAD-related diagnostic metrics such as luminal area (Uzu
+et al. [2019]), calcium morphology (Takahashi et al. [2021]), and plaque burden (Fischer et al. [2013], De Graaf et al.
+[2013], Brodoefel et al. [2009]). The majority of such studies (Takahashi et al. [2021], Fischer et al. [2013], Uzu et al.
+[2019], Brodoefel et al. [2009]) validate the performance of CCTA by manually co-registering image slices taken along
+the CCTA artery to intravascular imaging modalities such as intravascular ultrasound (IVUS) and optical coherence
+tomography (OCT); both providing higher-fidelity visualization of the lumen and surrounding tissue. There is also an
+increasing interest in validating CCTA-derived segmentation algorithms against co-registered intravascular imaging
+frames, again necessitating such multimodal image assessment (Lin et al. [2021], van Assen et al. [2019]).
+Manual co-registeration of CCTA and intravascular images is, however, a challenging and time consuming task.
+Typically, cross-sectional frames of the artery of interest are extracted from the CCTA images which then have to be
+matched with corresponding frames from an intravascular acquisition through an imaging catheter pullback procedure.
+Rigid registration in the longitudinal and rotational directions is usually achieved by matching single landmarks in both
+modalities, such as a large bifurcation (Takahashi et al. [2021]). However, the beating of the heart, the irregular motion
+of the imaging catheter, and the rotation of the catheter about its own axis create non-rigid distortions that accumulate
+along the length of the pullback (Tsiknakis et al. [2021]). Manually correcting for such artifacts is prohibitively
+time-consuming, requiring a cardiologist to manually mark fiduciary points in both images and shift images such that
+the annotated points sufficiently align (Carlier et al. [2014], Tu et al. [2011], Hebsgaard et al. [2015]). Although such
+techniques are accurate up to rigid translation, they require time investment from a trained expert to find matching
+features in both modalities, creating a need for computational algorithms that non-rigidly register CCTA images to
+corresponding intravascular data in an automatic fashion.
+Automatic co-registration techniques typically consist of discretely optimizing a constructed cost function over a set of
+longitudinal or rotational image shifts, where the cost function varies depending on the modalities being registered.
+Some proposed cost functions include metrics such as lumen diameters (Qin et al. [2021]), lumen contours (Molony
+et al. [2016], Karmakar et al. [2020]), calcium thickness (Gharaibeh et al. [2020], Molony et al. [2016]), and image
+pixel intensities (Tsiknakis et al. [2021]). Similarly, rigid rotational registration for intravascular pullbacks has also
+been based on extracted features such as luminal contours (Karmakar et al. [2020]), and calcium angle (Molony et al.
+[2016]). However, the registration accuracy of all rigid registration methods is compromised by inconsistent motor
+pullback speeds and rotational drift, which introduce non-rigid longitudinal and rotational distortions that misalign
+image features such as diseased plaque and bifurcations.
+To compensate for the longitudinal, rotational, and transverse motion of the catheter, several non-rigid registration
+approaches have been proposed, typically to be employed after initial rigid alignment. Currently, non-rigid registration
+of multiple intravascular imaging datasets has been predominantly performed through Dynamic Time Warping (DTW)
+and Dynamic Programming (DP) (Tsiknakis et al. [2021], Molony et al. [2016]). However, DTW introduces non-
+physiological assumptions into the registration process by discretely skipping or repeating intravascular frames, assumed
+to be evenly spaced along the longitudinal direction. As a result, DTW is not well suited for use for intravascular images,
+with pullback acquisitions sometimes rendering up to 10 repeated intravascular imaging frames at a time (Molony et al.
+[2016]). On the contrary, continuous non-rigid registration methods have been developed to model the longitudinal
+stretch and rotational drift between intravascular imaging frames using affine transforms and spline interpolation (Zhang
+et al. [2014], Uzu et al. [2019]). While such continuous non-rigid methods are more realistic, they extensively rely on
+manual annotations of all bifurcation zones for image registration, severely limiting their scalability. As such, there is
+2
+
+arXiv Template
+A PREPRINT
+no continuous non-rigid registration method as of yet that does not explicitly require fiduciary landmarks for rotational
+and longitudinal alignment. Further, there has been an increasing interest in machine learning approaches to image
+co-registration in which a neural network is trained to predict a spatial transform that maps a moving image onto a
+static target image (Balakrishnan et al. [2019], Fu et al. [2020]). Such approaches critically rely on a differentiable and
+continuous spatial transform allowing for back-propagation of gradients to adjust the neural network weights (Jaderberg
+et al. [2015]). While such continuous and differential spatial transforms are available for co-registration of 3D and 2D
+medical images, a similar framework that accounts for the unique variation in intravascular catheter motion has not
+been developed.
+Given the previous limitations noted in prior co-registration algorithms, we here propose a novel semi-automatic
+framework that takes as input an intravascular imaging pullback and a CCTA 3D image and aligns each intravascular
+image frame along the artery to the equivalent frame in the CCTA image. The proposed continuous registration
+methodology does not require manual matching of landmarks, with the only manual effort being the selection of viable
+intravascular imaging frames and the provision of a rough centerline within the CT image. Specifically, we explore the
+problem of reconstructing the path of a virtual catheter moving through and sampling from a 3D CCTA image such that
+the set of frames produced by the motion of the catheter optimally reflect the equivalent target intravascular pullback.
+Key contributions of this framework include:
+• We present the first continuous co-registration framework for rigid and non-rigid matching of CCTA images
+and intravascular images up to pixelwise alignment, with segmentations of the lumen and vessel wall as sole
+input.
+• We introduce a rigid registration approach that consists of our published longitudinal rigid registration
+algorithm, which uses lumen area in a multi-step decision process, and a rotational registration step that
+leverages the segmentation of the vessel wall to produce an initial rotational configuration for subsequent
+registration.
+• We introduce a novel non-rigid registration step, based only on the lumen segmentation, which is robust to
+physiological catheter motions. The registration is formulated in terms of finding the path of a virtual catheter,
+which translates the CCTA image into an intravascular-like image by sampling the segmentation along the
+virtual catheter path. The virtual catheter path is reconstructed by spatially deforming the CCTA centerline by
+B-spline deformations formulated in the longitudinal, rotational, and transverse directions, ensuring a smooth
+and physiological reconstruction of catheter motion.
+• Our non-rigid registration module being both continuous and differentiable, allows for easy integration into
+future machine-learning-based approaches for intravascular image registration.
+• We validate in a direct clinical setting, evaluating performance across a multimodal cohort of cardiac
+patients(n = 40) and benchmarking performance against previously developed state-of-the-art approaches.
+2
+Methodology
+An overview of the co-registration pipeline is detailed in Figure 1. In brief, bi-modality images are processed to produce
+binary segmentations of the lumen and vessel wall (section 2.1.1), which are first used in a rigid registration step,
+involving both longitudinal and rotational alignment (section 2.1.2). The rigid registration is then used as an initial
+estimate of a virtual catheter path forming the basis for a non-rigid registration (section 2.1.3). The virtual catheter
+path initially samples the geometry of the CT lumen to produce a virtual imaging pullback that is then compared to a
+Signed Distance Field (SDF) derived from the intravascular equivalent. A non-rigid transformation for the longitudinal,
+rotational, and transverse motion distortions is applied on the virtual catheter path and optimized to align the SDF’s in
+both modalities. The performance of our proposed co-registration algorithm is then validated on a clinical cohort of
+relevant cardiac patients (section 2.2.2)
+2.1
+Co-registration framework
+2.1.1
+Preprocessing
+As the basis for our co-registration pipeline, luminal segmentations from the two different image modalities are provided.
+Starting with the intravascular image set, luminal frame-by-frame segmentations are used to produce an SDF using
+a fast marching method (Treister and Haber [2016]), clamped to only have negative values (indicating that a pixel is
+inside the lumen). Further, the SDF is smoothed in the axial direction with a Gaussian convolutional kernel of size 3
+and standard deviation 0.1 in order to regularize the optimization process.
+3
+
+arXiv Template
+A PREPRINT
+OCT image
+CT image
+Rigid registration
+Non-rigid registration
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+CT non-rigid
+OCT
+Figure 1: Overview of the proposed registration pipeline. The imaging modalities are rigidly co-registered in the
+longitudinal and rotational directions, serving as the basis for the initialization of the virtual pullback trajectory. The
+virtual pullback trajectory is then used to sample a CT lumen signed distance field (SDF), used in direct comparison to
+the equivalent OCT SDF.
+Arc angle (degrees)
+Rigid longitudinal registration
+OCT image
+Lumen
+CT image
+Rigid rotational registration
+Vessel thickness (pixels)
+0
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+CT rigid
+OCT
+CT rigid
+OCT
+Figure 2: Overview of the proposed rigid registration pipeline. The lumen segmentation area vectors from both
+modalities are used to rigidly register the modalities in the longitudinal direction using a sliding window approach. The
+longitudinal registration is then used to match each equivalent frame for the rotational registration. The vessel wall
+segmentations are then converted to vessel thickness-arc angle plots and are used to determine an optimal rigid rotation.
+Coupled to the intravascular image set, a corresponding 3D SDF from the CCTA images is generated. Although several
+methods could potentially be applied for such, a convenient approach is to derive the SDF from a computational mesh
+of the coronary tree. Herein, to create an SDF a narrow band is defined within the object mesh boundary, subsequently
+used to compute exact Euclidean distances from each voxel center to the boundary. Outside the object boundary,
+the distance field values are then set to zero. Corresponding binary segmentations can then be produced by simple
+thresholding operations. Using these computational meshes, vessel centerlines are obtained using VMTK (Antiga et al.
+[2008]), generating an array ¯r representing n spatial positions with an axial spacing of 0.2mm. A spatial derivative
+is then applied to the centerline points ¯r, defining a tangent vector T for each point. The two vectors U and V that
+are orthogonal to the tangent vector can then be obtained through the parallel transport method (Guo et al. [2013]),
+ensuring that the vectors V and U remain stable between frames placed along the axial direction. The centerline points
+and the orthogonal vectors hence define a set of frames(¯r,T,U,and V) in 3D space that are used to sample the CCTA
+SDF along an equivalent virtual catheter pullback, with dimensions equalling the intravascular dataset (in our case:
+96x96xNframes with an in-plane resolution of 80 micrometers), all using a curved-planar reformation procedure
+(Kanitsar et al. [2002]). The resulting SDF is then smoothed in the axial direction with a Gaussian convolutional
+kernel of size 3 and standard deviation of 0.1. Through this method, virtual pullbacks of both the lumen and vessel
+segmentations were produced.
+4
+
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+80arXiv Template
+A PREPRINT
+2.1.2
+Rigid registration
+An overview of the rigid registration step can be seen in Figure 2. For the rigid longitudinal registration, the processed
+lumen segmentations are used to create an area vector of equal lengths, sampling the CT virtual pullback to correspond
+to the acquired intravascualr set. Here, We leverage our previous work to rigidly align the pullbacks using a multi-step
+sliding window method, minimizing the difference in area vectors (for details see (Karmakar et al. [2020])). Before
+registration, continuous segments of the OCT pullback with poor lumen segmentations due to residual blood or catheter
+housing were manually excluded.
+For rigid rotational registration, the luminal profiles were deemed unreliable for producing good alignment. Therefore,
+the vessel border segmentations were instead used for rotationally aligning the pullbacks. For each CT and intravascular
+frame, respectively, a thickness-arc angle vector is extracted by tracing a set of radial rays from the centroid of the
+vessel segmentation in increments of 12 degrees. The thickness vectors are then matched according to the result of the
+longitudinal registration, with non-overlapping frames subsequently cropped. The optimal rigid rotation angle is then
+obtained by sliding the set of CT thickness vectors over each equivalent intravascular image vector and minimizing the
+mean squared error across all frames.
+2.1.3
+Non-rigid registration
+The non-rigid registration process (Figure 3) consists of optimizing a set of frame variables (¯r, T, U, and V) representing
+a virtual catheter path moving through the CCTA image. The loss function to be optimized is defined as the mean squared
+error between the 3D SDF generated from the two image sets, with the CCTA-SDF sampled along the aforementioned
+virtual catheter path. The virtual pullback is initialized as the centerline that was calculated from the CCTA 3D model
+and longitudinally cropped and rotated according to the output of the rigid registration. After rigid registration a spline
+is defined based on the centerline points ¯r where the centerline points are fully described by their arclength values ¯s
+along the spline. Accordingly, every i-th frame can be manipulated by 4 variables, representing the arclength along
+the virtual catheter path si, the rotation angle of the frame θi about the catheter path T, and the in-plane transverse
+displacements du and dv along the frame vectors U and V respectively (see Figure 3).
+To regularize the motion of the virtual catheter to be smooth and physiological, the 4 frame manipulation variable
+sets are parametrized by a sparse set of control points controlling a B-spline deformation (Rueckert et al. [1999])
+independently acting on 4 nx1 vectors representing the frame manipulation variables ¯s, ¯θ, ¯du, and ¯dv. Thus, for a 1D
+control point grid of size N, the relation between a frame manipulation variable v and the control points p can be
+described by:
+v(s) =
+N
+�
+i=0
+Bi(s)pi,
+(1)
+where Bi(s) is a polynomial basis function of order 2. In matrix form, the same can be represented by:
+V = BP,
+(2)
+in which V ∈ Rn×1, B ∈ Rn×N, P ∈ RN×1 where n is the number of frames and N is the number of control points. B
+is the univariate B-spline tensor and can be pre-computed from the initial frame manipulation variable vectors, while P
+is the deformed control point grid vector that is optimized during co-registration.
+Instead of directly optimizing for the set of Ns = 30 control points P s controlling the arclength variables s for each
+frame, the control point deformations ∆P s
+i can be parametrized by a deformation vector Xs of size Ns − 1. dictating
+the relative displacement of each control point from its proximal neighbor, with the most proximal control point being
+fixed. This is done to account for the cumulative effect of catheter motor speed variations on the rest of the pullback.
+Therefore, the deformation of each control point can be defined as the cumulative sum of the relative deformations
+along the proximal control points. Moreover, to regularize the catheter motion and prevent backwards movement, the
+relative deformation of each control point is limited to a fraction (0.35) of the distance between control points.
+∆P s
+i = Xs
+i +
+i−1
+�
+j=0
+Xs
+j
+(3)
+Once the control points are deformed into a new configuration, the new arclength values for each frame ¯s is calculated
+through equation 2 and the frame vectors (T,U, and V) are then recalculated.
+¯s = BsP s
+(4)
+5
+
+arXiv Template
+A PREPRINT
+Rigid initialization
+Non-rigid transform
+Rotational transform
+Longitudinal transform
+Transverse transform
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+OCT: Target
+Loss
+CT: Moving
+Figure 3: Overview of the spatial deformation acting on the virtual catheter path. The longitudinal transform stretches
+and compresses the space between adjacent frames, at which point the frame vectors (T,U, and V) are recalculated. The
+rotational transform is then applied to the frame vectors orthogonal to the tangent (U and V) about T, and the transverse
+transform is then applied to shift the centerline points in the direction of the new frame vectors (U and V).
+2.1.4
+Non-rigid rotational registration
+Similar to the longitudinal registration, the set of Nθ = 20 control points P θ controlling the rotation of each frame
+about the catheter axis can be parameterized by a relative rotation vector Xθ of size Nθ. The rotation value for each
+control points is defined by:
+∆P θ
+i = Xθ
+i +
+i−1
+�
+j=0
+Xθ
+j
+(5)
+The rotation correction for each frame is applied after the non-rigid longitudinal transformation but before the non-rigid
+transverse transformation. Once the control points are deformed into a new configuration, the new rotation values for
+each frame ¯θ can be calculated through equation 2 and used to rotate frame vectors U and V about the tangent vectors T.
+¯θ = BθP θ
+(6)
+2.1.5
+Non-rigid transverse registration
+The virtual catheter was biased to stay close to the centerline by optimizing the Nd = 60 control points determining the
+in-plane transverse displacements du and dv directly. Thus the 2 orthogonal transverse displacements for each frame
+was calculated from the matrix relation:
+¯d = BdP d
+(7)
+Where for each frame the displacements along the vectors U and V were applied as a final step after the non-rigid
+longitudinal and rotational transforms.
+2.2
+Performance evaluation
+2.2.1
+Image data
+To evalute our proposed co-registration framework, a dataset consisting of n = 40 matched OCT and CT image pairs
+from 5 different clinical centers were selected, all originating from the Precise Percutaneous Coronary Intervention Plan
+(P3) study (Nagumo et al. [2021]). As each OCT pullback image consisted of 375 frames, the intravascular imaging
+dataset comprised of approximately 15,000 image frames. The OCT lumen in every frame was manually annotated by
+6
+
+arXiv Template
+A PREPRINT
+trained cardiologists. Further, the vessel wall borders in every OCT frame were segmented using a convolutional neural
+network, using the previously published U-net as base architecture (Ronneberger et al. [2015]). Details of the network,
+training, and validation can be found in Supplementary Material A. The lumen and vessel wall segmentations were then
+re-sampled to represent a 3D image of dimensions 96x96xNframes with an in-frame resolution of 80 micrometers
+and an out-of-frame resolution of 0.4 mm. All utilized intravascular pullbacks were manually deemed as of sufficient
+image quality, with appropriate quality lumen segmentations. For the CCTA data, a 3D model of the coronary tree for
+each patient was produced by HeartFlow using the CCTA image (Sonck et al. [2022]). The 3D model was then used to
+produce a 3D SDF with a resolution of 0.25mm along each axis with an image dimension of 768x768x482.
+2.2.2
+Co-registration accuracy
+In order to evaluate the performance of the non-rigid registration, 114 bifurcations were manually marked in the OCT
+pullback as well as in the rigid and non-rigid virtual pullback segmentations generated from the CCTA data. Bifurcations
+were defined as the last image frame before a visual coronary artery split into two branches could be seen. Bifurcations
+that were common to both modalities had their frame numbers recorded for validation of the non-rigid registration
+algorithm. Longitudinal validation was conducted by comparing the frame number of a bifurcation in the OCT data
+with the equivalent bifurcation frame number in the virtual pullback before and after non-rigid registration. In order to
+validate the non-rigid rotational registration, the bifurcation angle difference between OCT pullback and the virtual
+pullback was compared before and after rotational registration. As the bifurcation angle between bifurcation sections
+that were not longitudinally matched is expected to be uncorrelated, a separate analysis was conducted to characterize
+how angular mismatch varies when the bifurcations are longitudinally matched. Furthermore, only bifurcations that had
+a frame mismatch below 6 frames were considered for extensive analysis of rotational accuracy.
+2.2.3
+Comparison to alternative approaches
+The most common co-registration methodology employed for coronary artery registration has been discrete optimization
+approaches such as DTW and Dynamic Programming. Therefore, in order to evaluate the performance of our
+longitudinal and rotational co-registration framework against state-of-the-art discrete approaches, we applied the
+methodology described in Karmakar et. al (Karmakar et al. [2022]) on the same dataset used in this study. The approach
+utilizes DTW to longitudinally align two coronary imaging modalities and Dynamic Programming to rotationally align
+each frame. We utilized a window length of 4 (0.8mm) as implemented in the previous study and recorded identical
+alignment metrics for 114 matched bifurcations in the dataset. The non-rigid registration algorithm was applied after
+the rigid longitudinal registration step described in section 2.1.2. A substudy was also conducted in which the angular
+alignment of all bifurcations was compared to the angular alignment of longitudinally matched bifurcations.
+2.2.4
+Optimization details
+The gradient descent-based optimization procedure was implemented in PyTorch with the Adam optimizer (Kingma
+and Ba [2014]). A learning rate of 0.001 was used for the non-rigid longitudinal parameters and a rate of 0.01 was
+used for both the rotational and transverse parameters. Each co-registration procedure was run for a minimum of 200
+iterations to ensure convergence.
+3
+Results
+3.1
+Longitudinal Registration
+Longitudinal registration plots in Figures 4 and 7A1-2 show that using rigid registration alone (Figure 7A1), few
+bifurcations were longitudinally aligned within 6 (dotted line), 4 (dashed line), or 2 (solid line) frame distances.
+However, after non-rigid alignment (Figure 7A2), distinct improvement can be observed with a majority of bifurcations
+are aligned within 6 frames. These results are visualized by the longitudinal mismatch plot (Figure 5A), revealing
+that after rigid alignment, the percentage of bifurcations matched within 2, 4, and 6 frames are 26.3, 42.1, and 57.9%,
+respectively, while after non-rigid alignment, these values increase to 60.5, 78.9, and 86.8%. Moreover, the scatterplot
+for non-rigid registration (A2) demonstrates that the majority of bifurcations (86% shown in green) were enhanced
+in terms of frame alignment, while a negligible number of bifurcations had slightly (11.4 % shown in orange) or
+significantly (2.6 % shown in red) worse alignment after non-rigid registration. Table 2 further demonstrates the effect
+of non-rigid registration, in which the mean frame difference after rigid registration was 7.9 frames and subsequently
+decreased to 3.3 frames after non-rigid registration.
+7
+
+arXiv Template
+A PREPRINT
+A1
+B1
+C1
+D1
+E1
+F1
+G1
+A2
+B2
+C2
+D2
+E2
+F2
+G2
+A3
+B3
+C3
+D3
+E3
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+CT
+OCT
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+B
+C D
+E F
+G
+Figure 4: Qualitative results for a single co registered case. Top row shows area plot along the artery for the non-rigidly
+registered CT (green) and the OCT images. The bifurcation zones (Sections A-G) are marked and labeled for further
+analysis. Bifurcation frames from the CT, OCT, and overlapped segmentation maps are presented in the bottom row for
+qualitative analysis of the rotational and transverse co-registration.
+A
+B
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+0.0
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+Rigid
+Rigid+Non-rigid
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+Maximum angular mismatch (degrees)
+0.0
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+Matched bifucations (%)
+Rigid
+Rigid+Non-rigid
+Figure 5: Quantitative results comparing the quality of
+rigid and non-rigid co registration in longitudinal and ro-
+tational directions with varying degrees of misalignment.
+The mismatch plots exhibit the % of matched bifurca-
+tions with increasing longitudinal (A) and rotational (B)
+alignment mismatch criteria (x-axis).
+~10 degrees
+~20 degrees
+~30 degrees
+Angular Mismatch
+Figure 6: Grid plot showing multiple aligned bifurcation
+segmentations using an SDF-based loss. Angular mis-
+matches up to 10, 20, and 30 degrees are shown in the
+first, second, and third columns respectively.
+3.2
+Rotational Registration
+Examination of the individual bifurcating frames in figure 4 for the CT (row 1) and OCT (row 2) frames indicates
+excellent rotational and transverse alignment between both imaging modalities as evident from the raw images and the
+overlapped segmentations (row 3). Rotational registration plots in figure 7B1-2 demonstrate that few bifurcations are
+rotationally aligned within 30 (dotted line), 20 (dashed line), or 10 (solid line) degrees after rigid alignment (B1). After
+non-rigid alignment (Figure 7B2), a majority of bifurcations were aligned within 30 degrees, with a significant amount
+aligned within 20 and 10 degrees. Examination of the rotational mismatch plot (Figure 5B) quantitatively demonstrates
+an increase in the percentage of bifurcations aligned up to an angular mismatch of 10, 20, and 30 degrees from %
+values of 25.3, 40.4, and 52.3 to 51.5, 69.7, and 79.8% respectively. Similarly, the non-rigid registration scatterplot
+8
+
+arXiv Template
+A PREPRINT
+A1
+A2
+B1
+B2
+Bifurcation number
+0
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+10
+15
+20
+25
+30
+35
+40
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+Bifurcation number
+0
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+Non-rigid<0
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+Non-rigid>=2
+Bifurcation number
+0
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+Non-rigid<0
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+Non-rigid>=20
+Bifurcation number
+0
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+150
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+Angular misalignment
+Figure 7: Quantitative results comparing the quality of rigid and non-rigid co-registration in longitudinal and rotational
+directions. The first row compares bifurcation frame mismatch before (A1) and after (A2) non-rigid registration in
+the form of scatterplots. The second row compares bifurcation angular mismatch before (B1) and after (B2) non-rigid
+registration in the form of scatterplots. The scatterplot for the longitudinal and rotational non-rigid registration (A2 and
+B2) are color-coded to exhibit the change in alignment metric after non-rigid registration, where green represents an
+increase in alignment, orange represents a mild decrease in alignment, and red represents a strong decrease in alignment.
+Only bifurcations that were longitudinally matched within 6 OCT frames were analyzed for rotational alignment.
+demonstrates that the majority of bifurcations had their angular mismatch decreased after non-rigid alignment (66%
+shown in green) and only a minority had their angular mismatch values slightly (22% shown in orange) or significantly
+(12% shown in red) increased. The mean value of the angular mismatch before and after non-rigid alignment is reported
+in Table 2, in which the mean angular mismatch decreases from 36.0 to 28.6 degrees.
+3.3
+Comparison with previous approaches
+A direct comparison of the virtual catheter method with state-of-the-art discrete optimization approaches can be seen in
+Tables 2 and 3. Comparing the virtual catheter method to a discrete optimization approach for longitudinal registration,
+it was shown that DTW produces significantly poorer results in longitudinal registration, with the longitudinal mismatch
+of 11.7 frames being higher than rigid longitudinal registration average of 7.9 frames. Comparing the virtual catheter
+method to using Dynamic Programming for rotational registration, it was shown that such discrete optimization
+algorithms exhibit poor performance for CT-OCT rotational registration (angular mismatch of 77.9 degrees) which
+is higher than the angular mismatch after rigid rotational registration alone. Table 3 quantifies the angular mismatch
+in the case where non-rigid longitudinal registration is successful. For bifurcations with a maximum frame mismatch
+of 6 after non-rigid registration, the angular mismatch decreases from 77.9 to 65.2 for the Dynamic Programming
+approach, while for the virtual catheter method, the angular mismatch decreases from 28.6 to 24.8. In contrast, the
+angular mismatch for the rigid registration is unchanged after excluding non-matching bifurcations.
+9
+
+arXiv Template
+A PREPRINT
+Figure 8: Qualitative results comparing the alignment of calcium annotations between OCT (first row) and CT (third
+row) for selected frames with good luminal alignment. The middle row shows the calcium annotations for OCT (red)
+and CT (green) superimposed on each other.
+Table 1: Accuracy of alternative co-registration approaches, proposed for intravascular-intravascular image registration.
+Data is presented from left to right including evaluated co-registered modalities, dataset size, and overall methodological
+approach. Further, average errors are presented in both longitudinal (frames) and rotational (degree) directions.
+Ref.
+Modalities
+Dataset Size
+Methodology
+Longitudinal mismatch
+Angular mismatch
+Karmakar et al. [2022]
+OCT-OCT
+9 patients
+DTW + Dynamic Programming
+0.9 ± 0.8
+7.7 ± 6.7
+Tsiknakis et al. [2023]
+OCT-OCT
+21 patients
+DTW + Harmony Search
+5.6 ± 6.7
+1.2 ± 0.81
+Karmakar et al. [2022]
+OCT-IVUS
+7 patients
+DTW + Dynamic Programming
+1.45 ± 0.7
+29.1 ± 23.2
+Molony et al. [2016]
+OCT-IVUS
+12 patients
+DTW + Dynamic Programming
+5.0 ± 6.2
+17.8 ± 21.9
+Table 2: Accuracy of co-registration approaches applied to CT-OCT image registration. Data is presented from left to
+right including evaluated co-registered modalities, dataset size, and overall methodological approach. Further, average
+errors are presented in both longitudinal (frames) and rotational (degree) directions. All approaches in this table have
+been evaluated on the same dataset
+Ref.
+Modalities
+Dataset Size
+Methodology
+Longitudinal mismatch
+Angular mismatch
+Karmakar et al. [2022]
+CT-OCT
+40 patients
+DTW + Dynamic Programming
+11.7 ± 12.1
+77.9 ± 61.0
+Ours (Rigid)
+CT-OCT
+40 patients
+Virtual Catheter Method
+7.9 ± 7.1
+36.0 ± 31.9
+Ours (Rigid+Non-rigid)
+CT-OCT
+40 patients
+Virtual Catheter Method
+3.3 ± 3.9
+28.6 ± 40.9
+Table 3: Accuracy of co-registration approaches applied to CT-OCT image registration for bifurcations that are
+longitudinally matched. Data is presented from left to right including methodological approach and number of
+longitudinally matched bifurcations. Bifurcations are considered longitudinally matched when they have a maximum
+frame difference of 6 after non-rigid longitudinal registration. Further, average errors are presented for rotational
+direction in degrees.
+Ref.
+Methodology
+Matched Bifurcations
+Angular Mismatch
+Karmakar et al. [2022]
+DTW + Dynamic Programming
+52/114
+65.2 ± 72.9
+Ours (Rigid)
+Virtual Catheter Method
+99/114
+36.0 ± 33.0
+Ours (Rigid+Non-rigid)
+Virtual Catheter Method
+99/114
+24.8 ± 39.0
+10
+
+OUarXiv Template
+A PREPRINT
+4
+Discussion
+The aim of the current study was to develop a fully automatic registration algorithm to align CCTA and intravascular
+images. Specifically, we propose a novel registration process finding the optimal rigid and non-rigid spatial transforms
+using a virtual catheter path in the CCTA data, aligning the non-invasive modality to its invasive counterpart. Our results
+indicate excellent co-registration accuracy, with excellent agreement with reference manual landmark annotations
+(Figure 4). Further, our results underline the critical importance of a non-rigid registration step, with significant
+enhancement in both longitudinal and rotational alignments observed when comparing rigid vs. non-rigid alignments in
+Table 2. We demonstrate that for the vast majority of bifurcations, our framework is able to improve the longitudinal
+and rotational alignment of common bifurcations within the CT and OCT images. Lastly, we demonstrate the added
+value of our approach as compared to state-of-the-art alternatives, with a head-to-head comparison to previously
+developed discrete optimization alignment algorithms (Table 1). A head-to-head comparison demonstrates that discrete
+optimization approaches for longitudinal and rotational alignment suffer a significant drop in alignment quality when
+applied for the task of CT-OCT co-registration. Meanwhile, our approach maintains performance accuracy in line with
+simpler tasks such as intravascular-intravascular image registration.
+4.1
+Related work
+Currently, a majority of CCTA studies that validate their CT findings with intravascular images have used rigid manual
+registration based on fiduciary landmarks such as bifurcations or large calcifications (Carlier et al. [2014], Tu et al.
+[2011], Hebsgaard et al. [2015]). One of the few studies that attempted to align CT and intravascular data up to a non-
+rigid level is by Uzu et al. [2019] in which a B-spline deformation model was used to optimize the alignment of manually
+annotated bifurcation landmarks. Though powerful, such an approach is time-consuming due to the significant amount
+of manual processing required to process the OCT images, rigidly align the geometric models, and mark bifurcations
+within every artery. In comparison, our approach implicitly matches nearby bifurcations using longitudinally smoothed
+SDFs representing the CT and OCT lumens, respectively. Other approaches that register intravascular-to-intravascular
+modalities have in the past relied on DTW (Molony et al. [2016], Karmakar et al. [2022]), discretely optimizing the
+frame-wise progression of one intravascular pullback to maximize longitudinal and rotational alignment with another.
+Such methods, nevertheless, attempt to recapitulate the continuous motion distortions introduced by the catheter path
+with discrete non-physiological frames or repeats (Molony et al. [2016]). However, skipping or repeating several frames
+that the catheter motion is not smooth or continuous, which is an unrealistic assumption about the catheter path.
+Direct numerical comparison of reported co-registration accuracy across published approaches is inherently difficult
+as co-registration accuracy is highly dependent on the specific dataset explored as well as which modalities are being
+coregistered. For example, the simplest co-registration task would be represented by the alignment of same-modality
+images such as OCT-OCT image pairs. For such tasks, our previously published DTW and dynamic programming
+approach (Karmakar et al. [2022]) exhibits similar perfomance compared to other state-of-the-art algorithms (Tsiknakis
+et al. [2023]) (See Table 1). When applied to multimodality datasets, such as IVUS-OCT image pairs, our previously
+developed approach (Karmakar et al. [2022]) suffers distinctive drops in both longitudinal and rotational accuracy (see
+Table 1), however, still maintains comparative performance to similar Dynamic Programming approaches (Molony et al.
+[2016]). Thus, in order to facilitate a head-to-head comparison on the more challenging task of registering CT-OCT
+image pairs, we applied our previously developed discrete optimization algorithm (Karmakar et al. [2022]) on our
+multi-modal dataset of 40 patients. Doing so, we found that our previous approach produced significantly worse
+longitudinal and rotational alignment compared to the virtual catheter method, with both longitudinal and angular
+alignment being worse than simple rigid registration (Table 2). In contrast, our developed methodology achieves
+competitive results with even intravascular-intravascular registration studies (Table 1 and Table 2).
+4.2
+Methodological Adaptations
+From the above it can be seen that the task of co-registering CT and OCT images presents several unique difficulties for
+discrete registration algorithms. Our framework has several features that were designed to mitigate such challenges.
+First, the low resolution of CT images induces a circular bias in the lumen segmentations (see Figure 4), as well as a
+tendency to miss small bifurcations. Such circularly symmetric regions hence create zones of longitudinal and rotational
+ambiguity along the pullback. Our approach tries to minimize the dependency of such by formulating the longitudinal
+and rotational transforms acting on the virtual path in terms of a regularized and smooth B spline deformation. As
+such, the optimization procedure is mainly dominated by the alignment of prominent non-symmetric features such
+as bifurcations, rather than the circularly symmetric lumen segments. This ensures that the rotational alignment of
+all non-bifurcating lumen frames that are in proximity to matched bifurcations are properly matched due to B spline
+interpolation (Figure 4). Another significant issue faced in previous rotational co-registration algorithms (Karmakar
+11
+
+arXiv Template
+A PREPRINT
+et al. [2022], Molony et al. [2016]) is that lumen bifurcations are only able to contribute to rotational alignment if they
+exist within the same frame. As such, poor longitudinal alignment of bifurcations was a significant contributing factor
+to the poor performance of our previously developed dynamic programming algorithm for rotational co-registration
+(Table 2). Our framework, in contrast, minimizes this dependency through the use of a Gaussian smoothing kernel
+applied longitudinally over the SDF. Longitudinal smoothing allows single-frame bifurcations to appear in adjacent
+frames and smooths the loss surface such that bifurcations in the different modalities can be better aligned (Figures 4
+and 7). Another design choice that was found to increase training stability and co-registration quality was the use of
+SDF’s to determine alignment, as opposed to using a segmentation loss such as cross-entropy or Dice. This was due to
+the fact that when the lumen segmentations were fully overlapping, multiple rotational and transverse configurations
+contribute equally to a segmentation loss function, preventing the algorithm from making fine adjustments in the spatial
+transform. Figure 6 further demonstrates how the quality of rotational registration varies with angular mismatch, where
+the angular mismatch tends to occur due to the limitations of local optimization of pixel alignment. At less than 10
+degrees mismatch, the difference in alignment is minimal, while under 30 degrees, the difference in alignment can
+be attributed to differing lumen bifurcation shapes in the CT and OCT data. Lastly, many co-registration methods
+normalize the position of the lumen by the artery centroid (Uzu et al. [2019], Karmakar et al. [2020, 2022], Molony
+et al. [2016]). While such an approach manages to align CT and OCT frames with circularly symmetric lumen, it fails
+to effectively align equivalent frames with bifurcations as the segmentation can have different maximum diameters
+between the modalities and thus different centroids. Moreover, centering the image around the lumen centroids can
+cause the algorithm to align bifurcations 180-degrees from the correct orientation. It was empirically found that this
+phenomenon was found to be a significant contributing factor to the degradation of the co-registration performance of
+our previously developed discrete co-registration algorithm. In this framework, we instead choose to jointly optimize
+for the transverse displacements of the virtual path in addition to the longitudinal and rotational displacements, which
+allows for the bifurcations in both modalities to be anchored around the OCT catheter location and enables near
+pixelwise alignment of the lumen (Figures 4,6) and plaque constituents such as calcium (Figure 8).
+4.3
+Limitations
+Though very promising for clinical applications, our developed approach has a number of limitations. First, the
+non-rigid spatial transform acting on the virtual catheter path is found through gradient-based optimization, requiring
+that landmarks lie sufficiently close such that proper matching is ensured. For example, common bifurcations that
+have a frame mismatch of more than 6 frames (corresponding to the longitudinal smoothing kernel) are expected to
+be uncorrelated in terms of orientation. This issue can be mitigated by integrating deep learning networks which can
+accurately predict the spatial transform needed to align the two modalities. Another limitation is the dependence of
+non-rigid registration on the lumen. The lumen estimation is expected to be accurate for both modalities and as such,
+ensures good registration accuracy for regions that include many bifurcations. However, due to the poor resolution of
+CCTA images, the lumen estimation tends to be highly circular. Accordingly, it is expected that rotational co-registration
+certainty increases with bifurcation proximity but decreases in stenotic regions that contain highly circular CT luminal
+profiles. In the future, co-registration accuracy can likely be improved by including contextual information relating
+to the vessel wall such as lesion content and morphology as a supervisory signal in the loss function. Third, the use
+of a pixel-wise loss as a surrogate for luminal alignment may not necessarily result in optimal alignment of lumen
+bifurcations. As seen in Figure 6, a pixel-wise loss function can occasionally bias the spatial transform to align the
+central lumen body over aligning the bifurcation in scenarios where the bifurcation shapes are not perfectly matching.
+In the future, this issue can be mitigated by introducing an orientation loss to bias the spatial transform to rotationally
+align bifurcations. Lastly, regularizing the spatial transform and smoothing the SDF’s can create difficulties in localizing
+landmarks up to frame-wise precision. This can be seen in the area curve in Figure 4 section B with the slightly
+mismatched bifurcation and in Figure 7 A2 and B2 with a minor amount of bifurcations with increased frame and
+angular mismatch values. The localization capabilities of the algorithm can be improved by introducing multi-scale
+deformation steps where finer control point grids can be recursively used as the basis for the spatial transform.
+4.4
+Translational Benefits
+The development of automatic frame-wise matching algorithms for CT-OCT data fusion would enable the development
+of several research-based applications. First, intravascular imaging data can act as ground truth to validate the reliability
+of CT in delineating several morphological metrics of atherosclerosis, such as luminal area, lipid content, and calcium
+volume. Understanding when CT-derived morphological metrics are reliable is critical for both therapy planning and
+deciding when to rely on intravascular imaging. For example, studying the interaction between calcium blooming and
+measured lumen size in CT images necessitates that ground truth lumen measurements be available, which can only be
+provided by frame-wise co-registration algorithms. Figure 8 demonstrates that such a frame-by-frame comparison can
+be done provided that longitudinal and rotational co-registration is of sufficient quality. Second, multi-modal data fusion
+12
+
+arXiv Template
+A PREPRINT
+would allow for the enhanced generation of patient-specific digital twins from coronary images. There has been an
+increasing interest in the use of intravascular-images to create computational digital twins for the prediction of coronary
+pathophysiology and clinical decision-making (Kadry et al. [2021]). However, intravascular images, while providing
+excellent resolution within the imaging frame, do not provide sufficient information to create a fully physiological
+artery model. Intravascular images typically suffer from intra-frame motion drift artifacts in the longitudinal and
+rotational directions and cannot capture information on the three-dimensional centerline of the artery. On the other
+hand, CT suffers from poor resolution but is able to capture the three-dimensional nature of the artery with high
+accuracy. Combining both modalities would allow researchers to investigate the importance of longitudinal and
+rotational distortions, as well as modeling arterial tortuosity.
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+

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+1
+Transferable Energy Storage Bidder
+Yousuf Baker, Ningkun Zheng, Student Member, IEEE, Bolun Xu, Member, IEEE
+Abstract—Energy storage resources must consider both price
+uncertainties and their physical operating characteristics when
+participating in wholesale electricity markets. This is a challeng-
+ing problem as electricity prices are highly volatile, and energy
+storage has efficiency losses, power, and energy constraints. This
+paper presents a novel, versatile, and transferable approach
+combining model-based optimization with a convolutional long
+short-term memory network for energy storage to respond to
+or bid into wholesale electricity markets. We apply transfer
+learning to the ConvLSTM network to quickly adapt the trained
+bidding model to new market environments. We test our proposed
+approach using historical prices from New York State, showing
+it achieves state-of-the-art results, achieving between 70% to
+near 90% profit ratio compared to perfect foresight cases,
+in both price response and wholesale market bidding setting
+with various energy storage durations. We also test a transfer
+learning approach by pre-training the bidding model using
+New York data and applying it to arbitrage in Queensland,
+Australia. The result shows transfer learning achieves exceptional
+arbitrage profitability with as little as three days of local training
+data, demonstrating its significant advantage over training from
+scratch in scenarios with very limited data availability.
+Index Terms—Energy storage; Deep learning; Transfer learn-
+ing; Power system economics.
+I. INTRODUCTION
+Successful participation of energy storage resources in com-
+petitive electricity markets benefits storage investors and social
+welfare. Ancillary services such as frequency regulation have
+been the primary sources of profit for energy storage owners,
+but these markets have quickly saturated due to surging storage
+deployments and small market size [1]. In the meantime, the
+share of storage arbitraging in wholesale markets has tripled
+from a little less than 20% in 2016 to almost 60% in 2021 [1].
+Thus price arbitrage in wholesale markets will be the main
+focus for future grid-scale energy storage projects.
+Energy storage arbitrages price differences and earns rev-
+enues in wholesale energy markets, i.e., charging during low-
+price periods and discharging during high-price periods. At
+the same time, arbitrage from energy storage helps to reduce
+renewable curtailments, meet peak demands, mitigate extreme
+events, and reduce the cost of electricity [2], [3]. As countries
+and regions ramp up decarbonization efforts, energy storage
+resources are taking on an increasingly important role in future
+electricity markets and are becoming a cornerstone for cost-
+effective decarbonization [4], [5]. Thus, both energy storage
+owners and market organizers have significant economic and
+welfare drivers to evolve models and algorithms for energy
+storage arbitraging robustly and profitably.
+However, energy storage arbitrage is non-trivial due to
+highly volatile electricity prices and limited storage capacity.
+Y. Baker, N. Zheng, and B. Xu are with Columbia University, NY, USA
+(e-mail: {ykb2105, nz2343, bx2177}@columbia.edu).
+Various methods have been proposed in the literature to ad-
+dress energy storage participation in wholesale markets based
+on different theories, they require dedicated location-specific
+tuning and excessive computing power to achieve competitive
+arbitrage performance [6]. This paper proposes a novel end-to-
+end system for opportunity value calculation, prediction, and
+control, combining model-based dynamic programming with
+neural networks. Our approach innovates and provides several
+advantages as follows:
+• Our approach has reliable performance as it uses model-
+based dynamic programming to address physical con-
+straints in both training and control stages;
+• Our approach is extremely computation efficient as it uses
+dynamic programming to pre-process the training data,
+reducing the complexity of the learning module;
+• Our approach is transferable to different market en-
+vironments while maintaining competitive performance
+because of the integration of transfer learning;
+• Our approach is founded on dynamic programming value
+functions and adapts to different storage market designs
+and participation scenarios, including price response and
+market economic bidding;
+• Our approach achieves state-of-the-art arbitrage perfor-
+mance, achieving 70% to near 90% profit ratio compared
+to perfect foresight with various storage durations when
+tested using price data from New York, US, and Queens-
+land, Australia.
+The rest of the paper is organized as follows: Section II
+summarizes energy storage market participation and previous
+work using the learning method, Section III and IV elaborates
+on the arbitrage formulation and solution method, Section V
+presents the case study for price response and economic bid
+market rules in New York and the application of transfer
+learning for Queensland, and Section VII concludes the paper.
+II. LITERATURE REVIEW
+A. Energy Storage Price Response and Self-Schedule
+Energy storage price response assumes the storage partici-
+pant can observe the real-time price realization first and then
+decide on the operation privately without informing the system
+operator. The price response participation option primarily
+applies to small-scale behind-the-meter (BTM) storage re-
+sources (< 1 MW) [7]. Plenty of prior works have investigated
+energy storage price response using a variety of methods,
+including model-predictive control (MPC) [8], stochastic pro-
+gramming [9], approximate dynamic programming [10], and
+reinforcement learning [11]. Price response is comparably an
+easier problem than economic bids as the storage operator is
+not limited to market clearing models and can act after observ-
+ing new price signals. However, since price response mostly
+arXiv:2301.01233v1  [cs.LG]  2 Jan 2023
+
+2
+applies to small BTM storage projects, the revenue generated
+from arbitrage will unlikely justify any specialized computing
+hardware investments. Hence the arbitrage algorithm must be
+slim and efficient to minimize the computation cost.
+Alternatively, some markets allow energy storage operators
+to self-schedule and submit the operational schedule to the
+market operator. Still, this option is less frequently used in
+practice compared to participating by economic bids [12].
+Self-scheduled storage cannot update the operation based on
+the system clearing price, a key difference compared to price-
+response or economic bids, which often causes the storage to
+miss price spike opportunities and deliver fewer market profits.
+B. Energy Storage Economic Bids
+FERC Order 841, issued in 2018, ordered all system oper-
+ators in the US must allow storage to submit bids and cleared
+in spot markets [13]. In this case, the storage participant must
+submit charge and discharge bids to the system operator at a
+specific time period ahead of the market clearing, usually one
+hour (also called hour-ahead bidding). The storage participant
+must follow market clearing results to charge or discharge,
+unlike in the price response case in which the storage can
+privately decide the control decision after observing the price.
+The bid design adds another layer of complexity in arbitrag-
+ing, as optimal bid design requires mathematical tools due to
+storage SoC constraints. Wang et al. [14] formulate the energy
+storage look-ahead profit maximization problem as a bi-level
+optimization problem. A second approach for energy storage
+arbitrage control is backward dynamic programming [15],
+and then the evolution is approximate-dynamic programming.
+Jiang and Powell outline a general approximate-dynamic pro-
+gramming framework for policy generation for energy storage
+operating with a stochastic generation source in response to
+stochastic demand [16], and further introduce a “distribution-
+free” variant of the previous algorithm that does not make any
+assumption on the price process[10]. However, all of these
+methods are held back by large computational costs that make
+them hard to implement in real-world applications of arbitrage.
+There are other algorithms for energy storage real-time
+arbitrage control: Wang and Zhang [17] solve the arbitrage
+problem using reinforcement learning to come to an optimal
+arbitrage policy, and Zheng et al. [18] outline a computa-
+tionally efficient analytical stochastic dynamic programming
+algorithm (SDP) for the problem of real-time price arbitrage
+of energy storage. Krishnamurthy et al. [19] also propose an
+SDP algorithm for arbitrage under day-ahead and real-time
+price uncertainties. However, none of the methods outlined
+above demonstrate or address transferability between different
+ISO zones and geographic locations, or the hour-ahead bid
+submission requirements in most real-time markets.
+C. Machine Learning for Storage Arbitrage
+Recent efforts to apply machine learning for storage ar-
+bitrage can be grouped into two thrusts: the first is to use
+machine learning to generate price predictions and then in-
+tegrate them with MPC. In this case, the learning module
+is independent of the storage model. Sarafraz et al. [20]
+and Nwulu and Fahrioglu [21] outline two machine learning
+approaches for predicting locational marginal price (LMP)
+prediction using neuro-fuzzy logic and soft computing re-
+spectively, and Chaweewat and Singh [22] propose a residual
+neural network approach to price interval prediction. The
+main difficulty in combining price prediction with storage
+optimization is storage arbitrage requires a look-ahead of at
+least 24 hours to capture the daily price cycles [8], while most
+real-time prediction methods may only accurately generate a
+few steps ahead of time. To this end, existing MPC approaches
+rely on pre-scheduling storage using day-ahead prices but have
+to neglect the real-time price variability, which is significantly
+higher than in day-ahead prices [9].
+The second approach is to directly use machine learning,
+mainly reinforcement learning (RL), to learn the optimal
+control policy for storage arbitrage directly. Wang et al. [11]
+developed the first RL approach to arbitrage storage in real-
+time markets. Cao et al. [23] propose a deep reinforcement
+learning approach to learn an optimal control policy for energy
+storage arbitrage with consideration of battery degradation.
+Kwon et al. [24] demonstrated RL could optimize more
+sophisticated storage models in arbitrage by integrating battery
+degradation into the model. Yet, a common disadvantage of
+RL-based approaches is transferability, as the model must
+undergo time-consuming training to be adapted to a new price
+zone or market environment. Transferability is a crucial aspect
+of storage arbitrage due to spatial and temporal variations: a
+typical system consists of hundreds of price nodes, and system
+price behaviors evolve with changes in system resource mix
+and ambient climate conditions. While previous efforts have
+looked into combining transfer learning with RL [25] and its
+application in selected energy-related issues, including demand
+response prediction [26], event identification [27], and battery
+health forecast [28]. Yet, the transferability of the storage
+arbitrage model has not been previously studied.
+III. PROBLEM STATEMENT AND SYSTEM OUTLINE
+Our algorithm aims to predict the opportunity value at the
+current state of charge (SoC) of energy storage to maximize
+the price arbitrage profit. Our system is composed of three
+components: valuation, forecasting, and arbitrage. We will
+first present our methods for valuation and arbitrage and then
+combine them with our forecasting model to form our bidding
+algorithm. We define Qt(e) as the opportunity value function
+representing the monetary value of the SoC e at time step
+t. The problem formulation is adapted from
+[29], [30], in
+which the solution is formulated using dynamic programming
+as follows:
+max
+bt,pt,et
+∈E(et−1)
+λt(pt − bt) − cpt + ˆQ
+�
+et|θ, X)
+(1a)
+where the first term is arbitrage revenue which is the product of
+the real-time market price λt and the energy storage dispatch
+decision (pt − bt), where pt is the discharge power and bt
+is the charge power. The second term is the discharge cost,
+where c is the marginal discharge cost. The third term ˆQ is
+the predicted storage opportunity value function with respect
+
+3
+Fig. 1. The proposed structure of training opportunity value function prediction model.
+to SoC et. The dynamic programming approach evaluates the
+energy storage by back-propagation, which is not viable in the
+real-time market where we do not have price realization ahead
+of time. Thus, we need to directly predict the value function
+ˆQ using historical (and current) price data. ˆQ is dependent
+on the prediction model parameters θ and the prediction input
+features X over a look-back period.
+We denote that the storage charge and discharge power and
+the final storage SoC belong to a feasibility set E(et−1) which
+is dependent on the storage starting SoC et−1 at the start of
+time period t (same as by the end of time period t−1). E(et−1)
+is described with the following constraints:
+0 ≤ bt ≤ P, 0 ≤ pt ≤ P
+(1b)
+pt = 0 if λt < 0
+(1c)
+et − et−1 = −pt/η + btη
+(1d)
+0 ≤ et ≤ E
+(1e)
+where (1b) models the upper bound, P, and lower bound, 0,
+constraints on the storage charge and discharge power. (1c) is a
+relaxed form of the constraint that enforces the energy storage
+not charging and discharging simultaneously. Negative price
+is the necessary condition for storage to charge and discharge
+simultaneously in price arbitrage, hence by enforcing the stor-
+age to not discharge when the price is negative we eliminate
+simultaneous charging and discharging [29]. (1d) models the
+energy storage SoC evolution constraint with efficiency η and
+(1e) models the upper bound E and lower bound (we assume
+as 0) of the storage SoC level.
+Creating our proposed system amounts to solving the
+problem of optimizing the prediction model parameters θ
+to maximize storage arbitrage profit over a set of training
+price data and physical storage parameters. Intuitively, this
+problem can be formulated as a bi-level problem in which
+the upper level maximizes the total profit over the entire
+training time horizon. At the same time, the lower-level
+enforces a non-anticipatory decision-making process in which
+the storage dispatch decision only depends on the current
+price and the predicted value function as in (1). However, this
+problem quickly becomes computationally intractable since
+the prediction model is embedded in the lower-level problem,
+formulated as a constrained optimization problem. Therefore,
+strong duality is required to convert the bi-level problem into a
+single-level equivalent problem or to derive partial derivatives
+and calculate the back-propagation gradients for gradient-
+based approaches. However, gradient-based approaches are
+complicated by the inclusion of SoC constraints [31]. In
+either case, the computational complexity quickly becomes
+overwhelming as the lower-level can include thousands of
+problems representing the arbitrage over a particular price data
+point.
+Problem Statement. We consider an alternative two-stage
+training approach in which we first generate the optimal
+opportunity value function and then train the learning model
+to predict the generated value function. This is formulated as
+min
+θ
+�
+e∈S
+���
+���ˆqt
+�
+e|θ, X) − qt(e)
+���
+���
+2
+2
+(2a)
+subject to
+qt(e) = ∂
+∂eQt(e)
+(2b)
+Qt−1(et−1) =
+max
+bt,pt,et
+∈E(et−1)
+λt(pt − bt) − cpt + Qt(et)
+(2c)
+Note that (2c) is also subject to the storage operation con-
+straint set E(et−1) as described in (1b)–(1e). (2c) is a dynamic
+programming energy storage price arbitrage formulation in
+which the storage opportunity value is defined recursively
+as the maximized storage arbitrage profit including the profit
+from the current time step and the future opportunity values.
+This formulation fits a piece-wise linear approximation of
+the value function qt(e) based on the first order derivative
+of the optimal value function Qt, and e is from the set of
+SoC segments S. Note that in this formulation the prediction
+model parameters θ are not involved in (2c), hence this is a
+two-stage model in which we solve (2c) first and obtain all
+optimal value function results from Qt, and more specifically,
+their derivatives qt. We are then able to use (2a) to solve for
+the optimal value function at each time step, which we use to
+train the prediction model.
+IV. SOLUTION AND SYSTEM SETUP
+Our approach includes three steps: first, we use the deter-
+ministic price arbitrage dynamic programming approach from
+the previous section to generate the optimal storage opportu-
+nity value function segments using historical price data. We
+then train a learning model to predict the optimal storage
+opportunity value segments from past price data. Finally, we
+
+qe(e)
+Analytical Dynamic
+Programming
+Algorithm (model
+based)
+[X, Y] = [ADAP/AkP] Q] 
+Model-Based
+Arbitrage
+CNN-LSTM
+Price Pre-
+Processing
+Opportu
+qt (el0, X) - qt(e4
+test the learned model over unseen (future) price datasets.
+The system structure is shown in Fig. 1, which includes
+the dynamic programming solution and training method, with
+specifics on the data engineering in Section IV-A.
+A. Feature and Label Formatting
+In general, the spot price for energy exhibits long-term
+and short-term cycles according to cycling demand: the daily
+cycling between peak and non-peak hours and the long-term
+seasonal cycles; though events and the stochastic nature of
+price create differences in between. Thus we chose to use
+a convolutional long short-term memory (ConvLSTM) neural
+net, which can learn patterns in time series data. For learning
+timestep t, our network input/target pair could be [λt, qt] (or
+qt+hr, where +hr represents an hour time shift for the HA
+case). However, to better capture daily cycling, we elaborate
+our single-step input-output pair by constructing the following
+input-output matrices:
+{X, Y} = {[ΛDAP|ΛRTP], Q}
+ΛDAP =
+�
+����
+λDAP,t−m
+λDAP,t−m+1
+. . .
+λDAP,t
+λDAP,t−m−1
+λDAP,t−m
+. . .
+λDAP,t−1
+...
+...
+...
+λDAP,t−m−5hr
+λDAP,t−m−5hr+1
+. . .
+λDAP,t−5hr
+�
+����
+ΛRTP =
+�
+����
+λRTP,t−n
+λRTP,t−n+1
+. . .
+λRTP,t
+λRTP,t−n−1
+λRTP,t−n
+. . .
+λRTP,t−1
+...
+...
+...
+λRTP,t−n−5hr
+λRTP,t−n−5hr+1
+. . .
+λRTP,t−5hr
+�
+����
+Q =
+�
+����
+qt
+qt−1
+...
+qt−5hr
+�
+����
+where ΛDAP and ΛRTP are matrices made up of our day ahead
+and real-time price data, and m, n are a lookback window for
+the day ahead and real-time prices respectively, and 5hr is the
+number of timesteps that make up five hours in a given market
+resolution (60 in a 5 min resolution market). This allows the
+network to capture not only the information on past prices for
+the current value function but also the relationship between
+past value functions in a 5-hour lookback. We chose five hours
+here as it is long enough to capture cycles within a single day
+(e.g. peak vs non-peak demand and the transition between
+them). The inclusion of the day ahead price here serves as
+a more stable price reference for the corresponding hour’s
+spot price. Also of note is the cyclic symmetry of the price
+matrices along the diagonal, which allows the network to learn
+better the equivariant properties of the dataset [32]. Finally, the
+choice of a ConvLSTM, as opposed to a traditional LSTM, is
+to allow the network to capture the ”vertical” temporal relation
+between the five hours of data in each data block.
+Note that for DAP, the shift applied to t across rows
+corresponds to a step shift in the resolution of the real-time
+market (RTM). Meaning that if it is a 5-minute resolution
+RTM, the first 12 rows of ΛDAP will be the same since the
+day-ahead market (DAM) is hourly resolution.
+B. Model Selection and Transfer Learning
+The focus of this paper is to demonstrate the robustness
+of the approach across different market conditions and bat-
+tery durations and to show its transferability between zones.
+Thus we chose one general network architecture for testing.
+However, initial experimentation showed minimal gain-loss
+in network performance on minor parameter changes across
+cases. Further, to guarantee that the training converges to a
+well-performing set of weights, multiple networks were trained
+for each case. The weights achieving the most consistent and
+low validation error were saved for evaluation. Of those, the
+best model was chosen by the highest arbitrage profit. The
+network is trained over 100 epochs in the case where it is
+trained from scratch, and 25 epochs for the transfer learning
+training, with a learning rate of 10−3. Further, we use a
+callback function that saves the model weights only when the
+validation error improves, ensuring that the weights loaded for
+training are not overfitted. This callback also allows us to set
+our epochs with significant overhead to ensure convergence
+without over-fitting in all cases.
+Furthermore, we apply transfer learning to quickly adapt a
+trained model from one price zone to another. Our transfer
+learning approach freezes all model layers except the output
+layer and retraining on the dataset of the task to be transferred
+to [33]. The underlying assumption is that the output layer is
+more sensitive to data variability while the rest of the network
+captures persistent patterns in the data.
+C. Full Algorithm
+We lay out our workflow, which is a sequence of three
+algorithms. As a prerequisite for model training, we generate
+all value functions Q according to the dynamic programming
+solution in VII-A. After this, we construct our data set and
+train our LSTM prediction model according to Algorithm 1,
+which produces our trained model weights θ.
+Algorithm 1 Value Function Prediction Model Training
+1: Dataset Preparation: Pre-Process data according to IV-A
+2: Initialization: Initialize model parameters θ using random
+seed.
+3: i ← 0
+4: while stop criteria not true do
+5:
+for t ∈ [1, t] do
+6:
+x ← [ΛDAP|ΛRTP]
+7:
+y ← Q
+8:
+Calculate Loss Components by Eq. (2a)
+9:
+Update θ by backpropagation
+10:
+end for
+11:
+i ← i + 1
+12: end while
+13: return θ
+▷ Parameters of the prediction model
+After this, if the trained LSTM model produced by Algo-
+rithm 1 is to then be used by another zone, it can be retrained
+using the transfer learning approach outlined in Algorithm
+2. This is largely the same as the workflow of algorithm
+1, save that the training dataset is of the new zone and the
+
+5
+newly trained model weights are denoted θ∗. We differentiate
+between the two sets of model weights since we compare
+the two approaches of transferring (transfer learning, applying
+the model on new zones without retraining) later in the
+paper. Finally, algorithm 3 outlines the process of simulating
+Algorithm 2 Transfer Learning
+1: Initialization: Initialize model parameters θ∗ using ran-
+dom seed
+2: θ∗ ← θ (trained model parameters)
+3: Freeze all parameters except output layer parameters
+4: Repeat
+training
+loop
+using
+new
+region’s
+data
+set
+{[Λ∗
+DAP|Λ∗
+RTP], Q∗}
+5: return θ∗
+▷ Parameters of the prediction model
+arbitrage using our prediction model. The arbitrage simulation
+is as follows: use the prediction model trained in algorithm
+1 and/or algorithm 2 to predict value functions using the
+current real-time price and a look-back window (including the
+day ahead look-back) and then generate the bids according
+to VII-B. Once the bids are generated, use them to simulate
+arbitrage and market clearing as outlined in VII-C.
+Algorithm 3 Arbitrage with Value Function Prediction
+1: Initialization:
+2: Set energy storage parameters c, P, ηp, ηb, E.
+3: Initialize et−1 ← e0.
+4: for t ∈ [1, T] do
+5:
+Predict ˆv
+�
+et|θ, x
+�
+6:
+Solve single-period optimization (1)
+7:
+Return et, pt, dt
+8: end for
+V. CASE STUDY SET-UPS
+A. Market Participation Setting and Storage Parameters
+We consider the following four market designs and par-
+ticipation settings to demonstrate that our proposed approach
+fits a wide range of storage participation options and market
+designs:
+• HA-1 Energy storage owner submits single-segment bids
+one hour-ahead to real-time markets. This represents the
+current storage bidding model in most wholesale real-
+time markets in the US [10], [34] where energy storage
+submits one charge bid and one discharge bid one hour
+ahead of the market clearing. The storage can update its
+bid for each hour, but the bids must stay the same within
+each hour for multiple market clearings (for example,
+real-time markets clear every five minutes in NYISO, so
+one hour includes 12 real-time clearings).
+• HA-10 Same to HA-1 except the storage submits10-
+segment SoC-dependent charge and discharge bids. This
+is a new market design proposed by CAISO to econom-
+ically manage storage SoC in real-time [35], [36].
+• PR-10 The storage conducts price response in real-time,
+deciding the storage control after observing the published
+real-time price, instead of submitting bids [37]. The price
+response option is limited to behind-the-meter storage
+in which the associated demand is cleared in real-time
+market prices. In this case, the storage is not limited to
+any bidding models and can use any decision-making
+models. Yet, we assume the storage uses a 10-segment
+approximation of its opportunity value as it provides a
+good enough approximation to the actual value function.
+This also enables us to benchmark HA-10 and PR-10
+cases to demonstrate the economic cost of the hour-ahead
+bidding requirement.
+• PR-1 Same as PR-10 except the storage uses the average
+opportunity value (i.e., one segment approximation) for
+arbitrage control. This is not a realistic case as there is no
+motivation for the storage operator to limit itself to using
+a single-segment, less accurate approximation of its value
+function to conduct arbitrage. However, we include this
+case with the sole purpose to benchmark against the HA-
+1 case and PR-10 case.
+In all case studies, we consider storage with a 90% one-
+way efficiency and a 10$/MWh cost of discharge (excluding
+the opportunity cost), unless otherwise specified. We consider
+three storage durations including 2-hour, 4-hour, and 12-hour.
+Further, we adapt our base prediction model to predict the hour
+ahead case by adding an hour time shift to our ground truth
+training target value function, which corresponds to 12-time
+steps in 5-minute price resolution.
+We conduct the majority of our case studies over price
+data from New York ISO (NYISO) [38] for four price zones:
+NYC (Zone J), LONGIL (Zone K), NORTH (Zone D), and
+WEST (Zone A). We also use data from the Australian Energy
+Market Operator (AEMO) for Queensland to demonstrate the
+transferability of our approach using transfer learning [39].
+B. Market and Price Data
+We observe differences in price statistics and generation
+mix across zones from the same ISO, and in between zones
+from ISO’s in other states and even countries, summarized in
+Table I. In New York zones, these differences can be attributed
+to significant transmission congestion when comparing the
+two main zone groups in NY [40]. QUEENSLAND has the
+highest price volatility, which can potentially be attributed to
+the absence of a day-ahead market. Further, we see a clear
+tie between penetration rates of renewables into the zones and
+the price volatility [41]. We also see the highest occurrence
+of negative prices in NORTH (NY), which is due to the
+significantly higher penetration of wind when compared to
+NYC and LONGIL.
+TABLE I
+PRICE DATA STATISTICS
+,
+Zone
+Negative Price #
+STD
+Renewable %
+NYC (NY)
+208
+28.82
+0.93
+LONGIL (NY)
+190
+50.17
+0.93
+NORTH (NY)
+6334
+40.25
+13.06
+WEST (NY)
+633
+37.55
+13.06
+QNSLND (AUS)
+522
+243.00
+13.19
+
+6
+Fig. 2. Accumulated Profit over 2019 test set for NYISO Zones
+All code for valuation, network training, and arbitrage are
+written in python with Jupyter notebook and is available on
+GitHub1. All trials are run on a desktop computer with AMD
+Ryzen 9 processor and Nvidia GPU on Tensorflow 2.9.1 and
+with cuDNN and CUDA versions 8.1 and 11.2, respectively.
+All case studies using price data from NYISO were trained
+using data from 2017 to 2018, and tested over 2019 data.
+Each year of price data for each price zone has 8760 day-
+ahead price data points (hourly resolution) and 105,120 real-
+time price points (5-minute resolution). The look-back price
+window includes the last 36 real-time prices (3 hours) and
+24-day-ahead prices (one day). The maximum training time
+over two years of training price data, including the generation
+of historical optimal value functions and training of the neural
+network, is 390 seconds, a bit more than five minutes. The net-
+work consists of a Convolutional Block with three sequential
+time-distributed Convolutions+MaxPool layers, then an LSTM
+block with two sets of bi-directional LSTM+drop out layers,
+and then finally a Dense layer at the output end. The specific
+model hyperparameters and details can be found on GitHub.
+VI. RESULTS
+A. Benchmark with Competing Methods
+We first benchmark our proposed approach with other
+competing energy storage price arbitrage methods in a price
+response setting, in which storage can observe price first
+and act accordingly, without having to bid ahead into mar-
+kets. We benchmark the proposed method (DP-ConvLSTM)
+with a reinforcement learning method (RL) [17], a modified
+stochastic dynamic programming with day-ahead price updates
+(SDP) [37], the proposed method but implemented with a
+multilayer perceptron (DP-MLP) network [42], and perfect
+price predictions which provide the highest profit possible.
+In RL, we have 11 actions, 103 price states, and 121 SoC
+states, which takes more than 1 hour to train for 5-min
+resolution arbitrage. The RL approach uses a Markov decision
+process (MDP) model by discretizing the storage SoC, and it
+only works with perfect efficiency (100%). To provide a fair
+comparison, all methods in this case consider storage with
+perfect efficiency.
+1https://github.com/ybaker661/LSTM-Value-Prediction
+Fig. 2 shows the comparison result when trained using price
+data from 2017-2018 and tested in 2019 at price zones in NY-
+ISO. The result shows DP-ConvLSTM has a clear advantage
+over other methods in terms of profitability. Notably, the DP-
+ConvLSTM approach performs exceptionally well in capturing
+low-frequency extreme events, such as the surge in profits
+around June in LONGIL and WEST, where the ConvLSTM
+captures profit spikes that the RL benchmark misses, and the
+difference between the ConvLSTM profit value and the perfect
+prediction comes from the difference in arbitrage decision as
+a result of numerical saturation. In this context, numerical
+saturation means that the network learns to predict numerical
+values in the range of data it most frequently sees (value
+functions of stable prices), and so when it predicts on anomaly
+data (price spike value functions) that are numerically much
+larger, the network prediction saturates at the largest common
+numerical value it sees.
+B. Price Response
+In this subsection, we compare the price response arbitrage
+performance (PR-1 and PR-10) with different storage dura-
+tions. Table III shows the arbitrage profit ratio results.
+Overall, the result shows stable performance over the four
+price zones and three storage durations. In comparison, our
+previous work using SDP [37] and DP-MLP [42] have worse
+performance in LONGIL (more frequent price spikes) and
+NORTH (more frequent negative prices). The comparison
+between PR-1 and PR-10 shows that increasing the value
+function approximation from one to ten segments increased
+the profit ratio by around 3%. Considering different storage
+durations, the profit ratio is lower for long-duration energy
+storage (12hr), as the longer storage duration leads to a
+longer temporal correlation into the future, leading to higher
+prediction difficulties. Still, our method achieved around 75%
+profit ratio (HA-10) in the worst-case scenario in the NORTH
+zone.
+C. Hour-ahead Bidding
+We now investigate hour-ahead bidding which is the most
+common market design for energy storage owners operating in
+the real-time market, where the storage submits bids an hour
+
+NYC
+LONGIL
+NORTH
+WEST
+16
+... Perfect Prediction
+.... Perfect Prediction
+..... Perfect Prediction
+.... Perfect Prediction
+DP+LSTM
+DP+LSTM
+DP+LSTM
+DP+LSTM
+14
+14
+DP+MLP
+DP+MLP
+DP+MLP
+DP+MLP
+-- SDP
+25
+-- SDP
+--- SDP
+---- SDP
+- RL
+RL
+RL
+2
+ 20
+ 20
+Profit
+Cumulative F
+6
+8
+10 
+12
+0
+2
+6
+10
+12
+2
+6
+8
+10
+12
+0
+2
+6
+8
+10
+12
+months
+months
+months
+months7
+TABLE II
+PROFIT RATIO FOR HA PREDICTION QUEENSLAND AUS WITH DIFFERENT AMOUNTS OF TRAINING DATA
+HA-1
+HA-10
+Duration
+Training
+No Data
+3 Days
+1 Week
+1 Month
+1 Year
+No Data
+3 Days
+1 Week
+1 Month
+1 Year
+2hr
+T.L.
+77.24
+82.76
+82.85
+81.30
+85.35
+78.90
+84.00
+81.54
+79.93
+83.42
+No T.L.
+X
+48.22
+51.36
+78.59
+83.88
+X
+44.59
+44.88
+78.10
+86.95
+4hr
+T.L.
+81.21
+81.29
+81.31
+78.11
+79.12
+84.06
+87.44
+84.34
+77.79
+82.65
+No T.L.
+X
+62.40
+65.45
+74.92
+80.42
+X
+55.99
+55.99
+74.36
+83.11
+12hr
+T.L.
+92.43
+83.96
+81.32
+78.74
+82.73
+90.69
+80.78
+79.11
+78.84
+81.35
+No T.L
+X
+74.79
+75.53
+74.39
+75.43
+X
+73.65
+73.59
+74.56
+82.36
+TABLE III
+CAPTURED PROFIT RATIOS: PRICE RESPONSE
+Zone
+PR-1
+PR-10
+2hr
+4hr
+12hr
+2hr
+4hr
+12hr
+NYC
+80.83
+79.96
+73.54
+83.69
+83.63
+75.67
+LONGIL
+82.33
+80.61
+79.10
+82.98
+83.94
+82.38
+NORTH
+78.24
+75.87
+71.02
+79.52
+79.96
+74.29
+WEST
+84.43
+80.37
+83.65
+87.97
+87.44
+84.43
+TABLE IV
+CAPTURED PROFIT RATIOS: HOUR AHEAD
+Zone
+HA-1
+HA-10
+2hr
+4hr
+12hr
+2hr
+4hr
+12hr
+NYC
+73.99
+74.82
+77.00
+78.79
+80.61
+74.47
+LONGIL
+74.26
+76.56
+82.01
+75.30
+79.63
+81.89
+NORTH
+73.22
+71.71
+70.17
+75.83
+77.21
+73.16
+WEST
+78.79
+80.13
+84.17
+83.12
+83.94
+84.60
+ahead of time. Table IV shows the hour-ahead bidding profit
+ratio in the NYC case study. The profit ratio is lower than the
+price response as the storage owner must decide on the bids
+one hour before the actual time of arbitrage. The short-duration
+storage (2hr) cases have higher profit ratio reductions (up to
+7%) as the value function is more sensitive to recent market
+prices due to it’s shorter duration. On the other hand, the long-
+duration storage (12hr) is more resilient and the hour-ahead
+bidding has little impact on the profit ratio.
+Hour-ahead bidding results also restate our observation from
+the price response case, that multi-segment SoC bids are
+more beneficial for short-duration storage to better manage
+their SoC, but the improvement is not obvious for long-
+duration storage. Overall, our approach achieved a higher
+than 70% profit ratio in all hour-ahead cases, showing ro-
+bust performance under different market designs and storage
+technologies.
+D. Transfer Learning in AEMO
+We now demonstrate the effectiveness of applying transfer
+learning to quickly adapt a pre-trained value function predic-
+tion model from one market to a new market. In this case
+study, we pre-train the prediction model using NYC price
+data from 2017-2018 and conduct arbitrage in Queensland,
+Australia. In Queensland, we use selected data from 2019
+for training and the first 6 months of 2021 for evaluation.
+We skipped the year 2020 because of COVID-19’s impact.
+To present the sensitivity of transfer learning over a limited
+amount of data, we consider various durations of training
+datasets ranging from 3 days to one year. We present a
+sensitivity analysis comparing the performance of transfer
+learning versus training a model from scratch for the situations
+where we have access to training data for only 3 days, 1 week,
+1 month, and 1 year of data for the target zone. Thus this case
+study has the following steps:
+1) Use a pre-trained network (transfer learning) or a ran-
+domly initialized network (training from scratch).
+2) Use a limited duration of Queensland price data from
+2019, ranging from 3 days to 1 year, to train the model
+using transfer learning as outlined in algorithm 2, or
+normal training outlined in algorithm 1.
+3) Test the arbitrage performance to arbitrage, as outlined
+in algorithm 3, using the first six-month of data in
+Queensland, 2021.
+Table II shows the arbitrage profit ratio results for Queens-
+land. The main finding is that in the data scare scenarios,
+the transfer learning approach vastly outperforms training a
+model from scratch. We also see that adding more data to
+the transfer learning case does not necessarily increase perfor-
+mance, whereas training the model from scratch only becomes
+a viable option once a certain amount of data is available.
+For the 2-hour storage, training from scratch becomes viable
+around the point where you have 1 month of data available for
+the target zone. For the 4-hour storage, the model still needs
+about 1 month of data to reasonably perform when trained
+from scratch, though the model is able to capture higher profit
+ratios for 3 and 1 week of data than the 2-hour case. Compared
+to both of these, the 12-hour storage seems to be the easiest for
+the model to learn, only needing 3 days of data when training
+from scratch to achieve reasonable performance; however, the
+12-hour storage shows that the transfer learning approach
+outperforms training from scratch for all data scenarios for 1
+and 10 segments. However, since the 12-hr storage has lower
+opportunity cost and less significant change in the opportunity
+cost between sequential time steps, predicting the opportunity
+value might not be an effective method.
+The takeaway is that transfer learning beats out training the
+model from scratch when data scarcity is an issue. However,
+when the dataset size increases to a general size of 1 month,
+training from scratch becomes a viable option. Additionally,
+adding extra data, past 3 days or 1 week for transfer learning
+and past 6 months for training from scratch, does not necessar-
+ily yield better performance. As such, it is more useful to focus
+on stabilizing ConvLSTM’s volatile and initialization-sensitive
+training as well as other changes to the training process. We
+
+8
+see that in almost all cases, using the model trained on NY
+data without any retraining performs comparably or even better
+than transfer learning and even training a model from scratch.
+This indicates statistical robustness and generality in the NY
+zone data, and it also points to a unified generating distribution
+behind the price data/opportunity value of zones. However, we
+cannot conclude this for certain without further analysis of
+other permutations of transfer learning, and on testing different
+data duration permutations (including different data scarcity
+scenarios in the training of the base model along with in
+transfer learning).
+VII. CONCLUSION
+In this paper, we propose a computation-efficient, versatile,
+and transferable energy storage arbitrage model that fits both
+price response and market bidding. Our proposed approach
+achieves state-of-the-art profits compared to other methods and
+is both computation and data-efficient. We also demonstrate
+that by incorporating transfer learning, we can quickly adapt
+our bidding model to a new location with very limited training
+data. Our model suits a variety of arbitrage settings, including
+behind-the-meter price response and economic bids for utility-
+scale storage, and can be implemented using non-proprietary
+software and regular computing hardware. Our work would fa-
+cilitate storage participation in electricity markets and promote
+economic decarbonization of the electric power system.
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+APPENDIX
+A. Dynamic Programming Solution Algorithm
+We first solve the dynamic programming problem as listed
+in (2c) subject to constraints (1b)–(1e). We use results from our
+prior work [29] to solve the dynamic programming problem
+(2c) and obtain the full piece-wise linear approximation of the
+opportunity value function Qt for all time periods (i.e., one
+value function for each time step for an entire year, 105120
+for 5 min price resolution 35040 for 20 min price resolution).
+We start by defining qt as the derivative of storage opportunity
+value function Qt, which represents the marginal opportunity
+value of energy stored in the storage. Then we can use an
+analytical formulation to calculate the opportunity value qt(e)
+at any given energy storage SoC level.
+Our prior work proved qt−1 can be recursively calculated
+with next period value function qt, power rating P, and effi-
+ciency η. The value function calculated using the deterministic
+formulation is thus
+qt−1(e) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+qt(e + Pη)
+if λt ≤ qt(e + Pη)η
+λt/η
+if qt(e + Pη)η < λt ≤ qt(e)η
+qt(e)
+if qt(e)η < λt ≤ [qt(e)/η + c]+
+(λt − c)η
+if [qt(e)/η + c]+ < λt
+≤ [qt(e − P/η)/η + c]+
+qt(e − P/η)
+if λt > [qt(e − P/η)/η + c]+
+(3)
+and calculates the opportunity value function assuming the
+price follows a recursive computation framework. This deter-
+ministic formulation is what we will use in our investigation,
+and from this we are able to calculate opportunity value
+function qt(e) at any time step using backwards recursion by
+defining an end period value function qT . We then discretize qt
+by splitting the energy storage SoC level e into small equally
+spaced segments, which must be far smaller than power rating
+P. For any SoC level et, we can find the nearest segment and
+return the corresponding value.
+B. Bid generation
+We now design discharge and charge bids using the oppor-
+tunity valuation results based on our prior work [30], [35]. We
+consider generating time-varying SoC-dependent bids with a
+total number of J segments for charge bids Bt,j and discharge
+bids Ct,j. Note that these bids represent the combination of
+the discharge cost and the change in the opportunity value. We
+assume each bid segment j is associated with an SoC range
+Ej−1 to Ej. The discharge bids are thus calculated based on
+the average value function between the internal Ej−1 and Ej
+Ct,j = 1
+E
+� Ej
+Ej−1
+∂
+∂pt
+(cpt − Qt(et−1 − pt/η + btη))det−1
+= c + 1
+E
+� Ej
+Ej−1
+qt(et−1 − pt/η + btη)det−1/η
+≈ c + 1
+ηE
+� Ej
+Ej−1
+qt(e)de
+Similarly for charge bids
+Bt,j = 1
+E
+� Ej
+Ej−1
+∂
+∂bt
+(cpt − Qt(et−1 − pt/η + btη))det−1
+= 1
+E
+� Ej
+Ej−1
+qt(et−1 − pt/η + btη)det−1η
+≈ η
+E
+� Ej
+Ej−1
+qt(e)de
+In the special case of one segment, i.e., bids are not
+dependent on SoC (the current energy storage bidding model
+in most wholesale markets), Ej−1 is zero or the lowest allowed
+SoC and Ej is the highest allowed SoC value or the energy
+capacity. In this case the bids are simply based on the average
+marginal opportunity value ¯qt
+¯qt = 1
+E
+� E
+0
+qt(e)de
+(4)
+and the discharge bid is c + ¯qt/η, and the charge bid is ¯qtη.
+C. Real-time market clearing and arbitrage simulation
+We consider the following simplified real-time market clear-
+ing model with a generalized multi-segment energy storage
+bids
+min
+pt,j,s,dt,j,s
+Jt(gt) +
+�
+s
+�
+j
+(Ct,j,sdt,j,s − Bt,j,sbt,j,s) (5a)
+s.t.
+et,j,s − et−1,j,s = bt,j,sη − pt,j,s/η
+(5b)
+0 ≤ et,j,s ≤ Ej,s − Ej−1,s
+(5c)
+gt +
+�
+s
+�
+j
+pt,j,s = Dt +
+�
+s
+�
+j
+bt,j,s : λt
+(5d)
+where (5a) is the objective function minimizing total bidding
+costs. Note that we use aggregated generator supply curve
+Jt(gt) and total generation gt instead of modeling the bids
+from each individual generator for simplicity to focus on
+energy storage. The second term of the objective is the
+discharge bids and charge bids for each energy storage s and
+
+10
+each SoC segment j. (5b) models the SoC evolution under
+single-trip efficiency η for each SoC segment. (5c) models
+the upper and lower energy limit for each SoC segment, note
+that the minimum energy is always zero while the maximum
+energy for each segment is the difference between the upper
+and lower SoC range Ej,s − Ej−1,s. Finally, (5d) is the
+power balance constraint enforcing the sum of generation and
+storage charge/discharge equals to the total demand Dt over
+time period t, the associated dual variable is thus the market
+clearing price λt.
+Now in price-taker analysis, we use historical price data to
+simulate how the energy storage would have been cleared in
+the market. In this case, we perform a Lagrangian relaxation
+of (5d) and move it to the objective. This decomposes the
+optimization into independent sub-problems for each energy
+storage, and for each storage, the price-taker market clearing
+problem is equivalent to the following price arbitrage problem
+max
+pt,j,dt,j λt
+�
+j
+(dt,j − bt,j) −
+�
+j
+(Ct,jdt,j − Bt,jbt,j)
+(6)
+subject to the same storage unit constraints (5b) and (5c).
+Note that for this problem we omit the storage unit index
+s as the problem formulation is the same for each storage.
+Hence, price-taker market clearing simulation is equivalent to
+arbitrage using the same bidding cost model. While we did
+not consider the network model in this formulation, the price-
+taker market clearing model is the same should we use nodal
+prices.
+Note that the formulation in (6) applies to both price-taker
+market bidding (HA-1 and HA-10) and price response (PR-1
+and PR-10). The difference is that in HA cases, storage has
+to decide the bids (Ct,j and Bt,j) one hour before the market
+clearing period t, while in PR cases storage updates bids at
+the same time when observing the price.
+

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@@ -0,0 +1,1725 @@
+Anti-Exploration by Random Network Distillation
+Alexander Nikulin 1 Vladislav Kurenkov 1 Denis Tarasov 1 Sergey Kolesnikov 1
+Abstract
+Despite the success of Random Network Distilla-
+tion (RND) in various domains, it was shown as
+not discriminative enough to be used as an uncer-
+tainty estimator for penalizing out-of-distribution
+actions in offline reinforcement learning. In this
+paper, we revisit these results and show that, with
+a naive choice of conditioning for the RND prior,
+it becomes infeasible for the actor to effectively
+minimize the anti-exploration bonus and discrim-
+inativity is not an issue. We show that this lim-
+itation can be avoided with conditioning based
+on Feature-wise Linear Modulation (FiLM), re-
+sulting in a simple and efficient ensemble-free
+algorithm based on Soft Actor-Critic. We eval-
+uate it on the D4RL benchmark, showing that it
+is capable of achieving performance comparable
+to ensemble-based methods and outperforming
+ensemble-free approaches by a wide margin. 1
+1. Introduction
+In recent years, significant success has been achieved in ap-
+plying Reinforcement Learning (RL) to challenging and
+large-scale tasks such as Atari (Badia et al., 2020), Go
+(Schrittwieser et al., 2020), Dota 2 (Berner et al., 2019),
+and Minecraft (Baker et al., 2022). However, the online na-
+ture of such RL algorithms makes it difficult to apply them
+in the real world, where online collection of large amounts
+of exploratory data may not be feasible for safety or fi-
+nancial reasons. Offline Reinforcement Learning (Levine
+et al., 2020) promises a more controllable and data-driven
+approach, focusing on algorithms that can learn from a fixed,
+pre-recorded dataset without requiring additional environ-
+ment interactions.
+The use of ensembles for uncertainty-based penalization has
+proven to be one of the most effective approaches for offline
+RL. Ensemble-based algorithms, such as SAC-N, EDAC
+(An et al., 2021), and MSG (Ghasemipour et al., 2022)
+1Tinkoff, Moscow, Russia. Correspondence to: Alexander
+Nikulin <[email protected]>.
+1Our implementation is available at https://github.
+com/tinkoff-ai/sac-rnd
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+training steps
+1e6
+0
+20
+40
+60
+80
+100
+average normalized score
+CQL (ensemble-free)
+SAC-N (ensemble-based)
+SAC-RND (Naive)
+SAC-RND (Ours)
+Figure 1. Mean performance of SAC-RND variants on walker and
+hopper medium-* datasets, each averaged over 3 seeds. We plot
+performance for the naive version, which uses concatenation con-
+ditioning, and our final version, which is described in Section 5.
+We also plot the final scores for the ensemble-free CQL (Kumar
+et al., 2020) and the ensemble-based SAC-N (An et al., 2021). It
+can be seen that our version is a significant improvement over the
+naive version, achieving performance comparable to ensembles.
+currently achieve state-of-the-art results on most D4RL (Fu
+et al., 2020) datasets, outperforming ensemble-free methods
+by a wide margin. Unfortunately, in order to achieve the best
+performance, these algorithms may require tens or hundreds
+of ensemble members, leading to significant computational
+and memory overhead, as well as extended training duration
+(Nikulin et al., 2022).
+Recent research (Yang et al., 2022) has successfully reduced
+the ensemble size to tens of Q-networks in the worst-case
+scenarios. However, given the general trend for model scal-
+ing in offline RL (Kumar et al., 2022; Reed et al., 2022; Lee
+et al., 2022), efficiently training even ten Q-networks with
+80 million parameters each is not feasible. Furthermore,
+Ghasemipour et al. (2022) showed that methods for efficient
+ensemble training found in supervised learning literature
+do not deliver performance comparable to naive ensembles
+and can even worsen the results. Thus, further research
+on efficient uncertainty estimation for offline RL is needed,
+with the goal of reducing the size of the ensemble as much
+as possible or even fully removing it.
+In this work, we move away from ensembles and take an
+alternative approach to uncertainty estimation, proposing an
+arXiv:2301.13616v1  [cs.LG]  31 Jan 2023
+
+Anti-Exploration by Random Network Distillation
+efficient offline RL method with ensemble-free uncertainty
+estimation via Random Network Distillation (RND) (Burda
+et al., 2018). RND, a simple and fast ensemble competitor
+for epistemic uncertainty estimation (Ciosek et al., 2019),
+is an attractive choice for offline RL. However, previous
+research (Rezaeifar et al., 2022) found RND to be insuffi-
+ciently discriminative for good results.
+In our preliminary experiment (Section 3), we show that
+RND is discriminative enough to detect OOD actions, which
+contradicts the previous study (Rezaeifar et al., 2022). Nev-
+ertheless, our results show that the naive application of RND
+does indeed not lead to good results (see Figure 1). Building
+upon these findings, we further simplify the problem and
+analyze the reasons for this issue (Section 4). We discover
+that a naive choice of conditioning for the RND prior can
+hinder the minimization of the anti-exploration bonus by
+the actor, and that conditioning based on Feature-wise Lin-
+ear Modulation (FiLM) (Perez et al., 2018) is particularly
+effective in solving this problem.
+Based on our findings, we propose a new ensemble-free of-
+fline RL algorithm called SAC-RND (Section 5). We eval-
+uate our method on the D4RL (Fu et al., 2020) benchmark
+(Section 6), and show that SAC-RND achieves performance
+comparable to ensemble-based methods while outperform-
+ing ensemble-free approaches.
+2. Background
+Offline Reinforcement Learning. Reinforcement learning
+problem can be described as a Markov Decision Process
+(MDP) defined by the {S, A, P, R, γ} tuple with state space
+S ⊂ RN, action space A ⊂ RM, transition dynamics P :
+S × A → S, reward function R : S × A → R, and a
+discount factor γ. The goal of reinforcement learning in
+an infinite horizon setting is to produce a policy π(a|s)
+that maximizes the expected cumulative discounted return
+Eπ[�∞
+t=0 γtr(st, at)].
+In offline reinforcement learning, a policy must be learned
+from a fixed dataset D collected under a different policy or
+mixture of policies, without any environment interaction.
+This setting poses unique fundamental challenges (Levine
+et al., 2020), since the learning policy is unable to explore
+and has to deal with distributional shift and extrapolation
+errors (Fujimoto et al., 2019) for actions not represented in
+the training dataset.
+Offline RL as Anti-Exploration. There are numerous ap-
+proaches for offline RL, a substantial part of which constrain
+the learned policy to stay within the support of the train-
+ing dataset, thus reducing (Kumar et al., 2020) or avoiding
+(Kostrikov et al., 2021) extrapolation errors. For our work,
+it is essential to understand how such a constraint can be
+framed as anti-exploration (Rezaeifar et al., 2022).
+Similarly to online RL, where novelty bonuses are used as
+additive intrinsic rewards for effective exploration, in offline
+RL, novelty bonuses can induce conservatism, reducing the
+reward in unseen state-action pairs. Hence the name anti-
+exploration, since the same approaches from exploration
+can be used, but a bonus is subtracted from the extrinsic
+reward instead of being added to it.
+However, unlike online RL, subtracting a bonus from the
+raw reward would not be as useful, since the novelty bonus
+is, by design, close to zero for in-dataset state-action pairs.
+Therefore, it is more effective to apply it where the overesti-
+mation for OOD actions emerges — the temporal difference
+learning target:
+r + γEa′∼π(·|s′)[Q(s′, a′) − b(s′, a′)]
+(1)
+where the actor is trained to maximize the expected Q-value,
+as is usually done in off-policy actor-critic algorithms (Lil-
+licrap et al., 2015; Haarnoja et al., 2018). It can be shown
+that, theoretically, these approaches are equivalent, but the
+latter is more suited for use in offline RL (Rezaeifar et al.,
+2022).
+An illustrative example of how such framing can be effective
+are ensemble-based approaches such as SAC-N & EDAC
+(An et al., 2021) and MSG (Ghasemipour et al., 2022),
+which currently outperform their ensemble-free counterparts
+by a large margin on most D4RL (Fu et al., 2020) benchmark
+datasets. For the anti-exploration bonus, these methods use
+ensemble disagreement as a proxy for epistemic uncertainty.
+However, a large number of ensemble members is usually
+required for a competitive result.
+Random Network Distillation. Random network distilla-
+tion (RND) was first proposed in online RL (Burda et al.,
+2018) as a simple and effective exploration bonus. To this
+day, RND is still considered a strong baseline for explo-
+ration that can work well even in stochastic environments,
+contrary to some more modern approaches (Jarrett et al.,
+2022).
+RND consists of two neural networks: a fixed and randomly
+initialized prior network ¯f ¯
+ψ, and a predictor network fψ
+which learns to predict the prior outputs on the training data:
+∥fψ(s) − ¯f ¯
+ψ(s)∥2
+2
+(2)
+Both networks map states to embeddings in RK, and the
+gradient through prior is disabled. The interpretation of
+the novelty is straightforward: with the sufficiently diverse
+prior, the predictor must learn to match embeddings on data
+points similar to the training dataset, while failing to predict
+on new examples. A bonus in such a case may simply be a
+prediction error, as in Equation (2).
+
+Anti-Exploration by Random Network Distillation
+In a subsequent work, Ciosek et al. (2019) analyses the
+success of RND in a supervised setting, and shows that
+fitting random priors can be a competitive alternative to
+ensembles for estimating epistemic uncertainty.
+Note that in practice, the choice of predictor and prior having
+the same architecture and the estimation of novelty from
+states only are very common, but arbitrary. Moreover, for
+offline RL, we are interested in estimating the novelty of an
+action conditioned on the state, which is why in our work
+RND depends on both: fψ(s, a).
+Multiplicative Interactions. The most common way to
+fuse two different streams of information is feature con-
+catenation, which is straightforward but can be suboptimal
+(Dumoulin et al., 2018). Jayakumar et al. (2020) shows that
+multiplicative interactions provide a powerful inductive bias
+for fusing or conditioning from multiple streams and are
+superior in practice. We provide a brief review of those used
+in our work (excluding concatenation): gating, bilinear, and
+feature-wise linear modulation (FiLM).
+Gating. Simple conditioning with two linear layers and
+pointwise multiplication of the resulting features (Srivastava
+et al., 2019).
+f(a, s) = tanh(W1a + b1) ⊙ σ(W2s + b2)
+Bilinear. Bilinear layer in its most general form, as pro-
+posed by Jayakumar et al. (2020).
+f(a, s) = sT Wa + sT U + Va + b
+where W is a 3D tensor, U, V are regular matrices and b
+is a vector. However, in our work, we also use the imple-
+mentation as in PyTorch, which does not learn U, V by
+default.
+FiLM. Special case of a bilinear layer with low-rank weight
+matrices (Perez et al., 2018).
+f(h, s) = γ(s) ⊙ h + β(s)
+Usually, FiLM operates on hidden activations h before non-
+linearity between layers. Thus, the main network takes a as
+an input.
+3. Random Network Distillation is
+Discriminative Enough
+To better understand the possible difficulties of applying
+RND to offline RL, we first reproduce the main experiment
+from Rezaeifar et al. (2022), which showed that RND is not
+discriminative enough to be used as a novelty bonus. For
+convenience, we provide the original figure from Rezaeifar
+et al. (2022) in the Appendix A. We also compare RND with
+0.0
+2.5
+5.0
+7.5
+10.0
+standard deviation
+0
+20000
+40000
+Q-Ensemble
+0.0
+2.5
+5.0
+7.5
+10.0
+prediction error
+RND
+dataset
+uniform
+dataset + noise (std .25)
+dataset + noise (std .5)
+Figure 2. Anti-exploration bonus (Rezaeifar et al., 2022) on the
+walker2d-medium dataset for trained SAC-N (An et al., 2021),
+Q-ensemble (N = 25) and RND. Bonus is computed for state-
+action pairs from the original dataset and different perturbations of
+actions: random actions, dataset actions to which Gaussian noise
+is added with different scales. Both RND networks use simple
+state-action concatenation. The result is strikingly different from a
+similar figure in the Rezaeifar et al. (2022) (we provide the original
+figure in the Appendix A for convenience). Contrary to previous
+research, it can be seen that RND is capable of distinguishing ID
+from OOD actions and is comparable to a trained Q-ensemble.
+a trained Q-ensemble (N = 25) from the SAC-N algorithm
+(An et al., 2021). Similarly to Rezaeifar et al. (2022), we
+use simple state-action concatenation. Predictor and prior
+share the identical architecture of 4-layer MLPs.
+The goal of the experiment (see Figure 2) is to visually
+plot the anti-exploration bonus for ID state-action pairs
+and different perturbations of actions to model OOD data:
+random actions sampled from a uniform distribution and
+dataset actions to which Gaussian noise with different scales
+is added.
+To our surprise, the result on Figure 2 is strikingly different
+from previous work. It shows that RND is able to discrim-
+inate between ID and OOD actions with varying degrees
+of distributional shift and is comparable to a trained Q-
+ensemble. In contrast, Rezaeifar et al. (2022) hypothesizes
+that RND can only work well out of the box for discrete ac-
+tion spaces and visual features, and concludes that extending
+it to continuous action spaces is not straightforward.
+After further investigation of the open-sourced codebase2 in
+search of discrepancies with our implementation, we found
+that the only difference is that, contrary to the advice of
+Ciosek et al. (2019), Rezaeifar et al. (2022) sets the predictor
+smaller than prior by two layers during RND pretraining. It
+is important to make the predictor larger or comparable in
+capacity to the prior so that it can minimize the loss to zero
+on the training dataset (Ciosek et al., 2019). However, the
+actual RND hyperparameters used in the final publication
+were not listed, so we cannot draw a definitive conclusion
+about the reason for such different results.
+2https://github.com/shidilrzf/Anti-exploration-RL
+
+Anti-Exploration by Random Network Distillation
+4. Concatenation Prior Hinders Bonus
+Minimization
+A well-behaved anti-exploration bonus for continuous action
+spaces, be it RND or any other, should satisfy at least two
+criteria. First, it should be discriminative enough to detect
+novel actions and downweight their value estimates (see
+Equation (1)). Ideally, the bonus should be close to zero for
+ID data so that we do not bias the Q-function, as this can
+be detrimental to training. Second, it should allow the actor
+to easily minimize the bonus with gradient descent during
+training.
+In Section 3, we showed that RND can detect OOD ac-
+tions. Nevertheless, naive use of RND as an anti-exploration
+bonus on top of the Soft Actor Critic algorithm (Haarnoja
+et al., 2018) still does not provide satisfactory performance
+(see Figure 1) with scores lower than CQL (Kumar et al.,
+2020) and SAC-N (An et al., 2021). This gives us an hint
+that the problem may not be the discriminative power of
+RND, but that the actor cannot effectively minimize the
+anti-exploration bonus during training.
+To test our hypothesis that the actor cannot effectively min-
+imize the anti-exploration bonus, we further simplify the
+problem by removing the critic from the SAC algorithm
+but keeping the entropy bonus (see Algorithm 2 in the Ap-
+pendix). We expect that, in such a setting, the actor will be
+able to successfully minimize the anti-exploration bonus to
+the possible minimum, i.e. comparable to the bonus for the
+ground truth data at the end of the RND pretraining. As a
+consequence, since dataset actions provide the minimum
+bonus by design, we also expect that the distance from the
+agent to dataset actions should be small.
+We set predictor architecture to state-action concatenation.
+Additionally, we explore different conditioning schemes for
+the prior. We use the halfcheetah, walker2d and hopper
+medium datasets, with 3 seeds each. Figure 3 compares
+the anti-exploration bonus for dataset actions during RND
+pretraining (see Figure 3a) and for agent actions during
+training (see Figure 3b).
+As one can see for all prior architectures except one, the
+anti-exploration bonus during actor training is much higher
+than it should be according to the values on the dataset
+actions. These results confirm our hypothesis. Furthermore,
+we can note from Figure 3c that the actor cannot clone the
+behavioral policy, since the distance to the dataset actions
+can even increase during training.
+However, RND with the FiLM prior architecture allows the
+actor to effectively minimize the anti-exploration bonus and
+successfully clone the behavioral policy. This suggests that,
+with the right inductive bias for the prior, we can solve the
+problems of naive RND and possibly achieve better results.
+Table 1. Comparison of different RND predictors. Prior uses FiLM
+conditioning. Predictor uses conditioning in the first layer. All
+scores are averaged over 3 random seeds. Halfcheetah tasks are
+ommited, as we found them non-representative of the final perfor-
+mance on harder tasks.
+Task Name
+Concat
+Gating
+Bilinear
+FiLM
+hopper-medium-v2
+94.8
+39.7
+98.4
+86.3
+hopper-medium-expert-v2
+71.5
+59.3
+110.3
+102.7
+hopper-medium-replay-v2
+100.3
+51.3
+100.8
+100.3
+walker2d-medium-v2
+94.8
+82.3
+92.8
+95.1
+walker2d-medium-expert-v2
+86.1
+84.2
+108.9
+110.0
+walker2d-medium-replay-v2
+90.3
+87.5
+88.3
+75.7
+Average
+89.6
+67.3
+99.9
+95.0
+5. Anti-Exploration by Random Network
+Distillation
+We are now ready to present SAC-RND: a new offline RL
+method for continuous action spaces, based on our findings
+in Section 3 and Section 4. It is simple, ensemble-free and
+achieves state-of-the-art results comparable to ensemble-
+based methods.
+We have chosen the Soft Actor-Critic
+(Haarnoja et al., 2018) algorithm as the backbone of the
+method. In this section, we will explain how the RND is
+trained and how we define the anti-exploration bonus.
+Random Network Distillation. We pretrain RND with
+MSE loss between prior and predictor embeddings, stop-
+ping gradient through prior and freezing both networks after-
+wards during SAC training. We keep both networks similar
+in size to the agent and critic, which are 4 layer MLPs. Con-
+trary to Burda et al. (2018); Ciosek et al. (2019), we do not
+add additional layers to the predictor to prevent undesirable
+results. This is because, when the predictor size is bigger
+than prior on state-based tasks (not image-based as in orig-
+inal work by Burda et al. (2018)), we observe that it can
+sometimes overgeneralize to OOD prior embeddings.
+According to Section 4, for the prior, we use FiLM condi-
+tioning on penultimate layer before nonlinearity. In prin-
+ciple, the predictor can be arbitrary (Ciosek et al., 2019),
+but in practice, its architecture and conditioning type can
+also affect performance. We conduct a preliminary study
+on a small subset of the D4RL Gym tasks to select the best-
+performing conditioning. Based on the results in Table 1,
+we chose a predictor with bilinear conditioning in the first
+layer, as it showed the best performance.
+Anti-Exploration Bonus. We define the anti-exploration
+bonus similarly to RND loss as
+b(s, a) = ∥fψ(s, a) − ¯f ¯
+ψ(s, a)∥2
+2
+(3)
+and additionally divide it by RND loss running standard
+deviation (which is tracked during pretraining phase) to
+increase its scale uniformly among environments. Such
+
+Anti-Exploration by Random Network Distillation
+0
+50000
+100000
+150000
+200000
+250000
+300000
+350000
+training steps
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+RND bonus
+Concat
+FiLM
+Bilinear
+Gated
+(a) RND bonus for dataset actions
+0
+20000
+40000
+60000
+80000
+100000
+training steps
+0
+1
+2
+3
+4
+5
+6
+7
+RND bonus
+Concat
+FiLM
+Bilinear
+Gated
+(b) RND bonus for actor actions
+0
+20000
+40000
+60000
+80000
+100000
+training steps
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+mean squared error
+Concat
+FiLM
+Bilinear
+Gating
+(c) Distance to dataset actions
+Figure 3. Effect of different state-action conditioning in the prior of RND on actor training. We use the halfcheetah, walker2d and hopper
+medium datasets, with 3 seeds each. For training procedure, see Algorithm 2 in the Appendix. (a) Anti-exploration bonus for in-dataset
+actions during RND pretraining. We additionally divide the bonus by the RND loss running standard deviation to increase its scale
+(see Section 5) so the anti-exploration bonus increases slightly over time as standard deviation decreases. However, this does not affect
+minimization by the actor and is needed to highlight the differences. (b) Anti-exploration bonus for actor actions during training. Ideally,
+it should converge to values close to the final values in (a). (c) Distance of actor actions to true in-dataset actions during training. Ideally,
+it should decrease, as actions closer to the behavioral policy have the lowest bonus by design.
+scaling simplifies hyperparameter search, shrinking the pos-
+sible range of useful α coefficients that control the level of
+conservatism during training.
+For detailed training procedure and full SAC losses, we
+refer to Algorithm 1 in the Appendix (differences with the
+original SAC algorithm are highlighted in blue).
+6. Experiments
+In this section, we present an empirical evaluation of our
+method using the D4RL benchmark on the Gym domain
+(Section 6.1) and the more challenging AntMaze domain
+(Section 6.2). Next, we provide additional analysis and
+visual insight into why FiLM conditioning in the prior might
+be beneficial (Section 6.3). Finally, we present an ablation
+that compares more variations of conditioning for predictor
+and prior (Section 6.4). For each experiment, we also list the
+exact hyperparameters in Appendix D and implementation
+details in Appendix C. Additionally, we analyse sensitivity
+to hyperparameters in Appendix E.
+6.1. Evaluation on the Gym Domain
+Setup. We evaluate our method on all available datasets for
+the HalfCheetah, Walker2d and Hopper tasks in the Gym do-
+main of the D4RL benchmark. For ensemble-free baselines,
+we chose CQL (Kumar et al., 2020), IQL (Kostrikov et al.,
+2021), TD3+BC (Fujimoto & Gu, 2021), which show good
+results and are widely used in practice. For ensemble-based
+baselines, we chose SAC-N & EDAC (An et al., 2021) and
+the more recent RORL (Yang et al., 2022), which currently
+achieve state-of-the-art scores in this domain. We follow the
+An et al. (2021) and train for 3M gradient steps, evaluating
+on 10 episodes.
+Results. The resulting scores are presented in Table 2. We
+see that SAC-RND stands out from the ensemble-free meth-
+ods and outperforms them by a wide margin, achieving a
+mean score comparable to EDAC and only slightly behind
+RORL. Note that we do not use ensembles, whereas SAC-N
+can require up to 500 critics, EDAC up to 50 and RORL up
+to 20. In addition, we compare our proposed changes with
+the naive predictor and prior, confirming that our modifi-
+cations are essential for achieving good performance (see
+Figure 1).
+6.2. Evaluation on the AntMaze Domain
+Setup. We evaluate our method on all datasets available for
+the AntMaze domain of the D4RL benchmark. Ensemble-
+free baselines are the same as in Section 6.1. For ensemble-
+based baselines, we chose RORL (Yang et al., 2022) and
+MSG (Ghasemipour et al., 2022), the latter of which, to
+our knowledge, currently has the best mean score for this
+domain. We do not include SAC-N and EDAC, as there are
+no public results for them on this domain, and we were also
+unable to obtain a non-zero result. We follow the An et al.
+(2021) and train for 3M gradient steps, evaluating on 100
+episodes.
+Results. The resulting scores are presented in Table 3.
+Kostrikov et al. (2021) has shown that many offline RL
+methods that perform well on the Gym domain fail on the
+AntMaze domain. It can be seen that, on the AntMaze do-
+main, SAC-RND shows good results that are on par with
+ensembles, and outperforms ensemble-free methods. This
+also shows that our choice of predictor and prior generalises
+well to new domains. Note that, in addition to ensembles,
+both MSG and RORL require pre-training or supervision
+with behavioural cloning in order to achieve reported results,
+
+Anti-Exploration by Random Network Distillation
+Table 2. SAC-RND evaluation on the Gym domain. We report the final normalized score averaged over 4 random seeds on v2 datasets.
+TD3 + BC and IQL scores are taken from Lyu et al. (2022). CQL, SAC-N and EDAC scores are taken from An et al. (2021). RORL
+scores are taken from Yang et al. (2022).
+Ensemble-free
+Ensemble-based
+Task Name
+TD3+BC
+IQL
+CQL
+SAC-N
+EDAC
+RORL
+SAC-RND
+halfcheetah-random
+11.0 ± 1.1
+13.1 ± 1.3
+31.1 ± 3.5
+28.0 ± 0.9
+28.4 ± 1.0
+28.5 ± 0.8
+29.0 ± 1.5
+halfcheetah-medium
+48.3 ± 0.3
+47.4 ± 0.2
+46.9 ± 0.4
+67.5 ± 1.2
+65.9 ± 0.6
+66.8 ± 0.7
+66.6 ± 1.6
+halfcheetah-expert
+96.7 ± 1.1
+95.0 ± 0.5
+97.3 ± 1.1
+105.2 ± 2.6
+106.8 ± 3.4
+105.2 ± 0.7
+105.8 ± 1.9
+halfcheetah-medium-expert
+90.7 ± 4.3
+86.7 ± 5.3
+95.0 ± 1.4
+107.1 ± 2.0
+106.3 ± 1.9
+107.8 ± 1.1
+107.6 ± 2.8
+halfcheetah-medium-replay
+44.6 ± 0.5
+44.2 ± 1.2
+45.3 ± 0.3
+63.9 ± 0.8
+61.3 ± 1.9
+61.9 ± 1.5
+54.9 ± 0.6
+halfcheetah-full-replay
+-
+-
+76.9 ± 0.9
+84.5 ± 1.2
+84.6 ± 0.9
+-
+82.7 ± 0.9
+hopper-random
+8.5 ± 0.6
+7.9 ± 0.2
+5.3 ± 0.6
+31.3 ± 0.0
+25.3 ± 10.4
+31.4 ± 0.1
+31.3 ± 0.1
+hopper-medium
+59.3 ± 4.2
+66.2 ± 5.7
+61.9 ± 6.4
+100.3 ± 0.3
+101.6 ± 0.6
+104.8 ± 0.1
+97.8 ± 2.3
+hopper-expert
+107.8 ± 7.0
+109.4 ± 0.5
+106.5 ± 9.1
+110.3 ± 0.3
+110.1 ± 0.1
+112.8 ± 0.2
+109.7 ± 0.3
+hopper-medium-expert
+98.0 ± 9.4
+91.5 ± 14.3
+96.9 ± 15.1
+110.1 ± 0.3
+110.7 ± 0.1
+112.7 ± 0.2
+109.8 ± 0.6
+hopper-medium-replay
+60.9 ± 18.8
+94.7 ± 8.6
+86.3 ± 7.3
+101.8 ± 0.5
+101.0 ± 0.5
+102.8 ± 0.5
+100.5 ± 1.0
+hopper-full-replay
+-
+-
+101.9 ± 0.6
+102.9 ± 0.3
+105.4 ± 0.7
+-
+107.3 ± 0.1
+walker2d-random
+1.6 ± 1.7
+5.4 ± 1.2
+5.1 ± 1.7
+21.7 ± 0.0
+16.6 ± 7.0
+21.4 ± 0.2
+21.5 ± 0.1
+walker2d-medium
+83.7 ± 2.1
+78.3 ± 8.7
+79.5 ± 3.2
+87.9 ± 0.2
+92.5 ± 0.8
+102.4 ± 1.4
+91.6 ± 2.8
+walker2d-expert
+110.2 ± 0.3
+109.9 ± 1.2
+109.3 ± 0.1
+107.4 ± 2.4
+115.1 ± 1.9
+115.4 ± 0.5
+114.3 ± 0.6
+walker2d-medium-expert
+110.1 ± 0.5
+109.6 ± 1.0
+109.1 ± 0.2
+116.7 ± 0.4
+114.7 ± 0.9
+121.2 ± 1.5
+105.0 ± 7.9
+walker2d-medium-replay
+81.8 ± 5.5
+73.8 ± 7.1
+76.8 ± 10.0
+78.7 ± 0.7
+87.1 ± 2.4
+90.4 ± 0.5
+88.7 ± 7.7
+walker2d-full-replay
+-
+-
+94.2 ± 1.9
+94.6 ± 0.5
+99.8 ± 0.7
+-
+109.2 ± 1.8
+Average
+67.5
+68.9
+73.6
+84.4
+85.2
+85.7
+85.2
+while our method does not require any additional modifica-
+tions.
+Table 3. SAC-RND evaluation on AntMaze domain. We report
+the final normalized score averaged over 4 random seeds on v1
+datasets. IQL, CQL, MSG scores are taken from Ghasemipour
+et al. (2022). TD3+BC, RORL scores are taken from Yang et al.
+(2022).
+Ensemble-free
+Ensemble-based
+Task Name
+TD3+BC
+IQL
+CQL
+RORL
+MSG
+SAC-RND
+antmaze-umaze
+78.6
+87.5
+74.0
+97.7 ± 1.9
+97.8 ± 1.2
+97.2 ± 1.2
+antmaze-umaze-diverse
+71.4
+62.2
+84.0
+90.7 ± 2.9
+81.8 ± 3.0
+83.5 ± 7.7
+antmaze-medium-play
+10.6
+71.2
+61.2
+76.3 ± 2.5
+89.6 ± 2.2
+65.5 ± 35.7
+antmaze-medium-diverse
+3.0
+70.0
+53.7
+69.3 ± 3.3
+88.6 ± 2.6
+88.5 ± 9.2
+antmaze-large-play
+0.2
+39.6
+15.8
+16.3 ± 11.1
+72.6 ± 7.0
+67.2 ± 6.1
+antmaze-large-diverse
+0.0
+47.5
+14.9
+41.0 ± 10.7
+71.4 ± 12.2
+57.6 ± 22.7
+Average
+27.3
+63.0
+50.6
+65.2
+83.6
+76.6
+6.3. Why is FiLM Conditioning Beneficial for Bonus
+Minimization?
+In Section 4, we showed that FiLM conditioning in the RND
+prior significantly improved the actors’ ability to minimize
+the anti-exploration bonus. Since the issue occurred during
+actor training, we hypothesize that this may be related to
+the anti-exploration bonus optimization landscape. In this
+section, we analyze the anti-gradient fields for conditioning
+with concatenation or FiLM for the prior network.
+For the purpose of analysis, we design a toy dataset with
+only four categorical states and two-dimensional actions
+sampled uniformly in each corner of the grid (see Ap-
+pendix B for dataset visualization and generation details).
+We fix the hyperparameters and pretrain two RNDs that
+differ only in the type of prior conditioning. The predictor
+uses simple concatenation. Next, in Figure 4, we plot the
+two-dimensional anti-gradient field for the anti-exploration
+bonus conditioned on each state. The effect of FiLM be-
+comes more apparent in these plots. While the resulting
+anti-gradients for concatenation are noisy and only point
+in the direction of the minimum in a small neighbourhood,
+the directions for FiLM are smooth over the entire available
+action space and point to the correct global minimum for
+each state. While we cannot draw general conclusions from
+such a demonstration, based on the results of Section 4,
+we hypothesize that a similar phenomenon might exist in
+high-dimensional problems as well.
+6.4. Exploring More Conditioning Pairs
+One might wonder (1) how different types of conditioning
+for predictor and prior interact with each other and (2) where
+to introduce conditioning in terms of depth for it to be most
+beneficial.
+To answer these questions, we return to the experiment from
+Section 4 and generate more variations for each type (where
+it is possible): conditioning on the first layer, on the last
+layer, and on all layers. We also look at two variations
+of the bilinear layer: full, as presented in Jayakumar et al.
+(2020), and simplified, which is used by default in PyTorch.
+In Figure 5 we plot the final MSE between the resulting
+policy and the behavioural one on the training data. Two
+interesting observations can be made from these findings.
+
+Anti-Exploration by Random Network Distillation
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.0025
+0.0050
+0.0075
+0.0100
+0.0125
+0.0150
+0.0175
+0.0200
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.005
+0.010
+0.015
+0.020
+0.025
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.005
+0.010
+0.015
+0.020
+0.025
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.1
+0.2
+0.3
+0.4
+(a) State 0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+(b) State 1
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+(c) State 2
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+(d) State 3
+Figure 4. Actions’ anti-gradient field for the anti-exploration bonus conditioned on four categorical states at each corner for the toy problem
+introduced in Section 6.3. We visualize the dataset in Figure 7 in the appendix. The top row corresponds to RND with concatenation
+conditioning in the prior, while the bottom row corresponds to FiLM conditioning. As can be seen, the resulting anti-gradients for
+concatenation are noisy, while the directions for FiLM are smooth over the entire available action space.
+First, FiLM may not be the only architecture with the right
+inductive biases for the prior, and both bilinear types with
+conditioning on all layers can also achieve similar results.
+However, compared to FiLM, inner bilinear layers are much
+more computationally expensive, as they involve at least
+one 3D weight tensor and two additional 2D weight tensors,
+and the hidden dimensions are usually much higher than the
+input dimensions.
+Second, it appears that conditioning on the last layer is most
+beneficial for the predictor, while conditioning on all layers
+is beneficial for the prior. In spite of that, it is difficult to
+draw broad conclusions, as different types may work well
+for new problems and domains.
+7. Related Work
+Model-free offline RL. Most offline RL approaches focus
+on the distribution shift problem and overestimation bias
+of Q-values for OOD actions. Some researchers address
+this by imposing strict constraints for policy updates, pe-
+nalizing the divergence from the behavioral policy with KL
+divergence, maximum mean discrepancy (MMD) distance
+(Kumar et al., 2019; Wu et al., 2019), simple mean squared
+error (MSE) (Fujimoto & Gu, 2021), or by re-weighting
+behavioral policy actions with the estimated advantages
+(Nair et al., 2020). Others directly regularize Q-values
+by lowering return estimates for OOD actions, preventing
+gated
+concat_first
+concat_last
+concat_full
+bilinear_first
+bilinear_last
+bilinear_full
+torch_bilinear_first
+torch_bilinear_last
+torch_bilinear_full
+film_full
+film_first
+film_last
+prior
+gated
+concat_first
+concat_last
+concat_full
+bilinear_first
+bilinear_last
+bilinear_full
+torch_bilinear_first
+torch_bilinear_last
+torch_bilinear_full
+film_full
+film_first
+film_last
+predictor
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Figure 5. Mean squared error between actions of the actor trained
+with different conditioning for the predictor & prior and actions
+of the behavioral policy. We use the halfcheetah, walker2d and
+hopper medium datasets, with 3 seeds each. It can be seen that
+conditioning on each layer is beneficial for the priors, while for the
+predictors, it is better to condition on the last layer. Note that this
+experiment is in the setting of Section 4, that is, without a critic.
+overestimation for unseen actions. For instance, Kumar
+et al. (2020), Ghasemipour et al. (2022) and Rezaeifar et al.
+(2022) explicitly introduce an optimization term that lowers
+Q-values for OOD actions, while An et al. (2021) penalizes
+implicitly by utilizing the lower-confidence bound (LCB)
+of Q-values. Alternatively, the evaluation of OOD actions
+can be avoided altogether by using the upper expectile value
+
+Anti-Exploration by Random Network Distillation
+function (Kostrikov et al., 2021) or by policy optimization
+within a latent action space (Chen et al., 2022; Zhou et al.,
+2021; Akimov et al., 2022).
+In our work, we follow the anti-exploration approach
+(Rezaeifar et al., 2022). In contrast to An et al. (2021);
+Ghasemipour et al. (2022); Yang et al. (2022), we com-
+pletely eliminate ensembles for uncertainty estimation, thus
+reducing computational overhead without sacrificing perfor-
+mance. Moreover, unlike Rezaeifar et al. (2022), we have
+succeeded in using an RND for novelty detection in offline
+RL for continuous action spaces.
+Estimation bias in Q-learning. In both offline and on-
+line reinforcement learning, off-policy Q-learning methods
+suffer from an overestimation bias in the temporal differ-
+ence learning target (Van Hasselt et al., 2016; Fujimoto
+et al., 2018). This phenomenon is orthogonal to overes-
+timation due to unseen actions in offline RL, as it occurs
+even in the presence of strong conservatism constraints. It
+is mainly caused by target prediction errors for seen transi-
+tions and their propagation due to the maximum operation
+maxa′∈AQ(s′, a′). To address this problem, Fujimoto et al.
+(2018) introduced clipped double Q learning (Van Hasselt
+et al., 2016) in TD3, which uses a minimum of two critics.
+This approach was later used by Haarnoja et al. (2018) in
+SAC to improve stability and accelerate convergence.
+In our work, we use clipped double Q-learning (Fujimoto
+et al., 2018), since SAC-RND is based on SAC (Haarnoja
+et al., 2018), and found it beneficial for stability. However,
+to ensure that it does not introduce additional conservatism,
+which can be a confounding factor for the impact of RND,
+we always set the number of critics to two.
+Uncertainty estimation in offline RL. Uncertainty estima-
+tion is a popular technique in reinforcement learning and is
+used for a variety of purposes such as exploration, planning,
+and robustness. In offline RL, its use is mostly limited to
+modeling epistemic uncertainty (Clements et al., 2019), in-
+cluding measuring the prediction confidence of dynamics
+models (Yu et al., 2020; Kidambi et al., 2020) or critics (An
+et al., 2021; Rezaeifar et al., 2022). This approach can be
+further used to induce uncertainty-aware penalization during
+training.
+Alternatively, uncertainty can help overcome suboptimal
+conservatism by designing more flexible offline approaches,
+e.g., conditioning on different levels of confidence to dy-
+namically adjust the level of conservatism during evaluation
+(Hong et al., 2022) or using Bayesian perspective to design
+an optimal adaptive offline RL policy (Ghosh et al., 2022).
+In our work, we estimate epistemic uncertainty and use it as
+an anti-exploration bonus to induce conservatism. Unlike
+previous approaches, we do not use ensembles to estimate
+epistemic uncertainty.
+Efficient ensembles Ensembles are a powerful and sim-
+ple non-Bayesian baseline for uncertainty estimation that
+outperform Bayesian neural networks in practice (Lakshmi-
+narayanan et al., 2017). However, training deep ensembles
+can be both memory intensive and computationally demand-
+ing, making the design of efficient ensembles an attractive
+research direction for which numerous methods have been
+developed. For example, Gal & Ghahramani (2016) pro-
+posed to use dropout to approximate Bayesian inference in
+deep Gaussian processes, and Durasov et al. (2021) derived
+a method to interpolate between dropout and full ensembles
+with fixed masks and controllable overlap between them.
+Meanwhile, Wen et al. (2020) significantly reduced the cost
+by defining each weight matrix as the Hadamard product
+of a shared weight among all ensemble members and a
+rank-one matrix per member.
+Recently, Ghasemipour et al. (2022) showed that, in offline
+RL, none of the most popular approaches for efficient en-
+sembles are capable of delivering performance that is com-
+parable to naive ensembles, and that more work is needed in
+this research direction. In our work, we chose an alternative
+path for uncertainty estimation with RND, which was shown
+to a fast and competitive counterpart to ensembles (Ciosek
+et al., 2019).
+8. Conclusion
+In this work, we revisited the results from previous research
+(Rezaeifar et al., 2022), showing that with a naive choice
+of conditioning for the RND prior, it becomes infeasible
+for the actor to effectively minimize the anti-exploration
+bonus and discriminativity is not an issue. To solve this,
+we proposed conditioning based on FiLM, which led us
+to a new ensemble-free method called SAC-RND. We em-
+pirically validated that it achieves results comparable to
+ensemble-based methods and outperforms its ensemble-free
+counterparts. As such, we believe that our work is a valuable
+contribution to anti-exploration and uncertainty estimation
+in offline RL.
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+Y., Kay, J., Springenberg, J. T., et al. A generalist agent.
+arXiv preprint arXiv:2205.06175, 2022.
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+Bachem, O., Pietquin, O., and Geist, M. Offline rein-
+forcement learning as anti-exploration. In Proceedings
+of the AAAI Conference on Artificial Intelligence, vol-
+ume 36, pp. 8106–8114, 2022.
+Schrittwieser, J., Antonoglou, I., Hubert, T., Simonyan, K.,
+Sifre, L., Schmitt, S., Guez, A., Lockhart, E., Hassabis,
+D., Graepel, T., et al. Mastering atari, go, chess and shogi
+by planning with a learned model. Nature, 588(7839):
+604–609, 2020.
+Smith, L., Kostrikov, I., and Levine, S.
+A walk in the
+park: Learning to walk in 20 minutes with model-free
+reinforcement learning. arXiv preprint arXiv:2208.07860,
+2022.
+Srivastava, R. K., Shyam, P., Mutz, F., Ja´skowski, W., and
+Schmidhuber, J. Training agents using upside-down re-
+inforcement learning. arXiv preprint arXiv:1912.02877,
+2019.
+Tarasov, D., Nikulin, A., Akimov, D., Kurenkov, V., and
+Kolesnikov, S. CORL: Research-oriented deep offline
+reinforcement learning library. In 3rd Offline RL Work-
+shop: Offline RL as a ”Launchpad”, 2022. URL https:
+//openreview.net/forum?id=SyAS49bBcv.
+Van Hasselt, H., Guez, A., and Silver, D. Deep reinforce-
+ment learning with double q-learning. In Proceedings of
+the AAAI conference on artificial intelligence, volume 30,
+2016.
+Wen, Y., Tran, D., and Ba, J. Batchensemble: an alterna-
+tive approach to efficient ensemble and lifelong learning.
+arXiv preprint arXiv:2002.06715, 2020.
+Wu, Y., Tucker, G., and Nachum, O.
+Behavior regu-
+larized offline reinforcement learning. arXiv preprint
+arXiv:1911.11361, 2019.
+Yang, R., Bai, C., Ma, X., Wang, Z., Zhang, C., and Han,
+L. Rorl: Robust offline reinforcement learning via con-
+servative smoothing. arXiv preprint arXiv:2206.02829,
+2022.
+Yu, T., Thomas, G., Yu, L., Ermon, S., Zou, J. Y., Levine, S.,
+Finn, C., and Ma, T. Mopo: Model-based offline policy
+optimization. Advances in Neural Information Processing
+Systems, 33:14129–14142, 2020.
+Zhou, W., Bajracharya, S., and Held, D. Plas: Latent action
+space for offline reinforcement learning. In Conference
+on Robot Learning, pp. 1719–1735. PMLR, 2021.
+
+Anti-Exploration by Random Network Distillation
+A. Previous Research Results
+Figure 6. Anti-exploration bonus on walker2d-medium dataset for RND and CVAE. Note that figure taken from Rezaeifar et al. (2022) for
+a convenient comparison with our results in Figure 2.
+B. Toy Dataset
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+a1
+State
+0
+1
+2
+3
+Figure 7. Toy dataset visualization introduced in Section 6.3. This toy dataset consists of four categorical states for each corner of the
+limited 2D actions grid. For each state, we uniformly sample 4096 two-dimensional actions within a limited square. We use one-hot
+encoding for the states during RND training.
+C. Implementation Details
+In our experiments, we use hyperparameters from Table 4 where possible and sweep over α to pick the best value for each
+dataset. We implement all of our models using the Jax (Bradbury et al., 2018) framework. For the exact implementation
+of conditioning variants for predictor and prior networks, refer to our code at https://github.com/tinkoff-ai/
+sac-rnd. Similarly to Nikulin et al. (2022); Kumar et al. (2022); Smith et al. (2022), we add Layer Normalization (Ba
+et al., 2016) to the critic after each layer as it greatly improves stability and convergence speed. For SAC-N in Section 4 we
+use the implementation from the CORL library (Tarasov et al., 2022). All experiments were performed on V100 and A100
+GPUs. With our implementation, each training for 3 million training steps usually takes ∼ 40 minutes to run (∼ 15 minutes
+for the typical 1 million steps).
+Gym Domain.
+We use the v2 version of each dataset.
+We follow the An et al. (2021) approach and run our
+algorithms for 3 million training steps and report the final normalized average score over 10 evaluation episodes.
+For the final experiments, we use 4 seeds, while using less for hyperparameter tuning.
+We tune the α co-
+efficient over the {1.0, 3.0, 4.0, 5.0, 8.0, 9.0, 10.0, 13.0, 15.0, 20.0, 25.0} range for the walker and hopper datasets.
+We found that the halfcheetah datasets require a lower level of conservatism, which is why we tune over the
+{0.001, 0.1, 0.3, 0.5, 0.8, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0} range for these datasets while keeping the same number of candidates.
+We follow the Ghasemipour et al. (2022) approach and choose the best α for each dataset (see Table 5).
+AntMaze Domain. We use the v1 version of each dataset due to the fact that the v0 version has major problems and
+bugs during generation (e.g., some trajectories have discontinuities where the agent teleports from one part of the maze to
+
+CVAE
+RND
+300000-
+400000
+Dataset actions
+Shuffled actions
+Random actions
+300000
+Dataset actions + Gaussian noise (std = 0.25)
+200000
+Dataset actions + Gaussian noise (std = O.5)
+200000
+100000
+100000
+0
+0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+1e-7Anti-Exploration by Random Network Distillation
+another 3). We follow the An et al. (2021) approach and run our algorithms for 3 million training steps and report the final
+normalized average score over 100 evaluation episodes. Same as Chen et al. (2022), we scale the reward by 100.0. We found
+that actor and critic require different levels of conservatism in these tasks, which is why we chose to decouple α and use
+separate values (the same approach was used in Rezaeifar et al. (2022)). We tune the α for the actor in the {0.5, 1.0, 1.5}
+range, and α for the critic in the {0.001, 0.01, 0.1} range. We follow the Ghasemipour et al. (2022) approach and choose
+the best α for each dataset (see Table 6).
+D. Hyperparameters
+Table 4. SAC-RND general hyperparameters.
+Parameter
+Value
+optimizer
+Adam (Kingma & Ba, 2014)
+batch size
+1024 (256 on antmaze-*)
+learning rate (all networks)
+1e-3 (3e-4 on antmaze-*)
+tau (τ)
+5e-3
+hidden dim (all networks)
+256
+num layers (all networks)
+4
+RND embedding dim (all tasks)
+32
+target entropy
+-action_dim
+gamma (γ)
+0.99 (0.999 on antmaze-*)
+nonlinearity
+ReLU
+Table 5. SAC-RND best hyperparameters used in D4RL Gym domain.
+Task Name
+α
+halfcheetah-random
+0.1
+halfcheetah-medium
+0.3
+halfcheetah-expert
+6.0
+halfcheetah-medium-expert
+0.1
+halfcheetah-medium-replay
+0.1
+halfcheetah-full-replay
+3.0
+hopper-random
+5.0
+hopper-medium
+25.0
+hopper-expert
+20.0
+hopper-medium-expert
+15.0
+hopper-medium-replay
+8.0
+hopper-full-replay
+3.0
+walker2d-random
+1.0
+walker2d-medium
+8.0
+walker2d-expert
+4.0
+walker2d-medium-expert
+25.0
+walker2d-medium-replay
+8.0
+walker2d-full-replay
+3.0
+Table 6. SAC-RND best hyperparameters used in D4RL AntMaze domain.
+Task Name
+α (actor)
+α (critic)
+antmaze-umaze
+1.0
+0.1
+antmaze-umaze-diverse
+1.0
+0.1
+antmaze-medium-play
+0.5
+0.001
+antmaze-medium-diverse
+1.0
+0.01
+antmaze-large-play
+1.0
+0.01
+antmaze-large-diverse
+0.5
+0.01
+3https://github.com/Farama-Foundation/D4RL/issues/77
+
+Anti-Exploration by Random Network Distillation
+E. Sensitivty to Hyperparameters
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Policies Evaluated Online
+30
+40
+50
+60
+70
+80
+90
+Expected Online Performance
+HalfCheetah
+Hopper
+Walker2D
+AntMaze
+Figure 8. Expected Online Performance (Kurenkov & Kolesnikov, 2022) under uniform offline policy selection. It can be seen, that for
+satisfactory results in all domains a budget of at least five policies for online evaluations is needed.
+F. Pseudocode
+Algorithm 1 Soft Actor-Critic with Random Network Distillation (SAC-RND)
+Initialize policy parameters θ, Double Q-function parameters {φ1, φ2}, RND predictor and prior parameters {ψ, ψ′}, and
+offline replay buffer D
+for desired number of pretraining steps do
+Sample a mini-batch B = {(s, a)} from D
+Update RND predictor weights ψ with gradient descent using
+∇ψ
+1
+|B|
+�
+s∈B
+�
+∥fψ(s, a) − ¯f ¯
+ψ(s, a)∥2
+2
+�
+end for
+for desired number of training steps do
+Sample a mini-batch B = {(s, a, r, s′)} from D
+Compute target Q-values (shared by all Q-functions):
+y(r, s′) = r + γ
+�
+min
+j=1,2 Q ¯φi(s′, a′) − β log πθ(a′|s′) − αb(s′, a′)
+�
+where a′ ∼ πθ(·|s′) and b(s′, a′) is an anti-exploration bonus defined by Eq. (3).
+Update each Q-function Qφi with gradient descent using
+���φi
+1
+|B|
+�
+(s,a,r,s′)∈B
+�
+Qφi(s, a) − y(r, s′)
+�2
+Update policy with gradient ascent using
+∇θ
+1
+|B|
+�
+s∈B
+�
+min
+j=1,2 Qφi(s, ˜aθ(s)) − β log π(˜aθ(s)|s) − αb(s, ˜aθ(s))
+�
+where ˜aθ(s) is a sample from π(·|s) which is differentiable w.r.t. θ via the reparametrization trick.
+Update target networks with ¯φi ← (1 − ρ) ¯φi + ρφi
+end for
+
+Anti-Exploration by Random Network Distillation
+Algorithm 2 Simplified SAC-RND (without a critic) used in experiments for Section 4 and Section 6.4.
+Initialize policy parameters θ, RND predictor and prior parameters {ψ, ψ′}, and offline replay buffer D
+for desired number of pretraining steps do
+Sample a mini-batch B = {(s, a)} from D
+Update RND predictor weights ψ with gradient descent using
+∇ψ
+1
+|B|
+�
+s∈B
+�
+∥fψ(s, a) − ¯f ¯
+ψ(s, a)∥2
+2
+�
+end for
+for desired number of training steps do
+Sample a mini-batch B = {(s, a, r, s′)} from D
+Update policy with gradient descent using
+∇θ
+1
+|B|
+�
+s∈B
+�
+β log π(˜aθ(s)|s) + b(s, ˜aθ(s))
+�
+where ˜aθ(s) is a sample from π(·|s) which is differentiable w.r.t. θ via the reparametrization trick and b(s′, a′) is an
+anti-exploration bonus defined by Eq. (3).
+end for
+

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+page_content='FEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE TOMAS CABEZON PEDROSO1 and JINMO RHEE2 and DARAGH BYRNE3 1,2,3Carnegie Mellon University, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 1tcabezon@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='edu, 0000-0002-5483-2676 2jinmor@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='edu, 0000-0003-4710-7385 3daraghb@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='edu, 0000-0001-7193-006X  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' This paper compares the parametric design space with a feature space generated by the extraction of design features using deep learning (DL) as an alternative way for design space exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' In this comparison, the parametric design space is constructed by creating a synthetic dataset of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000 elements using a parametric algorithm and reducing its dimensions for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The feature space — reduced-dimensionality vector space of embedded data features — is constructed by training a DL model on the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' We analyze and compare the extracted design features by reducing their dimension and visualizing the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' We demonstrate that parametric design space is narrow in how it describes the design solutions because it is based on the combination of individual parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' In comparison, we observed that the feature design space can intuitively represent design solutions according to complex parameter relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Based on our results, we discuss the potential of translating the features learned by DL models to provide a mechanism for intuitive design exploration space and visualization of possible design solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Deep Learning, VAE, Design Space, Feature Design Space, Parametric Design Space, Design Exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Introduction  Parametric modeling has acquired widespread acceptance among creative practitioners as it allows the synthesis of various design options and solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Changing the parameters in this modeling process, either manually or randomly, can rapidly create a vast set of design variations (Toulkeridou, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Navigating the resulting parametric design space — where the design variants are topologically placed by their parameters — is part of the design exploration process — a crucial step in the development of new alternatives and design solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Exploration of the parametric design space allows creative practitioners  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' CABEZON PEDROSO, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' RHEE AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' BYRNE  many benefits: to reach satisfying solutions, better define design problems, and understand the opportunities and limitations of the possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Despite these benefits, design exploration is laborious within the parametric space and challenged along two fronts: comparison and selection (Fuchkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Parametric design exploration is an iterative process that focuses on the variation of these individual parameters, rather than on the relationship among them (Yamamoto and Nakakoji, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Hence, comparing one design solution with others by their parameters alone does not always result in a superior solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' for example, the variants generated by the local combination of parameters might not match the design requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Moreover, infinite alternative design solutions can be generated by inputting new parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Thus, the parametric design space consists of a huge amount of design variants that cannot be fully or sufficiently explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  We propose an alternative way to construct and examine the design space, by extracting features from a DL model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' By comparing and analyzing how the DL feature design space differs from the parametric design space, we illustrate the potential of feature design space for design practitioners during the design exploration process and provide a new way to compare, examine and select the design alternatives based on the exploration of a properly constrained design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  No previous approach to compare the parametric design space and feature design space as design exploration tools has been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' To demonstrate how the feature space compares to the parametric space, we designed an experiment to construct both a parametric design space and a feature design space using the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The dataset consists of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000 synthetic 3D models produced by a parametric algorithm with five parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' This parametric design space consists of five axes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' each axis corresponds to each of the parameters that are used as inputs of the parametric algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Subsequently, this same dataset is used to train a DL model to compress the data into a feature vector of 128 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Both the parametric space (five-axes) and the feature space (128 axes) are not directly visualizable due to their high dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Nevertheless, as visual feedback plays an important role in design exploration (Bradner, Iorio and Davis, 2014), we employ a dimensionality reduction algorithm (t-SNE) to the design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' We are able to illustrate the design exploration space, showing how the data is distributed across both the parametric and feature design spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  In the next section, we describe the generation of the dataset, as well as the construction of parametric design space and its visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', we illustrate how training a DL model resulted in a feature space for design exploration and comparison with the parametric approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Then, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', we will compare, contrast, and discuss the characteristics of the DL feature space and the parametric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=')  FEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The overall process of comparing parametric design space and feature space from deep learning  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Constructing Parametric Design Space 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' DATASET GENERATION  To conduct a design space comparison, a simple parametric modeling system was designed: a parametric algorithm for generating different styles of vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' As with handcraft of pottery wheel throwing, a simple Bezier curve with three control points was turned around an axis to generate each 3D digital vessels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' the form of each vessel is specified by the five parameters that were used as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' These parameters, as can be seen in Figure 2, are: the height of the vessel, the width of the base, the width of the top opening, and the horizontal and vertical coordinates of the central control point of the Bezier curve that are used to create the curve of the form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The five parameters are represented as a vector, and each vector corresponds to a specific 3D model of a vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Using this system, we created a 3D vessel dataset by randomly generating a total of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000 different vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The total shape of the parametric representation of the vessel dataset is [15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000, 5], however, as it will be explained in the next section,  Parametric 5 algorithm Parametric Data parameters Design Space Comparison Feature VAE DesignSpace ENCODER DECODERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' CABEZON PEDROSO, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' RHEE AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' BYRNE  only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000 vessels were used for the space exploration and visualization, so this will be a design space of size [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Upper: An illustration of the dataset parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Lower: Three illustrative examples from the dataset with the parameters and the resulting 3D form side-by-side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' DIMENSIONALITY REDUCTION  As a five-dimensional space makes it hard to compare models and to visualize and compare the characteristics, we employed a dimensionality reduction process to reduce the space to two-dimensions and enable the objects to be plotted and compared to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' shows the overall process of visualizing the space using t-Distributed Stochastic Neighbour Embedding (t-SNE) algorithm (van der Maaten and Hinton, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' t-SNE is a popular dimensionality-reduction algorithm for visualizing high-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The hyper-parameters used for this reduction are: perplexity: 30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' learning rate: 200;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and iterations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Illustration of the dimensional reduction process for the 3D vessel dataset, and the construction of a parametric design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  After dimensional  reduction, each point in the plot represents the corresponding embedding of a vessel in the parametric design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Each point is expressed as  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Pot top width 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Pot bottom width 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Control point horizontal coordinate 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Control point vertical coordinate 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Pot height 2TopLevelApp (400 x 400) Parameters: bottomWidth: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
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+page_content='755 Figure1 Press: d, to display the mesh bezier s, to save the,sti flle 75 50 25 0 -25 50 75 100 75 50 25 -100 0 -75 _50 -25 -25 0 50 25 75 50 75 100 100[3k,5] [3k,2] dimension 1 PLOT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' t-SNE x Parametric Dimensional dimension 0 Design Space ReductionFEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE a 2D image of the profile cut section of the corresponding vessel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' represents the reduced parametric design space of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' A 2D visualization of the parametric design space of the vessel dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Inset image: a detailed section for a subset of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Constructing the Feature Space  To construct the design space based on the features and not the parameters, we used a Variational Autoencoder (VAE) as a tool for extracting the morphological features of the vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' VAEs (Kingma and Welling, 2013) are a type of generative deep neural network used for statistical inference problems as they generalize a probabilistic distribution of the given dataset and synthesize new data samples from that distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' VAEs are composed of two modules: encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The encoder abstracts the input data into smaller dimensional vectors, latent vectors, and the decoder reconstructs the latent vector back into a 3D shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' During the encoding process, the network captures and extracts the features of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' These features can be topologically placed in the data space, namely, latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' In the latent space, the distance between two data points represents the degree of resemblance of data: the closer points, the more resembled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' We translate this latent space as the feature space for an alternative way to explore the design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' DATA PRE-PROCESSING  Different representations of 3D data have been used in DL research, like point clouds (Achlioptas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', 2018), meshes (Ranjan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', 2018), or voxels (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' As resolutions of the data is not key for our purpose rather than the extracted features of it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and because we will implement a VAE for this experiment that needs fixed space inputs for the Convolutional Neural Networks (CNNs), we will be representing our 3D data with voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Voxels are discretized three- dimensional grids containing a binary value of volumetric occupancy of an object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' they distinguish between the elements on the grid that are filled with material and those that are empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The size of the voxel will determine the number of divisions 1  4 I  -- I 1 1 1 1 1  I !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 1 I 1 1 1 I 1 1 1 I 1 I =T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' CABEZON PEDROSO, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' RHEE AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' BYRNE  of the grid, consequently, the resolution at which we represent our 3D models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' the larger size, the more detailed 3D models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' In this experiment, we used 32-sized voxels so that a 3D vessel model is represented by 32x32x32 grid, the shape of the entire dataset is [15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000, 32, 32, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Finally, the dataset was then divided into two groups: 80% of the dataset (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000 vessels) was used for training the DL model, and the remaining 20% (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000 vessels) was used for testing the model and the parametric and feature space analysis and comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' TRAINING  For training the model, we adopted the VAE architecture implemented in ‘Adversarial Generation of Continuous Implicit Shape Representations’ (Kleineberg, Fey and Weichert, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The encoder consists of four residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Each residual block is composed of a 3D convolution layer, followed by a batch normalization and a Leaky ReLu activation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The decoder, on the contrary, comprises four residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Each block starts with a batch normalization, followed by a Leaky ReLu activation layer, and finally a 3D transposed convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The following hyper-parameters are used for training the VAE with the voxelized vessel dataset: batch size 32,  Adam optimizer (Kingma and Ba, 2015), learning rate 5e-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The model was trained in Google Colab Pro using the Nvidia Tesla T4 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Training process losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  The model was trained for a total of 240 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' We early stopped the model before the model started to overfit, Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The loss function used during training was a combination of two losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The first one, is the Kullback–Leibler divergence (KLD) loss (Kullback and Leibler, 1951), with a weight in the total loss formula of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' This function is a measurement of the difference between two statistical distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The second loss is the Minimum Square Distance (MSE) loss (Sammut and Webb, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' It is used as the reconstruction loss and measures the error between the input voxels and the reconstructed output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' shows the reasonable quality of the reconstruction of the training result after 240 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' To ensure the performance of the model, it was evaluated using the test set and showed that the model maintained the accuracy with the new dataset, which shows  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='07 IMSE Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='06 KLD Loss Total Loss Total Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='01 0 50 100 150 200 250 EpochsFEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE that the model generalizes well to new data and is able to encode never seen before 3D vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Two examples of reconstructions from the trained VAE: the section slides and 3D voxels of the ground truth (the top row of each example) and the reconstruction (bottom of each example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' DIMENSIONALITY REDUCTION  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Feature space generation and visualization diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Once the VAE is trained, the encoder is used to extract the features of each vessel in the test dataset from 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='768 dimensions, the size of each voxelized vessel, into 128-dimensional vectors, the latent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Consequently, the entire test dataset of the vessels is represented into vectors whose total shape is [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='000, 128].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Like in the parametric case, 128 dimensions are non-visualizable so the same process as in Section 2 is followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' t-SNE algorithm is used to reduce the dimensionality of each vector and plot the resulting two dimensions in an image with the section of each of the vessels (Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The hyper-parameters used for this reduction are: perplexity: 50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' learning rate: 700;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and iterations: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' shows the results of distributed feature vectors in the reduced dimensional space, the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  1- Input (Ground truth) 10 10 10 20 30 05 20 20 20 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 Output 10 to 10 O1 1o 20 20 20 20 20 20 30 20 0 20 20 20 20 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 2- Input (Ground truth) 10 10 10 10 10 20 20 20 20 30 20 20 20 20 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 Output 10 OT 10 10 10 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='6 20 20 20 20 20 20 20 30 20 20 20 20 0 20 o 20 0 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='0[3k,128] [3k,2] dimension ENCODER PLOT voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='npy t-SNE : dimension 0 Latent Dimensional object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='stl Space ReductionT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' CABEZON PEDROSO, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' RHEE AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' BYRNE  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' A 2D visualization of the feature design space of the vessel dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Inset image: a detailed section for a subset of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Comparison Between the spaces  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' shows that similar vessels have been clustered together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Thinner vessels are located at the top right of the image, in contrast to the opposite lower bottom corner with the bigger vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' The figure illustrates how the VAE model is able to understand the relationship between the parameters and their influence on the output morphological shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  On the contrary, in the parametric space (Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' ), we can see how concave vessels were gathered at the bottom of the image, however, if the height of the vessels is considered, we can see that this parameter was not considered when clustering the vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Parametric space is based on each parameter independently, and not on the relationship among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Therefore, we observe that parametric design space insufficiently expresses the final form characteristics of the vessels by the combinations of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' On the contrary, in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', the feature space, a gradual change in the shape or concavity as well as height or width is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  To further examine and compare the characteristics of both design spaces, we used a clustering, algorithm: a Density-Based Spatial Clustering of Applications with Noise (DBSCAN) (Ester M et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=', 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' It is one of the most common clustering algorithms that finds core samples of high density and expands clusters with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' shows the results of this clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  The parametric design space has a total of seven clusters: three of them large, and four of them small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' It shows how the parametric design space doesn’t provide enough information to intuitively compare the design variants locally, this space shows extreme changes in vessel forms even in the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  "FEATURE SPACE EXPLORATION AS AN ALTERNATIVE FOR DESIGN SPACE EXPLORATION BEYOND THE PARAMETRIC SPACE The feature design space, on the contrary, has a total of nine clusters: six main big clusters, and three smaller ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' In the feature design space, we can trace smooth changes in the forms as we move through the different clusters (local changes) and along the whole image (global changes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Shorter vessels are located on the top, while taller ones are on the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' If we move on the horizontal axis, the curve that generates the vessels goes from a concave shape on the right to a convex shape on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' This gives the designer the ability to locally compare similar design alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Parametric Design Space:                        Feature Design Space:  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Final visualization and clusters of the parametric and feature design spaces with representative vessels of each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Conclusion and Future work  We constructed the parametric and the feature design spaces using a custom synthetic dataset and a VAE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' By comparing the parametric and feature design spaces, we observed improved distributions of design alternatives in the later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' When the multi-dimensional parametric design space is projected into a 2D space (Figures 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' left), the clusters are insufficiently relevant to the morphological characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' On the other hand, when the multi-dimensional feature space is projected into a 2D space (Figures 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' right), the clusters show sufficient relevance to the features of the data they represent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Based on this comparison, we conclude that combination of individual parameters in the parametric design space is limited in representing the morphological characteristics of the shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' However, we showed that DL models can be used to extract design features from 3D models and that the extracted features are more complex than the combinations of individual parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Hence, we conclude that the extracted features, that include information of the relationships between the parameters, can construct a well-distributed design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' For that reason, we propose feature design space as a tool for design space exploration that creative practitioners can use as a new way for looking at objects beyond the parametric design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' CABEZON PEDROSO, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' RHEE AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' BYRNE  Our results and implications are limited to a single dataset and DL model, however the results seem promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Future work will expand on this study with more diverse datasets generated by more complex parametric algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Accordingly, to perform the feature extraction, we would like to train other types of DL models to investigate different potentials of DL in design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
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+page_content='org/abs/1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='07584 (Accessed: 7 December 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' Yamamoto, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' and Nakakoji, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' (2005) ‘Interaction design of tools for fostering creativity in the early stages of information design’, International Journal of Human-Computer Studies, 63(4–5), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content=' 513–535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='  Available at: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='ijhcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}
+page_content='023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdFIT4oBgHgl3EQf_ywO/content/2301.11416v1.pdf'}

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+1
+A Fundamental Tradeoff Among Storage,
+Computation, and Communication for
+Distributed Computing over Star Network
+Qifa Yan Member, IEEE, Xiaohu Tang Senior Member, IEEE,
+Meixia Tao Fellow, IEEE, and Qin Huang Senior Member, IEEE
+Abstract
+Coded distributed computing can alleviate the communication load by leveraging the redundant
+storage and computation resources with coding techniques in distributed computing. In this paper, we
+study a MapReduce-type distributed computing framework over star topological network, where all
+the workers exchange information through a common access point. The optimal tradeoff among the
+normalized number of the stored files (storage load), computed intermediate values (computation load)
+and transmitted bits in the uplink and downlink (communication loads) are characterized. A coded
+computing scheme is proposed to achieve the Pareto-optimal tradeoff surface, in which the access point
+only needs to perform simple chain coding between the signals it receives, and information-theretical
+bound matching the surface is also provided.
+Index Terms
+Storage, coded computing, communication, MapReduce, star network
+I. INTRODUCTION
+The rapid growth of computationally intensive applications on mobile devices has attracted
+much research interest in designing efficient distributed computing frameworks. One of the most
+important programing models for distributed computing is MapReduce [1], [2], which has been
+utilized to deal with computation tasks with data sizes as large as tens of terabytes.
+MapReduce framework allows to assign multiple computation tasks to distributed nodes, where
+each node only stores a subset of files. This is done by decomposing each function to be computed
+into a set of “map” functions and a “reduce” function, where each map function can be computed
+from a batch of data, with the output called intermediate values (IVs), while the computation of
+a “reduce” function needs to collect the IVs from all the data as inputs. The whole procedure is
+composed of three phases, i.e., map, shuffle and reduce. In the map phase, each distributed node
+computes the map functions on its local file batch assigned by the server and generates output
+IVs; in the shuffle phase, the nodes exchange their computed IVs to facitate each node to obtain
+the IVs needed by its assigned reduce functions; in the reduce phase, each node computes its
+assigned reduce functions by decoding all the corresponding IVs.
+Q. Yan and X. Tang are with the Information Coding & Transmission Key Lab of Sichuan Province, CSNMT Int. Coop. Res.
+Centre (MoST), Southwest Jiaotong University, Chengdu 611756, China(email: [email protected], [email protected]).
+M.
+Tao
+is
+with
+the
+Department
+of
+Electronic
+Engineering,
+Shanghai
+Jiao
+Tong
+University,
+Shanghai
+200240,
+China(email:[email protected]).
+Q. Huang is with the School of Electronic and Information Engineering, Beihang University, Beijing 100191, China
+(email:[email protected]).
+arXiv:2301.03788v1  [cs.IT]  10 Jan 2023
+
+2
+Recently, a coded distributed computing (CDC) scheme was proposed by Li et al. [3], where
+the files are stored multiple times across the distributed nodes in the map phase. The IVs are also
+computed multiple times accordingly, such that multicast opportunities are created for the shuffle
+phase. As a result, the communication load was reduced significantly compared to traditional
+uncoded scheme. It was proved in [3] that the scheme achieves the optimal communication load
+for a given total storage requirements. Interestingly, the normalized number of files stored across
+the nodes was termed computation load by Li et al, because each node calculates all the IVs that
+can be obtained from the data stored at that node in the model therein, no matter if these IVs
+are used or not in the subsequent phases. Subsequently, Ezzeldin [4] and Yan et al [5], [6] found
+that some IVs are computed but not used in the model. For this reason, Yan et al reformulated
+the problem as a tradeoff between storage, computation, and communication loads in [7], which
+allows each node to choose any subset of IVs to compute from its stored files.
+Some interesting works that extend CDC have been proposed, for example, the technique was
+combined with maximum distance separable (MDS) code in matrix-vector multiplication tasks to
+resist stragglers in [8]; stragglers with general functions are considered in [9], [10]; the optimal
+resource allocations are considered in [11]; [12]–[14] investigated the iterative procedures of data
+computing and shuffling; [15] studied the case when each node has been randomly allocated
+files; [16] investigated the case with random connectivity between nodes.
+The coded distributed computing technique is extended to wireless distributed computing
+[17], [18], where the computation is typically carried out by the wireless devices. Due to the
+decentralized natural of the wireless networks, the nodes in wireless networks normally need a
+central Access Point (AP) to exchange data, which leads to uplink and downlink communications.
+For example, smart-phone end users typically communicate with each other through a base station
+in cellular networks, which operates in a star network. In [19] and [20], Li et al. investigated
+distributed computing in a wireless network where the nodes performs data shuffling through an
+AP. The optimal storage-communication tradeoff was characterized for both uplink and downlink
+transmissions.
+In this paper, following the conventions of Ezzeldin [4] and Yan et al [7], we investigate
+a distributed computing system with star network, where all nodes exchange IVs through an
+AP, but each node is allowed to choose any arbitrary subset of IVs to compute from its stored
+files. In particular, in addition to the storage and computation loads as considered in [7], the
+communication load includes both upload and download. The main contribution of this paper is
+the characterization of the Pareto-optimal surface in the storage-computation-upload-download
+space for distributed computing over star network. The idea is to form the same multicast
+signals as in CDC scheme but compute less IVs by ignoring the un-used IVs in the map phase
+in the uplink, and combine them through a simple chain coding to form the downlink signals at
+the AP. It turns out that, for any given storage-computation pair, both the optimal upload and
+download communication costs can be simultaneously achieved by a coded computing scheme
+that oriented from CDC. The information-theoretical bound matching the Pareto-optimal surface
+is also presented.
+Paper Organization: Section II presents the system model. Section III summarizes the main
+results. Section IV presents the coded computing scheme that achieves the optimal surface, and
+Section V provides information-theoretical bound. Finally, Section VI concludes the paper.
+Notations: Let N+ be the set of positive integers, and F2 be the binary field. For m, n ∈ N+,
+denote the n-dimensional vector space over F2 by Fn
+2, and the integer set {1, . . . , n} by [n]. If
+m < n, we use [m : n] to denote the set {m, m + 1, . . . , n}. We also use interval notations, e.g.,
+
+3
+Fig. 1: A Distributed Computing System with Star Network
+[a, b] ≜ {x : a ≤ x ≤ b} and [a, b) ≜ {x : a ≤ x < b} for real numbers a, b such that a < b. The
+bitwise exclusive OR (XOR) operation is denoted by ⊕. For sets we use upper case calligraphic
+font, e.g., A, and for collections (sets of sets) we use upper case Greek letters with bold font,
+e.g., Ω. We denote a point in two or three dimensional Euclidean space by an upper case letter.
+A line segment with end points A1, A2 or a line through the points A1, A2 is denoted by A1A2.
+A triangle with vertices A1, A2, A3 is denoted by △A1A2A3. A trapezoid with the four edges
+A1A2, A2A3, A3A4, and A4A1, where A1A2 is parallel to A3A4, is denoted by ⊟A1A2A3A4.
+Let F be a set of facets, if the facets in F form a continuous surface, then we refer to this
+surface simply as F.
+II. SYSTEM MODEL
+Let K, N, W, U, V be given positive integers. Consider a star network consisting of K dis-
+tributed computing nodes {1, . . . , K} that can communicate with each other through a common
+AP, as illustrated in Fig. 1. Each of the K nodes can transmit signals to the AP through an
+uplink channel, while the AP can broadcast signals to all the K nodes via a downlink channel.
+Each of the K nodes aims to compute an individual function from a set of N files,
+W = {w1, . . . , wN},
+wn ∈ FW
+2 , ∀ n ∈ [N],
+each of size W bits. Node k aims to compute an output function
+φk : FNW
+2
+→ FU
+2 ,
+which maps all the files to a bit stream
+uk = φk(w1, . . . , wN) ∈ FU
+2
+of length U. Assume that each output function φk decomposes as:
+φk(w1, . . . , wN) = hk(fk,1(w1), . . . , fk,N(wN)),
+(1)
+where
+
+Files 
+Map
+Node 3
+IVs
+Reduce
+The set of files
+X(X1, X2, X3)
+X1
+X2
+个
+Reduce
+Reduce
+Files 
+Files 
+Map
+Map
+IVs
+IVs
+Node 1
+Node 24
+• Each “map” function fk,n is of the form
+fk,n : FW
+2 → FV
+2 ,
+and maps the file wn into the IV
+vk,n ≜ fk,n(wn) ∈ FV
+2 .
+• The “reduce” function hk is of the form
+hk : FNV
+2
+→ FU
+2 ,
+and maps the IVs
+Vk ≜ {vk,n : n ∈ [N]}
+into the output stream
+uk = hk(vk,1, . . . , vk,N).
+Notice that one trivial decompositon is that, the map functions are identity functions and the
+reduce functions are the output functions, i.e., gk,n(wn) = wn, and hk = φk, ∀ n ∈ [N], k ∈ [K].
+But in practice, many output functions can be decomposed such that the main computation load is
+dominated by the map functions. For example, in federated learning, it typically needs to collect
+the sum of the gradients over all data blocks, where the map functions are used to compute the
+gradients of the loss functions over a data block, while the reduce function is the sum operation.
+The described structure of the output functions φ1, . . . , φK, allows the nodes to perform their
+computation in the following three-phase procedure.
+1) Map Phase: Each node k ∈ [K] chooses to store a subset of files Mk ⊆ W. For each file
+wn ∈ Mk, node k computes a subset of IVs
+Ck,n = {vq,n : q ∈ Zk,n},
+where Zk,n ⊆ [K]. Denote the set of IVs computed at node k by Ck, i.e.,
+Ck ≜
+�
+n:wn∈Mk
+Ck,n.
+(2)
+2) Shuffle Phase: The K nodes exchange some of their computed IVs through the AP via
+upload and download sub-phases:
+In the upload sub-phase, each node k generates a coded signal
+Xk = ϕk (Ck)
+of some length lk ∈ N and sends it to the AP, using a function
+ϕk : F|Ck|V
+2
+→ Flk
+2 .
+In the download sub-phase, receiving all the signals {X1, . . . , XK}, the AP generates a signal
+X = χ(X1, X2, . . . , XK)
+(3)
+of length l ∈ N, and broadcasts it to all nodes, where the encoding function is
+χ : Fl1+l2+...+lK
+2
+→ Fl
+2.
+
+5
+3) Reduce Phase: Using the received signal X broadcast from the AP in the shuffle phase
+and its own IVs Ck computed locally in the map phase, each node k now computes the IVs
+(vk,1, . . . , vk,N) = ψk (X, Ck) ,
+(4)
+for some function
+ψk : Fl
+2 × F|Ck|V
+2
+→ FNV
+2
+.
+Finally, it computes
+uk = hk(vk,1, . . . , vk,N).
+(5)
+To measure the storage, computation, and communication costs of the described procedure,
+following the convention in [7], we introduce the following definitions.
+Definition 1 (Storage load). Storage load r is defined as the total number of files stored across
+the K nodes normalized by the total number of files N:
+r ≜
+�K
+k=1 |Mk|
+N
+.
+(6)
+Definition 2 (Computation load). Computation load c is defined as the total number of map
+functions computed across the K nodes, normalized by the total number of map functions NK:
+c ≜
+�K
+k=1 |Ck|
+NK
+.
+(7)
+Definition 3 (Communication Load). The communication load is characterized by the tuple
+(L, D), where L (resp. D) is the upload (resp. download) defined as the total number of the bits
+sent by the K nodes (resp. AP) during the upload (resp. download) sub-phase, normalized by
+the total length of all intermediate values NKV :
+L ≜
+�K
+k=1 lk
+NKV ,
+D ≜
+l
+NKV .
+Remark 1 (Nontrivial Regime). In general, the non-trivial regime in our setup is
+1 ≤ c ≤ r ≤ K,
+(8a)
+0 ≤ D ≤ L ≤ 1 − r
+K .
+(8b)
+For completeness, we justify them by the following observations.
+• Justification of (8a): Since each IV needs to be computed at least once somewhere, we
+have c ≥ 1. Moreover, the definition of Ck in (2) implies that |Ck| ≤ |Mk|K, and thus by
+(6) and (7), c ≤ r. Finally, the regime r > K is not interesting, because in this case each
+node stores all the files, and can thus locally compute all the IVs required to compute its
+output function. In this case, c ≥ 1, D ≥ 0 and L ≥ 0, can be arbitrary.
+• Justification of (8b): D ≥ 0 is trivial. By (3), as the down-link signal X is created from the
+upload signals X1, . . . , XK, D = L is sufficient to communicate all the received information.
+Finally, each node k can trivially compute |Mk| of its desired IVs locally and thus only
+needs to receive N − |Mk| IVs from other nodes. Thus, such an uncoded manner requires
+an upload of L =
+�K
+k=1(N−|Mk|)V
+NKV
+= 1 − r
+K.
+
+6
+In the trivial case that the AP simply forwards all the receiving signals, i.e., X = (X1, . . . , XK),
+then D = L, and the model degrades to the distributed model without the AP as in [7], where
+the non-trivial region on the triple (r, c, L) was 1 ≤ c ≤ r ≤ K, 0 ≤ L ≤ 1 − r
+K.
+Definition 4 (Fundamental SCC Region). A Storage-Computation-Communication1 (SCC) quadru-
+ple (r, c, L, D) satisfying (8) is achievable if for any ϵ > 0 and sufficiently large N, W, V , there
+exist map, shuffle, and reduce procedures with storage load, computation load, upload and
+download less than r + ϵ, c + ϵ, L + ϵ and D + ϵ, respectively. The fundamental SCC region is
+defined as the set of all feasible SCC quadruple:
+R = {(r, c, L, D) : (r, c, L, D) is feasible}.
+Definition 5 (Optimal Tradeoff Surface). An SCC quadruple (r, c, L, D) is called Pareto-optimal
+if it is feasible and if no feasible SCC quadruple (r′, c′, L′, D′) exists so that r′ ≤ r, c′ ≤ c, L′ ≤ L
+and D ≤ D′ with one or more of the inequalities being strict. The set of all Pareto-optimal SCC
+quadruples is defined as the optimal tradeoff surface:
+O ≜ {(r, c, L, D) : (r, c, L, D) is Pareto-optimal}.
+The goal of this paper is to characterize the fundamental SCC region R and the optimal
+tradeoff surface O in our setup.
+III. MAIN RESULTS
+Before we present the main theorem, let us provide a toy example to illustrate the key idea
+of the proposed achievable scheme.
+A. An Toy Example for Achievable Scheme
+Consider the case, where there aorre K = 3 nodes and N = 6 files. Each node wants to
+compute an individual function from the N = 6 files as in (1). Fig. 2 illustrates the strategy
+achieving the Pareto-optimal point (r, c, L, D) = (2, 4
+3, 1
+6, 1
+9), where the uplink and downlink
+transmissions are illustrated in Fig.2(a) and 2(b), respectively.
+In Fig. 2, the three nodes are denoted by three boxes with red, green and blue edges re-
+spectively. The top-most lines in each of the three boxes indicate the files stored at the node.
+The rectangle below this line indicates the map functions at the node. The computed IVs
+are depicted below the rectangle, where red circles, green squares, and blue triangles indi-
+cate IVs {v1,1, · · · , v1,6}, {v2,1, · · · , v2,6}, and {v3,1, · · · , v3,6}, respectively. The dashed cir-
+cles/squares/triangles stand for the IVs that are not computed from the stored files. The last
+line of each box indicates the IVs that the node needs to learn during the shuffle phase.
+The N = 6 files
+W = {w1, w2, w3, w4, w5, w6}
+are partitioned into
+�K
+r
+�
+= 3 batches, i.e., {w1, w2}, {w3, w4}, {w5, w6}. In the map phase, the
+files {w1, w2} are simultaneously stored at nodes 1 and 3; the files {w3, w4} at nodes 1 and 2;
+and the files {w5, w6} at nodes 2 and 3. For each node, the computed IVs can be classified into
+two types: the IVs that will be used by its own reduce function (the first line below the “map”
+1The communication load includes both upload and download.
+
+7
+rectangle) and the IVs that will be used for transmission or decoding (the second and third lines
+below the “map” rectangle).
+In the shuffle phase, during the upload sub-phase, each node creates a coded signal by XORing
+two IVs and sends it to the AP as illustrated in Fig. 2(a), i.e., Nodes 1, 2 and 3 sends coded
+IVs v1,1 ⊕ v3,3, v3,4 ⊕ v2,5 and v6,2 ⊕ v2,1, respectively; during the download sub-phase, the AP
+combines the three received signals by a simple chain coding, i.e., the two downlink signals are
+formed by XORing the signals from nodes 1 and 2, and the signals from 2 and 3, respectively.
+The combined signals are sent to all the three nodes.
+In the reduce phase, for each node, since the two chain coded signals involve a coded signal
+transmitted by itself, the node can decode the two coded signals from the other two nodes.
+Moreover, from each of the coded singals, the node can further decode an IVs it needs, by
+XORing the coding signal with one of its computed IV. For example, Node 1 first decodes the
+two signals v3,4 ⊕ v2,5 and v2,6 ⊕ v1,2, then it can further decodes the IVs v2,5 and v2,6, since the
+IVs v3,4 and v1,2 have been computed locally. Finally, each node collects all IVs for its assigned
+reduce function, and computes the final output.
+B. Fundamental SCC Region and Optimal Tradeoff Surface
+For each i ∈ [K], define two SCC quadrules
+Pi ≜
+�
+i, i
+�
+1 − i − 1
+K
+�
+, 1
+i
+�
+1 − i
+K
+�
+,
+1
+i + 1
+�
+1 − i
+K
+��
+,
+Qi ≜
+�
+i, i, 1
+i
+�
+1 − i
+K
+�
+,
+1
+i + 1
+�
+1 − i
+K
+��
+.
+In the following, we will use P u
+i , Qu
+i , P d
+i , Qd
+i to denote the projections of Pi, Qi into the uplink
+and downlink SCC subspaces2, i.e.,
+P u
+i ≜
+�
+i, i
+�
+1 − i − 1
+K
+�
+, 1
+i
+�
+1 − i
+K
+��
+,
+Qu
+i ≜
+�
+i, i, 1
+i
+�
+1 − i
+K
+��
+,
+P d
+i ≜
+�
+i, i
+�
+1 − i − 1
+K
+�
+,
+1
+i + 1
+�
+1 − i
+K
+��
+,
+(9)
+Qd
+i ≜
+�
+i, i,
+1
+i + 1
+�
+1 − i
+K
+��
+.
+The main result of this paper is summarized in the following theorem, where the proofs are
+provided in the following sections.
+Theorem 1. The fundamental SCC region R is given by
+R =
+�
+(r, c, L, D) : 1 ≤ c ≤ r ≤ K, L∗(r, c) ≤ L ≤ 1 − r
+K , D∗(r, c) ≤ D ≤ L
+�
+,
+where L∗(r, c) is a function such that {(r, c, L∗(r, c)) : 1 ≤ c ≤ r ≤ K} forms the surface
+Fu ≜ △P u
+1 P u
+2 Qu
+2 ∪
+K−1
+∪
+i=2 △P u
+i−1P u
+i P u
+K ∪
+K−1
+∪
+i=2 ⊟P u
+i Qu
+i Qu
+i+1P u
+i+1
+2In this paper, we will refer r-c-L subspace as the uplink SCC subspace, and the r-c-D subspace the downlink SCC subspace.
+The superscripts “u” and “d” indicate “uplink” and “downlink”, respectively.
+
+8
+(a)
+(b)
+Fig. 2: Illustration of the CDC for star network: (a) Uplink (b) Downlink
+in the uplink SCC subspace, and D∗(r, c) is a function such that {(r, c, D∗(r, c)) : 1 ≤ c ≤ r ≤
+K} forms the surface
+Fd ≜ △P d
+1 P d
+2 Qd
+2 ∪
+K−1
+∪
+i=2 △P d
+i−1P d
+i P d
+K ∪
+K−1
+∪
+i=2 ⊟P d
+i Qd
+i Qd
+i+1P d
+i+1
+in the downlink SCC subspace. The optimal tradeoff surface is given by
+O =
+K−1
+∪
+i=2 {θ1Pi−1 + θ2Pi + θ3PK : θ1, θ2, θ3 ∈ [0, 1], θ1 + θ2 + θ3 = 1}.
+(10)
+In Fig. 3, the functions L∗(r, c) and D∗(r, c) are ploted for K = 10 nodes. Notice that, by
+setting r = c, we recover the optimal upload and download as investigated in [20]3, i.e.,
+3The measurement of communication load is up to a scalar “K” in [20] compared Definition 3, and a slightly difference in
+assumption in [20] is that each node has a fixed storage load.
+
+Files
+25
+6
+Map
+Node 3
+Computes
+12
+56
+1:256
+Needs
+3
+Files
+3
+2
+4
+3
+4
+Files
+T9
+1
+Map
+1④3
+Map
+3456
+Computes
+Computes 3 4 .5..6
+14
+3:4:60
+Needs
+Needs
+12
+Node 1
+Node 2Files
+25
+6
+Map
+Node 3
+Computes
+12
+1256
+Needs
+3
+2
+Files
+2
+3
+4
+Files
+3
+4
+5
+T9
+Map
+Map
+3451
+16
+Computes
+Computes 3 4 .5..6
+3:4:60
+Needs
+Needs
+12
+Node 1
+Node 29
+1) the optimal upload for given storage is given by
+L∗(r) ≜ Conv
+�1
+r
+�
+1 − r
+K
+��
+,
+which corresponds the curve formed by the line segments Qu
+1Qu
+2, Qu
+2Qu
+3, . . . , Qu
+K−1Qu
+K.
+2) the optimal download for given storage is given by
+D∗(r) ≜ Conv
+�
+1
+r + 1
+�
+1 − r
+K
+��
+,
+which corresponds to the curve formed by the line segments Qd
+1Qd
+2, Qd
+2Qd
+3, . . . , Qd
+K−1Qd
+K.
+Observe that, the line segments Qu
+i P u
+i in the uplink SCC space and Qd
+i P d
+i in the downlink
+space (i = 2, 3, . . . , K) are parellel to the c-axis, which indicate that the computation load can be
+saved to achieve L∗(r). The length of the line segments indicates the amount of the computation
+load that can be saved. Thus, with larger storage load r, the saving of computation load to
+achieve L∗(r) and D∗(r) is larger. It will be clear later that the saving on the computation load
+is due to the fact that, under the assumption that each not computes all IVs it can computes,
+some of the IVs computed are not used in neither generating the signal, nor in the decoding
+process.
+The projections of the Pareto-optimal surface O into the uplink and downlink SCC space
+correspond to the surfaces
+Ou ≜ {P u
+i−1P u
+i P u
+K : i ∈ [2 : K − 1]}
+and
+Od ≜ {P d
+i−1P d
+i P d
+K : i ∈ [2 : K − 1]},
+respectively. Observe that, for a given feasible (r, c) pair, the optimal upload is strictly larger
+than the optimal download. We will see that this is achieved by performing some simple chain
+coding at the AP to combine the signals from different nodes. Interestingly, both the upload and
+download can be simultaneously achieved for a fixed (r, c) pair.
+Remark 2 (Relation to Results in [7]). One can observe that, the surfaces composing L∗(r, c) and
+Ou concide with the optimal communication load and the Pareto-optimal SCC tradeoff surface
+in the setup where the nodes directly connect to each other through a shared link (c.f. [7, Fig.
+2]), respectively. It was showed in [7], by dropping the computations of the IVs that are not
+used in the CDC scheme [3] as in [4], the resultant coded computing scheme can achieve the
+corner points of the Pareto-optimal SCC tradeoff surface (which is same as Ou). In fact, in
+our proposed scheme, each node performs the same map procedures as in [4], but the signals
+are sent to the AP. The AP performs a simple chain coding on the received signals to further
+compress the length of the signals, which leads to a further decrease of the download compared
+to the upload. We will present the whole process in Section IV.
+IV. ACHIEVABILITY
+Since the set O is exactly all the Pareto-optimal points of the set R (Appendix A), we
+only need to prove the achievability of the hypersurface O. We will derive a coded computing
+scheme that achieves the SCC quadruple Pi. Moreover, for any fixed θ1, θ2, θ3 ∈ [0, 1] such that
+
+10
+Fig. 3: The functions L∗(r, c) and D∗(r, c).
+θ1+θ2+θ3 = 1, divide the N files into three groups of sizes4 θ1N, θ2N and θ3N. By applying the
+scheme achieving the points Pi−1, Pi and PK on the three groups of files, the resultant scheme
+achieves the point P = θ1Pi−1 + θ2Pi + θPK. Thus, we only need to prove the achievability of
+Pi, i ∈ [K].
+A. Coded Distributed Computing for Star Network
+We now describe the scheme achieving Pi for a fixed i ∈ [K].
+Define
+Ωi ≜ {T ⊆ [K] : |T | = i} ,
+∀ i ∈ [K].
+For i = K, PK = (K, 1, 0, 0) is trivial, since each node can simply store all the files and
+computes their IVs as well as their reduce functions locally, with no communication loads.
+Consider a fixed i ∈ [K − 1], the N files are partitioned into
+�K
+i
+�
+batches, each containing
+ηi = N
+�K
+i
+�
+(11)
+files. Each batch is then associated with a subset T of [K] of cardinality i, i.e., an element in
+Ωi. Let WT denote the batch of the ηi files associated with set T . Then,
+W = {w1, . . . , wN} =
+�
+T ∈Ωi
+WT .
+4This requires that θ1, θ2, θ3 have to be rational. If any one is irrational, one can replace it by a rational number arbitrarily
+close to it.
+
+Optimal Upload and Download, K = 10
+pu
+0.9 -
+..The Function L*(r, c)
+(/T)
+0.8 ~
+-- The Function D*(r,c)
+0.7
+-Plane r = c
+Communication load (
+0.6 ~
+Pf,
+0.5 
+Q
+0.4 ~
+0.3 ~
+Qu
+0.2 -
+0.1 
+0.
+Storage load (r)s
+Pl0/ P10
+2
+3
+5
+4
+10 Q1 /Qil 0
+6
+6
+8
+>
+Computation load (c)11
+Further let UT ,k be the set of IVs for output function φk that can be computed from the files in
+WT :
+UT ,k ≜ {vk,n : wn ∈ WT }.
+We now describe the map, shuffle, and reduce procedures.
+1) Map Phase: Each node k stores
+Mk =
+�
+T ∈Ωi:k∈T
+WT ,
+and computes the IVs
+Ck = C1
+k ∪ C2
+k,
+(12)
+where
+C1
+k =
+�
+T ∈Ωi:k∈T
+UT ,k,
+(13a)
+C2
+k =
+�
+T ∈Ωi:k∈T
+�
+q∈K\T
+UT ,q.
+(13b)
+In other words, for each batch T , each node k computes all the IVs for its own function
+k, and all the IVs for the function q if node q does not have the batch T .
+2) Shuffle Phase: For each element T ∈ Ωi and each index j ∈ K\T , we partition the set
+UT ,j into i smaller subsets
+UT ,j =
+�
+U k
+T ,j : k ∈ T
+�
+(14)
+of equal size.
+In the upload sub-phase, for each S ∈ Ωi+1 and k ∈ S, by (13b), node k can compute
+the signal
+Xk
+S ≜
+�
+l∈S\{k}
+U k
+S\{l},l
+from the IVs calculated during the map phase. Node k thus sends the multicast signal
+Xk =
+�
+Xk
+S : S ∈ Ωi+1 such that k ∈ S
+�
+to the AP R. Thus, the AP R receives the signals X1, . . . , XK.
+In the download sub-phase, for each S = {k1, . . . , ki+1} ∈ Ωi+1, the AP R creates a
+signal,
+XS ≜ (Xk1
+S ⊕ Xk2
+S , Xk2
+S ⊕ Xk3
+S , . . . , Xki
+S ⊕ Xki+1
+S
+).
+(15)
+Then the AP broadcast the signal
+X ≜ {XS : S ∈ Ωi+1}.
+(16)
+3) Reduce Phase: Notice that C2
+k only contains the IVs vq,n where q ̸= k. Thus, by (12) and
+
+12
+(13a), during the shuffle phase each node k needs to learn all the IVs in
+�
+T ∈Ωi : k/∈T
+UT ,k.
+Fix an arbitrary T ∈ Ωi such that k /∈ T . From the received multicast message XT ∪{k},
+since the signal Xk
+T ∪{k} is generated by node k, by (15), node k can decode Xj
+T ∪{k} for
+all j ∈ T , where the signal
+Xj
+T ∪{k} =
+�
+l∈T ∪{k}\{j}
+U j
+T ∪{k}\{l},l
+is sent by node j during the shuffle phase. For any fixed j ∈ T , node k can recover the
+missing IV U j
+T ,k through a simple XOR operation:
+U j
+T ,k = Xj
+T ∪{k} ⊕
+�
+l∈T \{j}
+U j
+T ∪{k}\{l},l,
+(17)
+where U j
+T ∪{k}\{l},l is calculated at node k by (13b) and (14) for all l ∈ T \{j}. Moreover,
+node k can decode UT ,k from
+�
+Xj
+T ∪{k} : j ∈ T
+�
+.
+by (14) and (17). After collecting all the missing IVs, node k can proceed to compute the
+reduce function (5).
+Remark 3 (Comparison with [20]). Compared to the coded computing scheme in [20], two
+differences of the above scheme are:
+1) In the map phase, each node only needs to compute the IVs described in (12) and (13),
+because only those IVs are useful for creating or decoding the coded signals, while in
+[20], all the IVs pertaining to the files in Mk are computed, i.e., node k computes
+�C ≜
+�
+T ∈Ωi,k∈T
+�
+q∈[K]
+UT ,q.
+(18)
+This scheme in fact achieves the point Qi, which is inferior to Pi for i > 0 in computation
+load. The idea of removing the redundancy has been proposed in the setup where the nodes
+connects to each other directly through a bus link by Ezzeldin [4] and Yan et al [7].
+2) In (15), for any node set S of size i + 1, we used a simple chain coding on the signals to
+form i signals, while in [20], it uses random coding on the signals {Xk
+S : k ∈ S} to form
+i coded signals. The advantage of chain coding in (15) is obvious:
+a) It has smaller encoding and decoding complexities;
+b) It can be operated on the binary field F2;
+c) The order of nodes in the chain can be arbitrary. It makes sense in some scenarios:
+the signals {Xk
+S : k ∈ S} may arrive at different time points. Consider the case that
+the signals arrive in the ordder Xk1
+S , Xk2
+S , . . . , Xki+1
+S
+, to perform the encoding (15), at
+any time the AP only needs to keep one signal in its buffer, because each coordinate
+in (15) only depends on two consecutive signals. While with random linear coding,
+the AP typically have to wait for all signals {Xk
+S : k ∈ S}. Thus, the chain coding
+can reduce the buffer size at the AP and the node to node delay.
+
+13
+Remark 4 (PDA framework). In [7], [21], [22], a coded computing scheme was derived based on
+placement delivery array (PDA), which was proposed in [23] to explore coded caching schemes
+with uncoded placement [24]. In particular, it turns out that the Maddah-Ali and Niesen’s coded
+caching scheme corresponds to a special structure of PDA (referred to as MAN-PDA). It was
+showed in [7] that, with any given PDA belonging to a special class (defined as PDA for
+distributed computing (Comp-PDA)), one can always obtain a coded computing scheme. The class
+of PDAs achieving the Pareto-optimal tradeoff surface was characterized in [7]. The advantage of
+establishing the PDA framework is, various known PDA structure, e.g., the constructions in [23],
+[25], [26] can be directly utilized to obtain coded computing schemes with low file complexity5.
+In our setup, similar connections between coded computing schemes and Comp-PDA can be
+established, by following the same steps as in [7] for upload singals, and incoporating the chain
+coding (15) on all multicast signals from the Comp-PDA for the downlink signals. For example,
+the scheme described in Fig. 2 can be derived from the PDA
+�
+�
+∗
+1
+∗
+1
+∗
+∗
+∗
+∗
+1
+�
+� ,
+for details of forming the upload signals in Fig. 2(a), one can refer to [7, Example 4].
+B. Performance Analysis
+We analyze the performance of the scheme.
+1) Storage Load: The number of batches in Mk is
+�K−1
+i−1
+�
+, each consisting of ηi files. Thus,
+the storage load is
+r = 1
+N · K ·
+�K − 1
+i − 1
+�
+· ηi = i.
+(19)
+2) Computation Load: Since C1
+k ∩ C2
+k = ∅, we have |Ck| = |C1
+k| + |C2
+k|. From (11), (13a), and
+(13b), we have
+|C1
+k| =
+�K − 1
+i − 1
+�
+· ηi = iN
+K ,
+|C2
+k| =
+�K − 1
+i − 1
+�
+· (K − i) · ηi
+=
+�
+1 − i
+K
+�
+· i · N.
+Thus, the computation load is
+c =
+�K
+k=1 |Ck|
+NK
+= i
+�
+1 − i − 1
+K
+�
+.
+(20)
+3) Communication Load: The number of signals that each node k transmits is
+�K−1
+i
+�
+, each
+of size ηi·V
+i
+bits. Thus, the length of the signal Xk is lk =
+�K−1
+i
+� ηi·V
+i
+bits. Therefore, the
+5The file complexity of a coded computing scheme is defined as the smallest number of files required to implement the
+scheme, e.g., the file complexity of the proposed scheme achiving Pi is
+�K
+i
+�
+.
+
+14
+upload is
+L =
+�K
+k=1 lk
+NKV
+= 1
+i ·
+�
+1 − i
+K
+�
+.
+(21)
+By (15) and (16), the AP R transmits
+� K
+i+1
+�
+· i signals, each of size
+ηi·V
+i
+bits, thus the
+download is
+D =
+1
+NKV ·
+� K
+i + 1
+�
+· i · ηi · V
+i
+=
+1
+i + 1
+�
+1 − i
+K
+�
+.
+(22)
+From (19), (20), (21) and (22), we show the achievability of the SCC quadruple Pi.
+V. CONVERSE
+We need to prove that for any achievable (r, c, L, D) satisfying (8),
+L ≥ L∗(r, c),
+(23a)
+D ≥ D∗(r, c).
+(23b)
+Consider a coded distributed computing scheme achieving (r, c, L, D), with file allocations M[K],
+IV allocations C[K], uplink signals X[K] and downlink signal X. By the decoding condition (4),
+H(Vk|X, Ck) = 0,
+∀ k ∈ [K].
+Thus for any k ∈ [K],
+H(Vk|X1, . . . , XK, Ck)
+(a)
+= H(Vk|X1, . . . , XK, X, Ck)
+≤ H(Vk|X, Ck)
+= 0,
+where (a) follows since the downlink signal X is determined by the uplink signals X[K] by (3).
+That is, with the signals X1, . . . , XK and the locally computed IVs Ck, node k can decode
+all the IVs it needs. As a result, the file allocations M[K], IV allocations C[K] and the uplink
+singals X[K] consisitute an valid scheme for the distributed computing system where the nodes
+are connected through a bus shared link directly, as investigated in [7]. Therefore, by the results
+in [7, Theorem 2], we have proved (23a).
+We proceed to prove the (23b). For any k ∈ [K] and nonempty S ⊆ [K]\{k}, define
+Bk,S ≜ {vk,n : vk,n is exclusively computed by the nodes in S},
+�Bk ≜ {vk,n : vk,n is the computed by node k}.
+Let bk,S be the cardinality of the set Bk,S and ˜bk be the cardinality of �Bk. Obviously, the subsets
+{Bk,S : S ⊆ [K]\{k}, S ̸= ∅} and �Bk form a partition of the IVs Vk, thus
+˜bk +
+�
+S⊆[K],S̸=∅
+bk,S = N.
+
+15
+For each j ∈ [K − 1], the set of IVs not computed locally but exclusively computed by j other
+nodes are
+Bj =
+�
+k∈[K]
+�
+S⊆[K]\{k},|S|=j
+Bk,S.
+Then the cardinality of set Bj is given by
+bj ≜
+�
+k∈[K]
+�
+S⊆[K]\{k},|S|=j
+bk,S,
+∀j ∈ [K − 1].
+(24)
+To prove the lower bound in (23b), we need the following two lemmas.
+Lemma 1. The entropy of the download signal X satisfy
+H(X) ≥ V
+K−1
+�
+j=1
+bj
+j + 1.
+Proof: Assume that the AP holds all IVs Vk, then the access point can create the signal X.
+Consider the data exchange problem6 formed by the AP and the K nodes, where only the AP
+sends the signal X to all the K nodes. Notice that, in this system, each bits in Bk,S is cached
+at the AP and the nodes in S, but only demanded by node k. Thus, by the lower bound in [27,
+Theorem 1],
+H(X) ≥ V
+�
+k∈[K]
+�
+S⊆[K]\{k}
+1
+(|S| + 1) + 1 − 1 · bk,S
+= V
+K
+�
+k=1
+K−1
+�
+j=1
+�
+S⊆[K]\{k},|S|=j
+1
+j + 1 · bk,S
+(a)
+= V
+K−1
+�
+j=1
+1
+j + 1
+K
+�
+k=1
+�
+S⊆[K]\{k},|S|=j
+bk,S
+= V
+K−1
+�
+j=1
+bj
+j + 1,
+where in (a), we utilized (24).
+The following lemma was proved in [7, Lemma 2].
+Lemma 2. The parameters b1, . . . , bK−1 defined in (24) satisfy
+K−1
+�
+j=1
+bj ≥ N(K − r),
+K−1
+�
+j=1
+(j − 1)bj ≤ (c − 1)NK.
+6Data exchange problem was defined in [27], where each of the nodes holds a subset of the information bits, and request
+another subset of information bits.
+
+16
+For a fixed r ∈ [1 : K], and each i ∈ [K], define
+ci ≜ 1 +
+�
+1 − r
+K
+�
+(i − 1).
+(25)
+Let λi, µi ∈ R+ such that
+λix + µi|x=ci−1 =
+1
+ci−1 + 1 − 2r/K ·
+�
+1 − r
+K
+�2
+= 1
+i
+�
+1 − r
+K
+�
+,
+(26a)
+λix + µi|x=ci =
+1
+ci + 1 − 2r/K ·
+�
+1 − r
+K
+�2
+=
+1
+i + 1
+�
+1 − r
+K
+�
+.
+(26b)
+From (26a) and (26b), the following relationships hold:
+λi = −
+1
+i(i + 1) < 0,
+(27a)
+µi = 2i − 1
+i(i + 1)
+�
+1 − r
+K
+�
++
+1
+i(i + 1) > 0,
+(27b)
+λi + µi = 2i − 1
+i(i + 1)
+�
+1 − r
+K
+�
+> 0.
+(27c)
+Moreover, by its convexity over x ∈ [1, ∞), the function
+1
+x + 1 − 2r/K
+�
+1 − r
+K
+�2
+− (λix + µi)
+must be nonnegative outside the interval formed by the two zero points, i.e.,
+1
+x + 1 − 2r/K
+�
+1 − r
+K
+�2
+≥ λix + µi,
+∀ x ∈ [1, ci−1] ∪ [ci, ∞).
+Therefore,
+1
+cj + 1 − 2r/K
+�
+1 − r
+K
+�2
+≥ λicj + µi,
+∀ j ∈ [K − 1].
+(28)
+Now, we are ready to derive the lower bound for the download D:
+D ≥ H(X)
+NKV
+(a)
+≥
+K−1
+�
+j=1
+bj
+NK ·
+1
+j + 1
+(b)
+=
+1
+N(K − r)
+K−1
+�
+j=1
+bj ·
+1
+cj + 1 − 2r/K
+�
+1 − r
+K
+�2
+(c)
+≥
+1
+N(K − r)
+K−1
+�
+j=1
+bj(λicj + µi)
+
+17
+(d)
+=
+1
+N(K − r)
+K−1
+�
+j=1
+bj
+�
+λi
+�
+1 +
+�
+1 − r
+K
+�
+(j − 1)
+�
++ µi
+�
+=
+λi
+NK ·
+K−1
+�
+j=1
+(j − 1)bj +
+λi + µi
+N(K − r) ·
+K−1
+�
+j=1
+bj
+(e)
+≥
+λi
+NK · (c − 1)NK +
+λi + µi
+N(K − r) · N(K − r)
+= λic + µi
+= −
+2i − 1
+Ki(i + 1)r −
+1
+i(i + 1)c +
+2
+i + 1.
+(29)
+where (a) follows from Lemma 1; (b) and (d) follow from the definition of ci in (25); (c) follows
+from (28); and (e) follows from Lemma 2 and the signs of λi and λi + µi in (27).
+Notice that the three points P d
+i−1, P d
+i and P d
+K defined in (9) satisfy (29) with equality. Thus, the
+inequalities above indicate that all the feasible points (r, c, L, D) must satisfy that the projection
+into the download SCC space (r, c, D) must lie above the plane containing △P d
+i−1P d
+i P d
+K.
+Furthmore, D should be lower bounded by the optimal download even if each node computes
+all the IVs that can be computed locally from their stored file, i.e., a similar setup as in [20].
+The converse in [20] indicates that L is lower bounded as follows in the r-D plane7:
+D ≥ Conv
+�1
+r
+�
+1 − r
+K
+��
+,
+r ∈ {1, 2, . . . , K}.
+(30)
+Finally, by the lower bounds in (29) for i = 2, 3, . . . , K − 1 and (30), D is lower bounded by
+D∗(r, c), i.e., the lower bound (23b) is proved.
+VI. CONCLUSION
+In this paper, the Pareto-optimal storage-computation-upload-download tradeoff surface is
+characterized for the MapReduce distributed computing system, where the nodes have to exchage
+intermediate values through an access point that can broadcast signals to all nodes. It turns
+out that, for a given storage-computation pair (r, c), the optimal upload and download can be
+simultaneously achieved. Information-theoretical bounds matching the achievable communication
+load are provided for both uplink and downlink.
+APPENDIX A
+THE RELATION OF HYPERSURFACE O AND REGION R
+We now prove that O is the Pareto-optimal surface of the region R. Obviously, all Pareto-
+optimal points must lie on the surface
+F = {(r, c, L∗(r, c), D∗(r, c)) : 1 ≤ c ≤ r ≤ K}.
+7Although the setup in [20] assumes a fixed storage capacity at each node, the proof the following inequality do not rely on
+this assumption.
+
+18
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+Fig. 4: The projections of Pi and Qi (i ∈ [K]) to the storage-computation subspace (r-c plane).
+Let the projections of points Pi and Qi into the r-c plane be P ′
+i and Q′
+i (i ∈ [K]), respectively8,
+i.e.,
+P ′
+i =
+�
+i, i
+�
+1 − i − 1
+K
+��
+,
+Q′
+i = (i, i).
+Let the projection of the surface F to the r-c plane be
+F′ ≜ {(r, c) : 1 ≤ c ≤ r ≤ K} = △P ′
+1P ′
+KQ′
+K.
+Notice that, here the “projection” map is one-to-one. Moreover, F′ can be decomposed into (see
+Fig. 4)
+F′ = △P ′
+1P ′
+2Q′
+2 ∪
+K−1
+∪
+i=2 △P ′
+i−1P ′
+iP ′
+K ∪
+K−1
+∪
+i=2 ⊟P ′
+iQ′
+iQ′
+i+1P ′
+i+1.
+Since the triangle △P u
+1 P u
+2 Qu
+2 and the trapezoids ⊟P u
+i Qu
+i Qu
+i+1P u
+i+1 in the uplink SCC space
+(i ∈ [2 : K − 1]) are parallel to c-axis, and so as the triangle △P d
+1 P d
+2 Qd
+2 and the trapezoids
+⊟P d
+i Qd
+i Qd
+i+1P d
+i+1 in the downlink SCC space, all the points (r, c, L∗(r, c), D∗(r, c)) ∈ F such
+that
+(r, c) ∈ △P ′
+1P ′
+2Q′
+2 ∪
+K−1
+∪
+i=2 ⊟P ′
+iQ′
+iQ′
+i+1P ′
+i+1\
+K−1
+∪
+i=2 △P ′
+i−1PiP ′
+K
+(31)
+cannot be Pareto-optimal. In the following, we prove that, all the points (r, c, L∗(r, c), D∗(r, c))
+such that
+(r, c) ∈
+K−1
+∪
+i=2 △P ′
+i−1P ′
+iP ′
+K
+(32)
+are Pareto-optimal.
+Now fix a quadruple (r1, c1, L∗(r1, c1), D∗(r1, c1)) ∈ F that satisfies (32). We show that it is
+Pareto-optimal. To this end, consider any other triple (r2, c2, L2, D2) ∈ R that satisfies
+r2 ≤ r1,
+c2 ≤ c1,
+(33a)
+L2 ≤ L∗(r1, c1),
+D2 ≤ D∗(r1, c1).
+(33b)
+We show by contradiction that all four inequalities must hold with equality. Notice that, (r2, c2)
+either satisfies (31) or (32).
+8Notice that the projections of P u
+i , Qu
+i and P d
+i , Qd
+i into the r-c plane are the same as the ones of the points Pi and Qi. As a
+result, the projections of △P u
+1 P u
+2 Qu
+2/△P d
+1 P d
+2 Qd
+2, △P u
+i−1P u
+i P u
+K/△P u
+i−1P u
+i P u
+K, and ⊟P u
+i Qu
+i Qu
+i+1P u
+i+1/⊟P d
+i Qd
+i Qd
+i+1P d
+i+1 into
+the r-c plane are △P ′
+1P ′
+2Q′
+2, △P ′
+i−1P ′
+iP ′
+K and ⊟P ′
+iQ′
+iQ′
+i+1P ′
+i+1, respectively.
+
+19
+1) Assume that (r2, c2) satisfies (32). If r2 < r1 or c2 < c1, then consider the uplink SCC
+subspace, one can verify that the points P u
+i−1, P u
+i and P u
+K are on the surface
+L = −
+1
+i(i − 1)c − 2
+Kir + 2i − 1
+i(i − 1).
+(34)
+Therefore, it must hold that
+L∗(r2, c2) > L∗(r1, c1),
+(35)
+because all the surfaces containing △P u
+i−1P u
+i P u
+K (i ∈ [2 : K − 1]) have positive directional
+derIVtives along (r2 − r1, c2 − c1) by (34). Since (r2, c2, L2, D2) ∈ R, we have L2 ≥
+L∗(r2, c2) and thus by (35), L2 > L∗(r1, c1), which contradicts (33). Therefore, it must
+hold that r2 = r1 and c2 = c1. Then obviously, L2 ≥ L∗(r2, c2) = L∗(r1, c1) and D2 ≥
+D∗(r2, c2) = D∗(r1, c1), thus all equalities in (33) hold.
+2) Assume now that (r2, c2) satisfies (31). Then, (r2, c2) must lie on at least one of the K −1
+facets
+△P ′
+1P ′
+2Q′
+2 or ⊟ P ′
+iQ′
+iQ′
+i+1P ′
+i+1,
+i ∈ [2 : K − 1],
+and it must not lie on the line segments P ′
+i−1P ′
+i, i ∈ [2 : K]. As the facets △P u
+1 P u
+2 Qu
+2,
+⊟P u
+i Qu
+i Qu
+i+1P u
+i+1 (i ∈ [2 : K −1]) in the uplink SCC subspace are all parellel to the c-axis,
+and so as the facets △P d
+1 P d
+2 Qd
+2, ⊟P d
+i Qd
+i Qd
+i+1P d
+i+1 (i ∈ [2 : K − 1]) in the downlink SCC
+facets, there exists c3 < c2 ≤ c1 such that (r2, c3) satisfies (32), and
+L∗(r2, c3) = L∗(r2, c2), D∗(r2, c3) = D∗(r2, c2).
+Therefore,
+L2 ≥ L∗(r2, c2) = L∗(r2, c3)
+(a)
+> L∗(r1, c1),
+(36)
+where (a) follows by proof step 1). But (36) contradicts with (33).
+From the above analysis, we conclude that, the set of all Pareto-optimal points of R is exactly
+all the quadruples (r, c, L∗(r, c), D∗(r, c)) ∈ F satisfying (32). Notice that those points are exactly
+the surface O defined in (10).
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+

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+arXiv:2301.13645v1  [math.AP]  31 Jan 2023
+Existence, uniqueness and
+L2
+t(H2
+x) ∩ L∞
+t (H1
+x) ∩ H1
+t (L2
+x) regularity of the
+gradient flow of the Ambrosio-Tortorelli functional
+Tommaso Cortopassi ∗
+Abstract
+We consider the gradient flow of the Ambrosio-Tortorelli functional at
+fixed ǫ > 0, proving existence and uniqueness of a solution in dimension 2.
+The strategy of the proof essentially follows the one in the first part of [9],
+but as suggested in a footnote by the authors of [9] it employs a different
+and simpler technique, which is used for a different equation in [4] and in the
+end it allows to prove better estimates than the ones obtained in the original
+article. In particular we prove that if U ⊂ R2 is a bounded Lipshitz domain,
+the initial data (u0, z0) ∈ [H1(U)]2 and 0 ≤ z0 ≤ 1, then for every T > 0
+there exists a unique gradient flow (u(t), z(t)) of the Ambrosio-Tortorelli
+functional such that
+(u, z) ∈ [L2(0, T; H2(U)) ∩ L∞(0, T; H1(U)) ∩ H1(0, T; L2(U))]2.
+The basic difference from [9], as already said, is a better regularity result
+and a simpler proof: while in [9] they used a localization argument based
+on an idea by Struwe (see [19]), here crucial estimates on the fourth powers
+of the L4
+t (L4
+x) norms of the gradients will be obtained employing a suitable
+version of the Meyers theorem due to Gallouet and Monier (see [15], [11]).
+1
+Introduction
+The Mumford-Shah functional, introduced in [16], is defined as
+E(u, Γ) = 1
+2
+�
+U\Γ |∇u|2 + (u − g)2dx + H1(Γ),
+(1.1)
+where u ∈ H1(U \ Γ), U ⊂ R2 open and bounded, Γ ⊂ U is closed and H1
+is the one dimensional Hausdorff measure. This functional has been extensively
+studied for its applications in image segmentation and fracture mechanics (a de-
+tailed survey can be found in [5]). The definition of the functional depends on the
+model it is used for, in (1.1) we gave the original definition of [16] which is suited
+∗Scuola Normale Superiore, 56126 Pisa, Italy. E-mail: [email protected]
+1
+
+for image segmentation models, the interested reader can see the seminal paper
+[10] about the fracture mechanic case. The idea is rather simple: given a gray
+image g (i.e. a scalar function), we want to find (˜u, ˜Γ) such that
+(˜u, ˜Γ) = arg min
+u∈H1(U\Γ)
+Γ⊂U closed
+E(u, Γ).
+(1.2)
+In this case, the function u will approximate in a “smooth way” the image g
+while Γ will be the contour set. From a theoretical point of view, in [6] the authors
+proved the existence of a solution for (1.2) by restricting to functions u ∈ SBV (U)
+(see [1]) and Γ = Su, i.e. the set of discontinuity jump points of u.
+From a numerical point perspective, since the functional involves the measure
+of the singular jump set of a (unknown) function u, its direct numerical implemen-
+tation is often not possible or not feasible. A standard approach is to minimize a
+more regular functional proposed by Ambrosio and Tortorelli in [2] defined as
+1
+2
+�
+U[(ηǫ + z2)|∇u|2 + (u − g)2] +
+�
+U
+�(1 − z)2
+4ǫ
++ ǫ|∇z|2
+�
+(1.3)
+which approximates the Mumford-Shah functional in Γ-convergence as ǫ → 0
+in an even more general setting, i.e. not being restricted to the 2 dimensional case,
+but considering U ⊂ Rn open and the (n − 1)-dimensional Hausdorff measure for
+Γ. The Ambrosio-Tortorelli functional is the way with which one usually finds
+(approximate) minima points for (1.1), in particular one can use a gradient flow
+approach for (1.3). As in [9], we’ll consider the gradient flow of (1.3), which is
+given by
+
+
+
+
+
+
+
+
+
+
+
+
+
+∂tu = div((ηǫ + z2)∇u) − (u − g) in (0, T) × U
+∂tz = 2ǫ∆z − z|∇u|2 + 1−z
+2ǫ in (0, T) × U
+∂
+∂nu =
+∂
+∂nz = 0 in (0, T) × ∂U
+u(0, ·) = u0 and z(0, ·) = z0 in {0} × U
+(1.4)
+with ηǫ, ǫ > 0 fixed. As already mentioned, our goal is to study the existence,
+uniqueness and regularity of solutions of (1.4), and the novelty lies in an approach
+(already suggested in a footnote of [9] and used in [4] for a different equation)
+which simplifies the proof of [9] while also gaining better regularity results. In
+particular in [9] they only manage to prove the L2
+t(H2
+x) regularity for a short time
+T1, since the crucial estimate in their argument is a local energy estimate, inspired
+by a technique due to Struwe [19], which only holds for sufficiently small times.
+2
+
+2
+Main result
+Let U ⊂ R2 be a Lipshitz bounded domain and let (u0, z0) ∈ [H1(U)]2 with
+0 ≤ z0 ≤ 1, g ∈ L2(U). As stated in the introduction, we want to prove existence,
+uniqueness and regularity of solutions of the gradient flow of (1.3), given by
+
+
+
+
+
+
+
+
+
+
+
+
+
+∂tu = div((ηǫ + z2)∇u) − (u − g) in (0, T) × U
+∂tz = 2ǫ∆z − z|∇u|2 + 1−z
+2ǫ ) in (0, T) × U
+∂
+∂nu =
+∂
+∂nz = 0 in (0, T) × ∂U
+u(0, ·) = u0 and z(0, ·) = z0 in {0} × U.
+(2.1)
+However we will mostly work with the slightly modified system
+
+
+
+
+
+
+
+
+
+
+
+
+
+∂tu = div((ηǫ + φ(z)2)∇u) − (u − g) in (0, T) × U
+∂tz = 2ǫ∆z − φ′(z)φ(z)|∇u|2 + 1−z
+2ǫ in (0, T) × U
+∂
+∂nu =
+∂
+∂nz = 0 in (0, T) × ∂U
+u(0, ·) = u0 and z(0, ·) = z0 in {0} × U
+(2.2)
+where φ is a cutoff. The reason to introduce such a cutoff, as we will show later,
+is to bound the L∞ norm of ηǫ+φ(zN)2 when considering Galerkin approximations
+zN. To be precise, a good working definition of φ might be
+φ(s) =
+
+
+
+
+
+
+
+−1 if s ≤ −1
+s if − 1 < s < 2
+2 if s ≥ 2
+but you could actually do the cuts at levels −δ1 and 1 + δ2 for every δ1, δ2 > 0
+and nothing would change. A modified Ambrosio-Tortorelli functional (of which
+(2.2) is the gradient flow of) will be denoted as ATǫ and defined as
+ATǫ(u(t), z(t)) = 1
+2
+�
+U[(ηǫ+φ(z(t))2)|∇u(t)|2+(u(t)−g)2]+
+�
+U
+�(1 − z(t))2
+4ǫ
++ ǫ|∇z(t)|2
+�
+.
+(2.3)
+First of all, let’s define the notion of strong solution, following [9].
+Definition 2.1. Strong solution
+A couple (u, z) is a strong solution of (2.1) if it satisfies the system in a dis-
+tributional sense, i.e. for every ψ ∈ H1(U) it holds
+�
+U
+u(t)ψdx =
+�
+U
+u0ψdx −
+� t
+0
+�
+U
+(ηǫ + z(s)2)⟨∇u(s), ∇ψ⟩dxds −
+� t
+0
+�
+U
+(u(s) − g)ψdxds
+�
+U
+z(t)ψdx =
+�
+U
+z0ψdx − 2ǫ
+� t
+0
+�
+U
+⟨∇z(s), ∇ψ⟩dxds −
+� t
+0
+�
+U
+z(s)ψ|∇u(s)|2 + 1 − z(s)
+2ǫ
+ψdxds
+for almost every t ∈ (0, T), and moreover we require
+(u, z) ∈ [L2(0, T; H2(U)) ∩ L∞(0, T; H1(U)) ∩ H1(0, T; L2(U))]2.
+We start with a proposition which shows how a solution of (2.2) is also solution
+of (2.1) if 0 ≤ z0 ≤ 1.
+3
+
+Proposition 2.1. Let 0 ≤ z0 ≤ 1. If a strong solution (u, z) for (2.2) exists, it
+must be 0 ≤ z(t, x) ≤ 1 for every t ∈ (0, T) and for a.e. x ∈ U. So (u, z) will also
+be a solution of (2.1).
+Proof. If we test the equation for z in (2.2) with z itself it gives
+1
+2
+d
+dt||z(t)||2
+L2(U) = −2ǫ||∇z(t)||2
+L2(U)−
+�
+U φ′(z(t))φ(z(t))z(t)|∇u(t)|2ds+
+�
+U
+1 − z(t)
+2ǫ
+z(t)ds
+(2.4)
+where the existence of the time derivative is ensured by Lions-Magenes lemma
+(see [14]). Consider f0(z) = max{−z, 0} and f1(z) = max{z − 1, 0}, so f0 and f1
+are defined as
+f0(s) =
+
+
+
+−s if s ≤ 0
+0 if s > 0
+and f1(s) =
+
+
+
+0 if s ≤ 1
+s − 1 if s > 1
+.
+It’s easy to check that:
+1
+2
+d
+dt||f0(z(t))||2
+L2(U) =
+�
+U f0(z(t))f ′
+0(z(t))∂tz(t)dx =
+�
+U χ{z(t)<0} z(t) ∂tz(t)dx,
+and noting that χ{z(t)<0}z(t) = −f0(z(t)) is a Sobolev function (see Lemma 7.6
+in [12]) we are allowed to use it as a test function in (2.2), getting:
+1
+2
+d
+dt||f0(z(t))||2
+L2(U) = ⟨∂tz(t), −f0(z(t))⟩L2(U) =
+= χ{z(t)<0}
+�
+−2ǫ||∇z(t)||2
+L2(U) −
+�
+U φ′(z(t))φ(z(t))z(t)|∇u(t)|2 +
+�
+U
+1 − z(t)
+2ǫ
+z(t)
+�
+≤ 0.
+Since f0(z(0)) = 0 because z0 ≥ 0, we have f0(z(t)) ≡ 0 for every t. In the
+same way we can prove z ≤ 1 by considering f1(z(t)).
+The strategy will be to make use of Galerkin approximates. Consider an or-
+thogonal basis of H1(U) composed of eigenfunctions of −∆ on U with homoge-
+neous Neumann boundary conditions normalized with respect to the L2(U) norm,
+and denote it as {ei}i∈N. So
+
+
+
+−∆ei = λiei in U
+∂nei = 0 in ∂U
+and ||ei||L2(U) = 1 for every i.
+We want to find Galerkin approximates (uN, zN) such that they solve (in dis-
+tributional sense) in
+VN = Span({e1, . . . , eN})
+(2.5)
+the system
+4
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+∂tuN = πN[div((ηǫ + φ(zN)2)∇uN)] − (uN − gN) in (0, T) × U
+∂tzN = 2ǫ∆zN − πN[φ′(zN)φ(zN)|∇uN|2] + 1−zN
+2ǫ
+in (0, T) × U
+∂
+∂nuN =
+∂
+∂nzN = 0 in (0, T) × ∂U
+uN(0, ·) = πN[u0] and zN(0, ·) = πN[z0] in {0} × U
+(2.6)
+with πN the orthogonal projection on VN. Notice that by the orthogonality
+properties of {ei}N
+i=1 this is actually a 2N system in u(1)(t), . . . , u(N)(t), z(1)(t), . . . , z(N)(t),
+with
+uN(t, x) =
+N
+�
+i=1
+u(i)(t)ei(x) and zN(t, x) =
+N
+�
+i=1
+z(i)(t)ei(x)
+which by Cauchy-Lipshitz admits a unique local solution. Indeed if we test
+with ei it holds that:
+u(i)(t) = (u0)(i) −
+� t
+0
+��
+U(ηǫ + φ(zN(s))2)⟨∇uN(s), ∇ei⟩ + (uN(s) − gN)eidx
+�
+ds
+z(i)(t) = (z0)(i) −
+� t
+0
+��
+U 2ǫ⟨∇zN(s), ∇ei⟩ − φ′(zN(s))φ(zN(s))|∇uN(s)|2ei + 1 − zN(s)
+2ǫ
+eidx
+�
+ds
+for every 1 ≤ i ≤ N. However there are strong non linearities at play, so a
+priori for every N we only have a local solution in [0, tN) without being able to
+extend it immediately to [0, T]. In order to gain existence in [0, T] of Galerkin
+approximates we’ll use the following a priori estimates, which hold in a slightly
+more general situation with less regular initial data:
+Proposition 2.2. Existence of weak approximate solutions in [0,T]
+Given (u0, z0) ∈ [L2(U)]2 and a solution (uN, zN) of (2.6) in VN, it holds
+sup
+0≤t≤T[||uN(t)||2
+L2(U)] +
+� T
+0 ||∇uN(s)||2
+L2(U)ds ≤ C
+and
+sup
+0≤t≤T[||zN(t)||2
+L2(U)] +
+� T
+0 ||∇zN(s)||2
+L2(U)ds ≤ C
+with C a positive constant independent of N.
+Proof. Test the equation for uN in (2.6) with uN itself, to get
+d
+dt||uN||2
+L2(U) = −
+�
+U(ηǫ + φ(zN)2)|∇uN|2dx −
+�
+U uN(uN − gN)dx.
+(2.7)
+So
+d
+dt||uN||2
+L2(U) ≤ ||gN||L2(U)||uN||L2(U) ≤ ||g||L2(U)(1 + ||uN||2
+L2(U))
+5
+
+and you get uniform boundedness of ||uN||L2(U) by Gronwall’s lemma. Going
+back to (2.7) you easily conclude by integrating in time. The same holds if we test
+the equation in zN with zN itself, obtaining
+sup
+0≤t≤T[||zN(t)||2
+L2(U)] + 2ǫ
+� T
+0
+≤ T
+8ǫ.
+By orthogonality of the ei s this can be rewritten as
+||uN(t)||2
+L2(U) =
+N
+�
+i=1
+[u(i)(t)]2 and ||zN(t)||2
+L2(U) =
+N
+�
+i=1
+[z(i)(t)]2
+so we cannot have a blow-up in finite time and we have thus proved existence
+up to time T for every T > 0.
+Now that we have existence of solutions of (2.6) in [0, T] for every N, let’s prove
+stronger inequalities exploiting the variational characterization of the problem.
+Proposition 2.3. A priori energy estimates
+Assume (u0, z0) ∈ [H1(U)]2, then
+sup
+t∈[0,T]
+[ATǫ(uN(t), zN(t))] +
+� T
+0 ||∂tuN(s)||2
+L2(U) + ||∂tzN(s)||2
+L2(U)ds =
+= sup
+t∈[0,T]
+[ATǫ(uN(t), zN(t))] +
+� T
+0 ||πN[∇ATǫ(uN, zN)]||2
+L2(U)ds ≤
+≲ ATǫ(u0, z0) + ||g||2
+L2(U).
+In particular we have (∂tuN, ∂tzN) ∈ [L2(0, T; L2(U))]2 and (uN, zN) ∈ [L∞(0, T; H1(U))]2,
+both uniformly bounded independently from N.
+Proof. Derive ATǫ(uN, zN) in time and get:
+d
+dtATǫ(uN, zN) = −||πN∇ATǫ||2
+L2(U) = −||∂tuN||2
+L2(U) − ||∂tzN||2
+L2(U).
+Integrating this equality:
+ATǫ(uN, zN) +
+� T
+0 ||∂tuN(s)||2
+L2(U) + ||∂tzN(s)||2
+L2(U)ds = ATǫ(πNu0, πNz0) ≤ C.
+(2.8)
+Notice that a priori we have no control in N for the quantity
+�
+U(ηǫ + πN[z0]2)|∇πN[u0]|2dx.
+But since we truncated with |φ| ≤ 2 we have
+�
+U(ηǫ + 4)|∇πN[u0]|2dx ≤
+�
+U(ηǫ + 4)|∇u0|2dx.
+6
+
+At this point we still can’t prove the weak convergence of the non linear parts of
+the equation, in particular div((ηǫ+φ(zN)2)∇uN) and φ′(zN)φ(zN)|∇u|2. Stronger
+estimates are needed. In the next Proposition we’ll prove uniform L2(0, T; H2(U))
+boundedness of (uN, zN).
+Proposition 2.4. Uniform L2(0, T; H2(U)) estimates
+Let U ⊂ R2 be a bounded Lipshitz domain and let (u0, z0) ∈ [H1(U)]2. Consider
+solutions (uN, zN) of (2.6). It holds that
+sup
+t∈[0,T]
+[||uN(t)||2
+H1(U) + ||zN(t)||2
+H1(U)] +
+� T
+0 ||∆uN||2
+L2(U) + ||∆zN||2
+L2(U) ≤ C
+for some C > 0 independent of N.
+Proof. First of all, we want to prove an estimate like
+sup
+0≤t≤T[ATǫ(uN(t), zN(t))] +
+� T
+0 ||∆uN(s)||2
+L2(U) + ||∆zN(s)||2
+L2(U)ds ≲
+≲ C +
+� T
+0 ||∇uN(s)||4
+L4(U) + ||∇zN(s)||4
+L4(U)ds.
+(2.9)
+The idea is to expand the energy equality (2.8) obtained in Proposition 2.3
+with ||πN∇ATǫ(uN, zN)||2
+L2(U). For the sake of readability we omit writing time
+dependence, abbreviate φ(z) as φ and omit the subscripts too:
+|∇ATǫ(u, z)|2 = ∂uATǫ(u, z)2 + ∂zATǫ(u, z)2 =
+= [(η + φ2)∆u + 2φφ′⟨∇z, ∇u⟩ − (u − g)]2 +
+�
+2ǫ∆z − φφ′|∇u|2 + 1 − z
+2ǫ
+�2
+=
+= (η + φ2)2(∆u)2
+�
+��
+�
+1
++ 4φ2(φ′)2⟨∇z, ∇u⟩2
+�
+��
+�
+2
++(u − g)2 + 4φφ′(η + φ2)⟨∇z, ∇u⟩∆u
+�
+��
+�
+3
+−
+−4φφ′(u − g)⟨∇z, ∇u⟩
+�
+��
+�
+4
+−2(u − g)(η + φ2)∆u
+�
+��
+�
+5
++ 4ǫ2(∆z)2
+�
+��
+�
+6
++(φ′)2φ2|∇u|4 + (1 − z)2
+4ǫ2
+−
+−4ǫφφ′|∇u|2∆z
+�
+��
+�
+7
+−φφ′(1 − z)|∇u|2
+ǫ
+�
+��
+�
+8
++ 2(1 − z)∆z
+�
+��
+�
+9
+,
+where we highlighted all the terms we will manipulate. The strategy is very
+simple: use estimates like
+ab ≥ − 1
+2δa2 − δ
+2b2
+(2.10)
+to get
+7
+
+C ≥ sup
+0≤t≤T[ATǫ(u(t), z(t))] +
+� T
+0 ||πN∇ATǫ(uN(s), zN(s))||2
+L2(U)ds ≳
+sup
+0≤t≤T[ATǫ(u(t), z(t))] +
+� T
+0 ||∆uN(s)||2
+L2(U) + ||∆zN(s)||2
+L2(U)ds−
+−
+� T
+0 ||∇uN(s)||4
+L4(U) + ||∇zN(s)||4
+L4(U)ds − 1,
+and from this recover the desired inequality (2.9). The idea is to use (2.10)
+with different suitable δ on all highlighted terms except
+1
+and
+6
+which will
+absorb squared laplacians. You can easily see that each term can be estimated
+with a sum like in (2.10) of two of the following quantities:
+• A term −(∆u)2 and/or −(∆z)2;
+• A term −|∇u|4 and/or −|∇z|4;
+• A term which by Proposition 2.3 we know to be uniformly bounded.
+The only tedious part (which we skip) is to choose δ wisely each time so that
+in the end you remain with
+c1(∆u)2 + c2(∆z)2 − C1|∇u|4 − C2|∇z|4 + h
+with c1, c2 > 0 and h a sum of functions in L∞(0, T; L2(U)). In fact we can
+reduce to estimating the L2(0, T; H2(U)) norm of uN. Testing the equation in zN
+with −∆zN in (2.6) we get
+1
+2
+d
+dt||∇zN||2
+L2(U) = −2ǫ||∆zN||2
+L2(U) +
+�
+U φ′(zN)φ(zN)|∇uN|2∆zNdx −
+�
+U
+1 − zN
+2ǫ
+∆zNdx
+and from this, using again ab ≤ (δ/2)a2 + (1/2δ)b2, it’s clear we can estimate
+the L2(0, T; H2(U)) norm of zN with the L4(0, T; L4(U)) norm of ∇uN. Moreover
+by Gagliardo-Niremberg inequality:
+||∇zN||4
+L4(U) ≤ C(1 + ||∇zN||2
+L2(U)||∇2zN||2
+L2(U)) ≤ C(1 + ||∇2zN||2
+L2(U))
+thanks to the L∞(0, T; H1(U)) estimates on zN. Then:
+� T
+0 ||uN(t)||2
+H2(U)dt ≲ 1 + sup
+t∈[0,T]
+[ATǫ(uN(t), zN(t))]+
++
+� T
+0 ||∆uN(t)||2
+L2(U) + ||∆zN(t)||2
+L2(U)dt ≲ 1 +
+� T
+0 ||∇uN(t)||4
+L4(U) + ||∇zN(t)||4
+L4(U)dt ≲
+≲ 1 +
+� T
+0 ||∇uN(t)||4
+L4(U) + ||∇2zN(t)||2
+L2(U)dt ≲ 1 +
+� T
+0 ||∇uN(t)||4
+L4(U)dt.
+(2.11)
+8
+
+The goal will be to obtain an estimate like
+� T
+0 ||∇uN(t)||4
+L4(U) ≲
+�� T
+0 ||uN(t)||2
+H2(U)
+�q/2
+(2.12)
+for some q < 2 so that we can get uniform bounds in (2.11) and conclude.
+Notice that in (2.12) the estimate is non homogeneous, i.e. we are estimating
+a fourth power with something of homogeneity q < 2. The reason why this is
+possible is that the constants we are omitting in (2.12) actually depend on uN in
+a way such that the homogeneity is preserved, as we will see later.
+Considering the time fixed (we will thus omit writing the dependence on t
+for the moment) we focus on the first equation of (2.6) and we consider uN the
+solution of:
+
+
+
+−div((ηǫ + φ(zN)2)∇uN) = f
+∂nuN = 0
+where f = −∂tuN − (uN − gN).
+Now we use a procedure used in [4] and suggested as a possible alternative
+proof in a footnote in [9], that is using Meyers theorem (see [15]) to get H2 esti-
+mates. Meyers theorem was originally proved for homogeneous Dirichlet boundary
+conditions on ∂U, but in [11] it has been generalised (among others) to the case
+of homogeneous Neumann boundary conditions. The proof in [11] for the Neu-
+mann case consists in a series of strategies (i.e. partition of unity, extension of the
+functions, etc.) in order to go back to the case of the original Meyers theorem. In
+particular we want to use Theorem 2 in [11], so we consider
+G : L2
+m(U) �→ H1
+m(U)
+such that G(f) = ϕ with
+
+
+
+−∆ϕ = f in U
+∂nϕ = 0 in ∂U
+(2.13)
+and
+f ∈ L2
+m(U) =
+�
+g ∈ L2(U)
+����
+�
+U g = 0
+�
+;
+ϕ ∈ H1
+m(U) = H1(U) ∩ L2
+m(U).
+Notice that problem (2.13) admits a unique solution in H1
+m(U) if and only if
+�
+U f = 0 (see [8]), which is our case. In particular it holds
+⟨∇G(f), ∇φ⟩L2 = ⟨f, φ⟩L2 for all φ ∈ H1(U).
+(2.14)
+By Theorem 2 in [11] (up to multiplicative constants we are neglecting):
+||∇uN||Lp(U) ≤ ||∇G(f)||Lp(U) for some p ∈ (2, +∞).
+(2.15)
+9
+
+Remark 1. Actually, the precise statement of Theorem 2 in [11] would give the
+estimate:
+||uN||W 1,p(U) ≲ ||f||W 1,q(U)′,
+with 1
+p + 1
+q = 1, 2 < p < +∞ and W 1,q(U)′ denoting the dual. But then we
+readily have:
+||f||W 1,q(U)′ =
+sup
+||φ||W 1,q(U)=1
+⟨f, φ⟩L2 =
+sup
+||φ||W 1,q(U)=1
+⟨∇G(f), ∇φ⟩L2 ≤
+≤
+sup
+||φ||W 1,q(U)=1
+||∇G(f)||Lp(U)||∇φ||Lq(U) ≤ ||∇G(f)||Lp(U)
+and so we have the estimate (2.15). The reason why we consider ∇G(f) instead
+of dealing with f is because we’ll make use of Gagliardo-Niremberg inequality and
+estimates from elliptic regularity theory on ∇G(f).
+Using Gagliardo-Niremberg inequality we get
+||∇uN||Lp(U) ≤ C(1 + ||∇G(f)||2/p
+L2(U)||∇2G(f)||
+p−2
+p
+L2(U)).
+(2.16)
+Now notice that up to modification by an additive constant we can consider
+without loss of generality −∇G(f) = (ηǫ + φ(zN)2)∇uN. Indeed:
+−∆G(f) = div(−∇G(f)) = f = div((ηǫ + φ(zN)2)∇uN),
+so −∇G(f) = (ηǫ + φ(zN)2)∇uN ∈ L∞(0, T; L2(U)) thanks to Proposition 2.3.
+Integrate in time the inequality (2.16) raised to the power 2p/(p − 2) to get:
+� T
+0 ||∇uN(t)||
+2p
+p−2
+Lp(U)dt ≲ 1 +
+� T
+0 ||∇2G(f)(t)||2
+L2(U)dt ≲ 1 +
+� T
+0 ||f(t)||2
+L2(U)dt ≤ C,
+(2.17)
+where we used standard elliptic regularity theory to pass from the L2 norm of
+∇2G(f) to the L2 norm of f. We can assume without loss of generality that 2 < p <
+4, otherwise if we had p > 4 we could conclude directly by the above estimates,
+indeed 2p/(p − 2) < 4 and by Hölder, (2.17) and the uniform L∞(0, T; L2(U))
+bounds on ∇uN:
+� T
+0 ||∇uN(t)||4
+L4(U)dt =
+� T
+0
+��
+U |∇uN(t)|
+2p
+p−2|∇uN(t)|
+2p−8
+p−2 dx
+�
+dt ≤
+≤
+� T
+0 ||∇uN(t)||2p/(p−2)
+Lp(U)
+��
+U |∇uN(t)|2dx
+� p−4
+p−2 dt ≤
+≤ C
+� T
+0 ||∇uN(t)||2p/(p−2)
+Lp(U)
+dt ≤ C.
+(2.18)
+Of course we also assume p ̸= 4, or the thesis would follow trivially. So assume
+2 < p < 4. Applying again Gagliardo-Nirenberg, Hölder and (2.17):
+10
+
+� T
+0 ||∇uN(t)||4
+L4(U)dt ≤
+� T
+0 ||∇uN(t)||p
+Lp(U)||uN(t)||4−p
+H2(U)dt ≤
+≤
+�� T
+0 ||∇uN||
+2p
+p−2
+Lp(U)
+� p−2
+p
+�� T
+0 ||uN||2
+H2(U)
+� 4−p
+2
+≲
+�� T
+0 ||uN||2
+H2(U)
+� 4−p
+2
+,
+(2.19)
+and we can conclude since (4 − p)/2 < 2.
+Remark 2. Notice the assumption n = 2 is needed in order to have the necessary
+Gagliardo-Niremberg estimates in the previous Proposition. Also, notice how the
+homogeneity of degree 4 is preserved both in (2.18) and in (2.19), where to conclude
+we uniformly bound some quantities depending on uN, namely (
+�
+U |∇uN(t)|2dx)
+p−4
+p−2
+and
+�� T
+0 ||∇uN||
+2p
+p−2
+Lp(U)
+� p−2
+p
+.
+The estimates obtained in the previous Proposition actually yield uniform esti-
+mates of uN and zN in L2(0, T; H2(U)) thanks to the classical fact that ||u||L2(U) +
+||∆u||L2(U) is an equivalent norm for H2(U). To recapitulate, we have (up to a
+subsequence we will not rename):
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+(uN, zN) weakly- ∗ converging in L∞(0, T; H1(U))
+(uN, zN) weakly converging in L2(0, T; H2(U))
+(∂tuN, ∂tzN) weakly converging in L2(0, T; L2(U))
+(uN, zN) converging in C(0, T; L2(U))
+(uN, zN) converging in the strong topology in L2(0, T; H1(U))
+(2.20)
+where the compact embeddings in C(0, T; L2(U)) and L2(0, T; H1(U)) are ob-
+tained by applying the Aubin-Lions lemma (see [3], [13], [18]). We are now ready
+to prove the main result.
+Theorem 2.1. Existence and uniqueness of strong solutions
+Let U ⊂ R2 be a bounded Lipshitz domain and let (u0, z0) ∈ [H1(U)]2 with
+0 ≤ z0 ≤ 1. Then there exists a unique strong solution (u, z) of (2.1).
+Proof. Let (u, z) be the weak limit of (uN, zN) in L2(0, T; H2(U)), let’s see how
+the pair is a solution of (2.2). This will be sufficient to prove the thesis thanks to
+Proposition 2.1. Let ψ ∈ VM = Span{e1, . . . , eM} be a test function for (2.6) with
+N > M, so it holds:
+�
+U uN(t)ψ
+�
+��
+�
+1
+=
+�
+U πN[u0]ψ −
+� t
+0
+�
+U(ηǫ + φ(zN)2)∇uN∇ψ
+�
+��
+�
+2
+−
+� t
+0
+�
+U(uN − gN)ψ
+�
+U zN(t)ψ
+�
+��
+�
+3
+=
+�
+U πN[z0]ψ − 2ǫ
+� t
+0
+�
+U ∇zN∇ψ −
+� t
+0
+�
+U φ′(zN)φ(zN)|∇uN|2ψ
+�
+��
+�
+4
++
+� t
+0
+�
+U
+1 − zN
+2ǫ
+ψ,
+(2.21)
+11
+
+and we want to show we can pass to the limit in every highlighted term, since
+for the others it’s trivial by weak convergence.
+• As for 1 and 3 , we can pass to the limit thanks to the compactness in
+C(0, T; L2(U)).
+• For 2 , we have (by dominated convergence) strong convergence in L2(0, T; L2(U))
+of (ηǫ + φ(zN)2), and weak convergence of ∇uN. So their product weakly
+converges and we can pass to the limit.
+• We already saw in Proposition 2.4 how ∇uN is uniformly bounded in L4(0, T; L4(U)),
+which is the same as saying |∇uN|2 is uniformly bounded in L2(0, T; L2(U)).
+Then, up to taking another subsequence, |∇uN|2 ⇀ |∇u|2 in L2(0, T; L2(U)).
+Since φ′(zN)φ(zN) → φ′(z)φ(z) in the strong L2(0, T; L2(U)) topology by
+dominated convergence, their product weakly converges and we can pass to
+the limit in 4 .
+It only remains to prove that (u, z) satisfy the homogeneous Neumann bound-
+ary conditions of (2.1).
+To do that we first have to make sense of ∂nu for any u ∈ H2(U). We define
+∂n : H2(U) → H1/2(∂U)
+as
+∂nu(ψ) =
+�
+U ∆uΨdx +
+�
+U ∇u∇Ψdx,
+where ψ ∈ H1/2(∂U) and Ψ ∈ H1(U) is an extension of ψ to the whole U. In
+particular Ψ will be chosen according to the trace extension operator, i.e. Ψ = Eψ,
+where E is defined as:
+Theorem 2.2. Trace extension operator, see [17]
+Given a bounded, Lipshitz domain Ω ⊂ Rn and 1 < p < +∞, there exists a
+linear and bounded trace extension operator
+E : W 1− 1
+p ,p(∂Ω) → W 1,p(Ω)
+such that Tr(Eu) = u for every u ∈ W 1− 1
+p(∂Ω).
+The operator ∂n just defined is continuous, indeed using ||Eψ||H1(U) ≤ C||ψ||H1/2(U):
+||∂nu||H1/2(∂U) =
+sup
+ψ∈H1/2(∂U)
+�
+∂nu(ψ)
+||ψ||H1/2(∂U)
+�
+≤
+≤ C
+sup
+ψ∈H1/2(∂U)
+�
+1
+||Eψ||H1(U)
+�
+U ∆uEψdx +
+�
+U ∇u∇Eψdx
+�
+≤
+≤ C(||∆u||L2(U) + ||∇u||L2(U)).
+Moreover it is known that W 1−1/p,p(∂U) compactly embeds into Lp(∂U) (see
+[7]), so
+12
+
+∂n : H2(U) �→ L2(∂U) is weak-strong continuous,
+meaning it sends weakly converging sequences in strong converging ones. We
+have to prove that ∂nu(t) = ∂nz(t) = 0 for almost every t. Since the argument is
+the same we’ll only show that the boundary conditions hold for u, moreover for
+simplicity we assume that u(t) ∈ H2(U) for every t. By weak-strong continuity of
+∂n we have that for every t ∈ [0, T], modulo a subsequence (which depends on t):
+uN(t)
+H2(U)
+−−−⇀ u(t) =⇒ ∂nuN(t)
+L2(∂U)
+−−−−→ ∂nu(t) as N → +∞,
+but since ∂nuN(t) ≡ 0 for every N we have the thesis.
+References
+[1] Luigi Ambrosio, Nicola Fusco, and Diego Pallara. Functions of bounded vari-
+ation and free discontinuity problems. Courier Corporation, 2000.
+[2] Luigi Ambrosio and Vincenzo Maria Tortorelli. Approximation of functional
+depending on jumps by elliptic functional via gamma-convergence. Commu-
+nications on Pure and Applied Mathematics, 43(8):999–1036, 1990.
+[3] Jean-Pierre Aubin.
+Analyse mathematique-un theoreme de compacite.
+Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences,
+256(24):5042, 1963.
+[4] John W Barrett, Xiaobing Feng, and Andreas Prohl. Convergence of a fully
+discrete finite element method for a degenerate parabolic system modelling
+nematic liquid crystals with variable degree of orientation. ESAIM: Mathe-
+matical Modelling and Numerical Analysis, 40(1):175–199, 2006.
+[5] Guy David. Singular sets of minimizers for the Mumford-Shah functional,
+volume 233. Springer Science & Business Media, 2006.
+[6] E De Giorgi, M Carriero, and A Leaci. Existence theorem for a minimum
+problem with free discontinuity set. Ennio De Giorgi, page 654, 1989.
+[7] Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci. Hitchhiker’s
+guide to the fractional sobolev spaces. Bulletin des sciences mathématiques,
+136(5):521–573, 2012.
+[8] Lawrence C Evans. Partial differential equations, volume 19. American Math-
+ematical Soc., 2010.
+[9] Xiaobing Feng and Andreas Prohl. Analysis of gradient flow of a regular-
+ized mumford-shah functional for image segmentation and image inpaint-
+ing. ESAIM: Mathematical Modelling and Numerical Analysis, 38(2):291–320,
+2004.
+13
+
+[10] Gilles A Francfort and J-J Marigo.
+Revisiting brittle fracture as an en-
+ergy minimization problem. Journal of the Mechanics and Physics of Solids,
+46(8):1319–1342, 1998.
+[11] Thierry Gallouet and Alexis Monier. On the regularity of solutions to elliptic
+equations. Rend. Mat. Appl.(7), 19(4):471–488, 1999.
+[12] David Gilbarg, Neil S Trudinger, David Gilbarg, and NS Trudinger. Elliptic
+partial differential equations of second order, volume 224. Springer, 1977.
+[13] Jacques-Louis Lions.
+Quelques méthodes de résolution de problemes aux
+limites non linéaires. 1969.
+[14] Jacques Louis Lions and Enrico Magenes. Non-homogeneous boundary value
+problems and applications: Vol. 1, volume 181. Springer Science & Business
+Media, 2012.
+[15] Norman G Meyers. An Lp-estimate for the gradient of solutions of second
+order elliptic divergence equations. Annali della Scuola Normale Superiore di
+Pisa-Classe di Scienze, 17(3):189–206, 1963.
+[16] David Bryant Mumford and Jayant Shah. Optimal approximations by piece-
+wise smooth functions and associated variational problems. Communications
+on pure and applied mathematics, 1989.
+[17] Jindrich Necas. Les méthodes directes en théorie des équations elliptiques.
+1967.
+[18] Jacques Simon. Compact sets in the space Lp(0, T; B). Annali di Matematica
+pura ed applicata, 146(1):65–96, 1986.
+[19] Michael Struwe. Geometric evolution problems. Nonlinear partial differential
+equations in differential geometry, 2:257–339, 1996.
+14
+

EdFRT4oBgHgl3EQfyTjG/content/tmp_files/load_file.txt

@@ -0,0 +1,352 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf,len=351
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='13645v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='AP]  31 Jan 2023  Existence, uniqueness and  L2  t(H2  x) ∩ L∞  t (H1  x) ∩ H1  t (L2  x) regularity of the  gradient flow of the Ambrosio-Tortorelli functional  Tommaso Cortopassi ∗  Abstract  We consider the gradient flow of the Ambrosio-Tortorelli functional at  fixed ǫ > 0, proving existence and uniqueness of a solution in dimension 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  The strategy of the proof essentially follows the one in the first part of [9],  but as suggested in a footnote by the authors of [9] it employs a different  and simpler technique, which is used for a different equation in [4] and in the  end it allows to prove better estimates than the ones obtained in the original  article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In particular we prove that if U ⊂ R2 is a bounded Lipshitz domain,  the initial data (u0, z0) ∈ [H1(U)]2 and 0 ≤ z0 ≤ 1, then for every T > 0  there exists a unique gradient flow (u(t), z(t)) of the Ambrosio-Tortorelli  functional such that  (u, z) ∈ [L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U)) ∩ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H1(U)) ∩ H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U))]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  The basic difference from [9], as already said, is a better regularity result  and a simpler proof: while in [9] they used a localization argument based  on an idea by Struwe (see [19]), here crucial estimates on the fourth powers  of the L4  t (L4  x) norms of the gradients will be obtained employing a suitable  version of the Meyers theorem due to Gallouet and Monier (see [15], [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  1  Introduction  The Mumford-Shah functional, introduced in [16], is defined as  E(u, Γ) = 1  2  �  U\\Γ |∇u|2 + (u − g)2dx + H1(Γ),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1)  where u ∈ H1(U \\ Γ), U ⊂ R2 open and bounded, Γ ⊂ U is closed and H1  is the one dimensional Hausdorff measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' This functional has been extensively  studied for its applications in image segmentation and fracture mechanics (a de-  tailed survey can be found in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The definition of the functional depends on the  model it is used for, in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1) we gave the original definition of [16] which is suited  ∗Scuola Normale Superiore, 56126 Pisa, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' E-mail: tommaso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='cortopassi@sns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='it  1  for image segmentation models, the interested reader can see the seminal paper  [10] about the fracture mechanic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The idea is rather simple: given a gray  image g (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' a scalar function), we want to find (˜u, ˜Γ) such that  (˜u, ˜Γ) = arg min  u∈H1(U\\Γ)  Γ⊂U closed  E(u, Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2)  In this case, the function u will approximate in a “smooth way” the image g  while Γ will be the contour set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' From a theoretical point of view, in [6] the authors  proved the existence of a solution for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2) by restricting to functions u ∈ SBV (U)  (see [1]) and Γ = Su, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' the set of discontinuity jump points of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  From a numerical point perspective, since the functional involves the measure  of the singular jump set of a (unknown) function u, its direct numerical implemen-  tation is often not possible or not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' A standard approach is to minimize a  more regular functional proposed by Ambrosio and Tortorelli in [2] defined as  1  2  �  U[(ηǫ + z2)|∇u|2 + (u − g)2] +  �  U  �(1 − z)2  4ǫ  + ǫ|∇z|2  �  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3)  which approximates the Mumford-Shah functional in Γ-convergence as ǫ → 0  in an even more general setting, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' not being restricted to the 2 dimensional case,  but considering U ⊂ Rn open and the (n − 1)-dimensional Hausdorff measure for  Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The Ambrosio-Tortorelli functional is the way with which one usually finds  (approximate) minima points for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1), in particular one can use a gradient flow  approach for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' As in [9], we’ll consider the gradient flow of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3), which is  given by  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∂tu = div((ηǫ + z2)∇u) − (u − g) in (0, T) × U  ∂tz = 2ǫ∆z − z|∇u|2 + 1−z  2ǫ in (0, T) × U  ∂  ∂nu =  ∂  ∂nz = 0 in (0, T) × ∂U  u(0, ·) = u0 and z(0, ·) = z0 in {0} × U  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='4)  with ηǫ, ǫ > 0 fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' As already mentioned, our goal is to study the existence,  uniqueness and regularity of solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='4), and the novelty lies in an approach  (already suggested in a footnote of [9] and used in [4] for a different equation)  which simplifies the proof of [9] while also gaining better regularity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In  particular in [9] they only manage to prove the L2  t(H2  x) regularity for a short time  T1, since the crucial estimate in their argument is a local energy estimate, inspired  by a technique due to Struwe [19], which only holds for sufficiently small times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  2  2  Main result  Let U ⊂ R2 be a Lipshitz bounded domain and let (u0, z0) ∈ [H1(U)]2 with  0 ≤ z0 ≤ 1, g ∈ L2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' As stated in the introduction, we want to prove existence,  uniqueness and regularity of solutions of the gradient flow of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3), given by  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∂tu = div((ηǫ + z2)∇u) − (u − g) in (0, T) × U  ∂tz = 2ǫ∆z − z|∇u|2 + 1−z  2ǫ ) in (0, T) × U  ∂  ∂nu =  ∂  ∂nz = 0 in (0, T) × ∂U  u(0, ·) = u0 and z(0, ·) = z0 in {0} × U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1)  However we will mostly work with the slightly modified system  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∂tu = div((ηǫ + φ(z)2)∇u) − (u − g) in (0, T) × U  ∂tz = 2ǫ∆z − φ′(z)φ(z)|∇u|2 + 1−z  2ǫ in (0, T) × U  ∂  ∂nu =  ∂  ∂nz = 0 in (0, T) × ∂U  u(0, ·) = u0 and z(0, ·) = z0 in {0} × U  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2)  where φ is a cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The reason to introduce such a cutoff, as we will show later,  is to bound the L∞ norm of ηǫ+φ(zN)2 when considering Galerkin approximations  zN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' To be precise, a good working definition of φ might be  φ(s) =  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  −1 if s ≤ −1  s if − 1 < s < 2  2 if s ≥ 2  but you could actually do the cuts at levels −δ1 and 1 + δ2 for every δ1, δ2 > 0  and nothing would change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' A modified Ambrosio-Tortorelli functional (of which  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2) is the gradient flow of) will be denoted as ATǫ and defined as  ATǫ(u(t), z(t)) = 1  2  �  U[(ηǫ+φ(z(t))2)|∇u(t)|2+(u(t)−g)2]+  �  U  �(1 − z(t))2  4ǫ  + ǫ|∇z(t)|2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3)  First of all, let’s define the notion of strong solution, following [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Strong solution  A couple (u, z) is a strong solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1) if it satisfies the system in a dis-  tributional sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' for every ψ ∈ H1(U) it holds  �  U  u(t)ψdx =  �  U  u0ψdx −  � t  0  �  U  (ηǫ + z(s)2)⟨∇u(s), ∇ψ⟩dxds −  � t  0  �  U  (u(s) − g)ψdxds  �  U  z(t)ψdx =  �  U  z0ψdx − 2ǫ  � t  0  �  U  ⟨∇z(s), ∇ψ⟩dxds −  � t  0  �  U  z(s)ψ|∇u(s)|2 + 1 − z(s)  2ǫ  ψdxds  for almost every t ∈ (0, T), and moreover we require  (u, z) ∈ [L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U)) ∩ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H1(U)) ∩ H1(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U))]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  We start with a proposition which shows how a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2) is also solution  of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1) if 0 ≤ z0 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  3  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Let 0 ≤ z0 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' If a strong solution (u, z) for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2) exists, it  must be 0 ≤ z(t, x) ≤ 1 for every t ∈ (0, T) and for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' x ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' So (u, z) will also  be a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' If we test the equation for z in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2) with z itself it gives  1  2  d  dt||z(t)||2  L2(U) = −2ǫ||∇z(t)||2  L2(U)−  �  U φ′(z(t))φ(z(t))z(t)|∇u(t)|2ds+  �  U  1 − z(t)  2ǫ  z(t)ds  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='4)  where the existence of the time derivative is ensured by Lions-Magenes lemma  (see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Consider f0(z) = max{−z, 0} and f1(z) = max{z − 1, 0}, so f0 and f1  are defined as  f0(s) =  \uf8f1  \uf8f2  \uf8f3  −s if s ≤ 0  0 if s > 0  and f1(s) =  \uf8f1  \uf8f2  \uf8f3  0 if s ≤ 1  s − 1 if s > 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  It’s easy to check that:  1  2  d  dt||f0(z(t))||2  L2(U) =  �  U f0(z(t))f ′  0(z(t))∂tz(t)dx =  �  U χ{z(t)<0} z(t) ∂tz(t)dx,  and noting that χ{z(t)<0}z(t) = −f0(z(t)) is a Sobolev function (see Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6  in [12]) we are allowed to use it as a test function in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2), getting:  1  2  d  dt||f0(z(t))||2  L2(U) = ⟨∂tz(t), −f0(z(t))⟩L2(U) =  = χ{z(t)<0}  �  −2ǫ||∇z(t)||2  L2(U) −  �  U φ′(z(t))φ(z(t))z(t)|∇u(t)|2 +  �  U  1 − z(t)  2ǫ  z(t)  �  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Since f0(z(0)) = 0 because z0 ≥ 0, we have f0(z(t)) ≡ 0 for every t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In the  same way we can prove z ≤ 1 by considering f1(z(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  The strategy will be to make use of Galerkin approximates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Consider an or-  thogonal basis of H1(U) composed of eigenfunctions of −∆ on U with homoge-  neous Neumann boundary conditions normalized with respect to the L2(U) norm,  and denote it as {ei}i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' So  \uf8f1  \uf8f2  \uf8f3  −∆ei = λiei in U  ∂nei = 0 in ∂U  and ||ei||L2(U) = 1 for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  We want to find Galerkin approximates (uN, zN) such that they solve (in dis-  tributional sense) in  VN = Span({e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' , eN})  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='5)  the system  4  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∂tuN = πN[div((ηǫ + φ(zN)2)∇uN)] − (uN − gN) in (0, T) × U  ∂tzN = 2ǫ∆zN − πN[φ′(zN)φ(zN)|∇uN|2] + 1−zN  2ǫ  in (0, T) × U  ∂  ∂nuN =  ∂  ∂nzN = 0 in (0, T) × ∂U  uN(0, ·) = πN[u0] and zN(0, ·) = πN[z0] in {0} × U  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6)  with πN the orthogonal projection on VN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Notice that by the orthogonality  properties of {ei}N  i=1 this is actually a 2N system in u(1)(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' , u(N)(t), z(1)(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' , z(N)(t),  with  uN(t, x) =  N  �  i=1  u(i)(t)ei(x) and zN(t, x) =  N  �  i=1  z(i)(t)ei(x)  which by Cauchy-Lipshitz admits a unique local solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Indeed if we test  with ei it holds that:  u(i)(t) = (u0)(i) −  � t  0  ��  U(ηǫ + φ(zN(s))2)⟨∇uN(s), ∇ei⟩ + (uN(s) − gN)eidx  �  ds  z(i)(t) = (z0)(i) −  � t  0  ��  U 2ǫ⟨∇zN(s), ∇ei⟩ − φ′(zN(s))φ(zN(s))|∇uN(s)|2ei + 1 − zN(s)  2ǫ  eidx  �  ds  for every 1 ≤ i ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' However there are strong non linearities at play, so a  priori for every N we only have a local solution in [0, tN) without being able to  extend it immediately to [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In order to gain existence in [0, T] of Galerkin  approximates we’ll use the following a priori estimates, which hold in a slightly  more general situation with less regular initial data:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Existence of weak approximate solutions in [0,T]  Given (u0, z0) ∈ [L2(U)]2 and a solution (uN, zN) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6) in VN, it holds  sup  0≤t≤T[||uN(t)||2  L2(U)] +  � T  0 ||∇uN(s)||2  L2(U)ds ≤ C  and  sup  0≤t≤T[||zN(t)||2  L2(U)] +  � T  0 ||∇zN(s)||2  L2(U)ds ≤ C  with C a positive constant independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Test the equation for uN in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6) with uN itself, to get  d  dt||uN||2  L2(U) = −  �  U(ηǫ + φ(zN)2)|∇uN|2dx −  �  U uN(uN − gN)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='7)  So  d  dt||uN||2  L2(U) ≤ ||gN||L2(U)||uN||L2(U) ≤ ||g||L2(U)(1 + ||uN||2  L2(U))  5  and you get uniform boundedness of ||uN||L2(U) by Gronwall’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Going  back to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='7) you easily conclude by integrating in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The same holds if we test  the equation in zN with zN itself, obtaining  sup  0≤t≤T[||zN(t)||2  L2(U)] + 2ǫ  � T  0  ≤ T  8ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  By orthogonality of the ei s this can be rewritten as  ||uN(t)||2  L2(U) =  N  �  i=1  [u(i)(t)]2 and ||zN(t)||2  L2(U) =  N  �  i=1  [z(i)(t)]2  so we cannot have a blow-up in finite time and we have thus proved existence  up to time T for every T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Now that we have existence of solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6) in [0, T] for every N, let’s prove  stronger inequalities exploiting the variational characterization of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' A priori energy estimates  Assume (u0, z0) ∈ [H1(U)]2, then  sup  t∈[0,T]  [ATǫ(uN(t), zN(t))] +  � T  0 ||∂tuN(s)||2  L2(U) + ||∂tzN(s)||2  L2(U)ds =  = sup  t∈[0,T]  [ATǫ(uN(t), zN(t))] +  � T  0 ||πN[∇ATǫ(uN, zN)]||2  L2(U)ds ≤  ≲ ATǫ(u0, z0) + ||g||2  L2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  In particular we have (∂tuN, ∂tzN) ∈ [L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U))]2 and (uN, zN) ∈ [L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H1(U))]2,  both uniformly bounded independently from N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Derive ATǫ(uN, zN) in time and get:  d  dtATǫ(uN, zN) = −||πN∇ATǫ||2  L2(U) = −||∂tuN||2  L2(U) − ||∂tzN||2  L2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Integrating this equality:  ATǫ(uN, zN) +  � T  0 ||∂tuN(s)||2  L2(U) + ||∂tzN(s)||2  L2(U)ds = ATǫ(πNu0, πNz0) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='8)  Notice that a priori we have no control in N for the quantity  �  U(ηǫ + πN[z0]2)|∇πN[u0]|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  But since we truncated with |φ| ≤ 2 we have  �  U(ηǫ + 4)|∇πN[u0]|2dx ≤  �  U(ηǫ + 4)|∇u0|2dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  6  At this point we still can’t prove the weak convergence of the non linear parts of  the equation, in particular div((ηǫ+φ(zN)2)∇uN) and φ′(zN)φ(zN)|∇u|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Stronger  estimates are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In the next Proposition we’ll prove uniform L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U))  boundedness of (uN, zN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Uniform L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U)) estimates  Let U ⊂ R2 be a bounded Lipshitz domain and let (u0, z0) ∈ [H1(U)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Consider  solutions (uN, zN) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' It holds that  sup  t∈[0,T]  [||uN(t)||2  H1(U) + ||zN(t)||2  H1(U)] +  � T  0 ||∆uN||2  L2(U) + ||∆zN||2  L2(U) ≤ C  for some C > 0 independent of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' First of all, we want to prove an estimate like  sup  0≤t≤T[ATǫ(uN(t), zN(t))] +  � T  0 ||∆uN(s)||2  L2(U) + ||∆zN(s)||2  L2(U)ds ≲  ≲ C +  � T  0 ||∇uN(s)||4  L4(U) + ||∇zN(s)||4  L4(U)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='9)  The idea is to expand the energy equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='8) obtained in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3  with ||πN∇ATǫ(uN, zN)||2  L2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' For the sake of readability we omit writing time  dependence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' abbreviate φ(z) as φ and omit the subscripts too:  |∇ATǫ(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' z)|2 = ∂uATǫ(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' z)2 + ∂zATǫ(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' z)2 =  = [(η + φ2)∆u + 2φφ′⟨∇z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' ∇u⟩ − (u − g)]2 +  �  2ǫ∆z − φφ′|∇u|2 + 1 − z  2ǫ  �2  =  = (η + φ2)2(∆u)2  �  ��  �  1  + 4φ2(φ′)2⟨∇z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' ∇u⟩2  �  ��  �  2  +(u − g)2 + 4φφ′(η + φ2)⟨∇z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' ∇u⟩∆u  �  ��  �  3  −  −4φφ′(u − g)⟨∇z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' ∇u⟩  �  ��  �  4  −2(u − g)(η + φ2)∆u  �  ��  �  5  + 4ǫ2(∆z)2  �  ��  �  6  +(φ′)2φ2|∇u|4 + (1 − z)2  4ǫ2  −  −4ǫφφ′|∇u|2∆z  �  ��  �  7  −φφ′(1 − z)|∇u|2  ǫ  �  ��  �  8  + 2(1 − z)∆z  �  ��  �  9  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  where we highlighted all the terms we will manipulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The strategy is very  simple: use estimates like  ab ≥ − 1  2δa2 − δ  2b2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='10)  to get  7  C ≥ sup  0≤t≤T[ATǫ(u(t), z(t))] +  � T  0 ||πN∇ATǫ(uN(s), zN(s))||2  L2(U)ds ≳  sup  0≤t≤T[ATǫ(u(t), z(t))] +  � T  0 ||∆uN(s)||2  L2(U) + ||∆zN(s)||2  L2(U)ds−  −  � T  0 ||∇uN(s)||4  L4(U) + ||∇zN(s)||4  L4(U)ds − 1,  and from this recover the desired inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The idea is to use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='10)  with different suitable δ on all highlighted terms except  1  and  6  which will  absorb squared laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' You can easily see that each term can be estimated  with a sum like in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='10) of two of the following quantities:  A term −(∆u)2 and/or −(∆z)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  A term −|∇u|4 and/or −|∇z|4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  A term which by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3 we know to be uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  The only tedious part (which we skip) is to choose δ wisely each time so that  in the end you remain with  c1(∆u)2 + c2(∆z)2 − C1|∇u|4 − C2|∇z|4 + h  with c1, c2 > 0 and h a sum of functions in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In fact we can  reduce to estimating the L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U)) norm of uN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Testing the equation in zN  with −∆zN in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6) we get  1  2  d  dt||∇zN||2  L2(U) = −2ǫ||∆zN||2  L2(U) +  �  U φ′(zN)φ(zN)|∇uN|2∆zNdx −  �  U  1 − zN  2ǫ  ∆zNdx  and from this, using again ab ≤ (δ/2)a2 + (1/2δ)b2, it’s clear we can estimate  the L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U)) norm of zN with the L4(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L4(U)) norm of ∇uN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Moreover  by Gagliardo-Niremberg inequality:  ||∇zN||4  L4(U) ≤ C(1 + ||∇zN||2  L2(U)||∇2zN||2  L2(U)) ≤ C(1 + ||∇2zN||2  L2(U))  thanks to the L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H1(U)) estimates on zN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Then:  � T  0 ||uN(t)||2  H2(U)dt ≲ 1 + sup  t∈[0,T]  [ATǫ(uN(t), zN(t))]+  +  � T  0 ||∆uN(t)||2  L2(U) + ||∆zN(t)||2  L2(U)dt ≲ 1 +  � T  0 ||∇uN(t)||4  L4(U) + ||∇zN(t)||4  L4(U)dt ≲  ≲ 1 +  � T  0 ||∇uN(t)||4  L4(U) + ||∇2zN(t)||2  L2(U)dt ≲ 1 +  � T  0 ||∇uN(t)||4  L4(U)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='11)  8  The goal will be to obtain an estimate like  � T  0 ||∇uN(t)||4  L4(U) ≲  �� T  0 ||uN(t)||2  H2(U)  �q/2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='12)  for some q < 2 so that we can get uniform bounds in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='11) and conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Notice that in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='12) the estimate is non homogeneous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' we are estimating  a fourth power with something of homogeneity q < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The reason why this is  possible is that the constants we are omitting in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='12) actually depend on uN in  a way such that the homogeneity is preserved, as we will see later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Considering the time fixed (we will thus omit writing the dependence on t  for the moment) we focus on the first equation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6) and we consider uN the  solution of:  \uf8f1  \uf8f2  \uf8f3  −div((ηǫ + φ(zN)2)∇uN) = f  ∂nuN = 0  where f = −∂tuN − (uN − gN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Now we use a procedure used in [4] and suggested as a possible alternative  proof in a footnote in [9], that is using Meyers theorem (see [15]) to get H2 esti-  mates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Meyers theorem was originally proved for homogeneous Dirichlet boundary  conditions on ∂U, but in [11] it has been generalised (among others) to the case  of homogeneous Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The proof in [11] for the Neu-  mann case consists in a series of strategies (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' partition of unity, extension of the  functions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=') in order to go back to the case of the original Meyers theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In  particular we want to use Theorem 2 in [11], so we consider  G : L2  m(U) �→ H1  m(U)  such that G(f) = ϕ with  \uf8f1  \uf8f2  \uf8f3  −∆ϕ = f in U  ∂nϕ = 0 in ∂U  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='13)  and  f ∈ L2  m(U) =  �  g ∈ L2(U)  ����  �  U g = 0  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  ϕ ∈ H1  m(U) = H1(U) ∩ L2  m(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Notice that problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='13) admits a unique solution in H1  m(U) if and only if  �  U f = 0 (see [8]), which is our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In particular it holds  ⟨∇G(f), ∇φ⟩L2 = ⟨f, φ⟩L2 for all φ ∈ H1(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='14)  By Theorem 2 in [11] (up to multiplicative constants we are neglecting):  ||∇uN||Lp(U) ≤ ||∇G(f)||Lp(U) for some p ∈ (2, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='15)  9  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Actually, the precise statement of Theorem 2 in [11] would give the  estimate:  ||uN||W 1,p(U) ≲ ||f||W 1,q(U)′,  with 1  p + 1  q = 1, 2 < p < +∞ and W 1,q(U)′ denoting the dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' But then we  readily have:  ||f||W 1,q(U)′ =  sup  ||φ||W 1,q(U)=1  ⟨f, φ⟩L2 =  sup  ||φ||W 1,q(U)=1  ⟨∇G(f), ∇φ⟩L2 ≤  ≤  sup  ||φ||W 1,q(U)=1  ||∇G(f)||Lp(U)||∇φ||Lq(U) ≤ ||∇G(f)||Lp(U)  and so we have the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' The reason why we consider ∇G(f) instead  of dealing with f is because we’ll make use of Gagliardo-Niremberg inequality and  estimates from elliptic regularity theory on ∇G(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Using Gagliardo-Niremberg inequality we get  ||∇uN||Lp(U) ≤ C(1 + ||∇G(f)||2/p  L2(U)||∇2G(f)||  p−2  p  L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='16)  Now notice that up to modification by an additive constant we can consider  without loss of generality −∇G(f) = (ηǫ + φ(zN)2)∇uN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Indeed:  −∆G(f) = div(−∇G(f)) = f = div((ηǫ + φ(zN)2)∇uN),  so −∇G(f) = (ηǫ + φ(zN)2)∇uN ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U)) thanks to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Integrate in time the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='16) raised to the power 2p/(p − 2) to get:  � T  0 ||∇uN(t)||  2p  p−2  Lp(U)dt ≲ 1 +  � T  0 ||∇2G(f)(t)||2  L2(U)dt ≲ 1 +  � T  0 ||f(t)||2  L2(U)dt ≤ C,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='17)  where we used standard elliptic regularity theory to pass from the L2 norm of  ∇2G(f) to the L2 norm of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' We can assume without loss of generality that 2 < p <  4, otherwise if we had p > 4 we could conclude directly by the above estimates,  indeed 2p/(p − 2) < 4 and by Hölder, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='17) and the uniform L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U))  bounds on ∇uN:  � T  0 ||∇uN(t)||4  L4(U)dt =  � T  0  ��  U |∇uN(t)|  2p  p−2|∇uN(t)|  2p−8  p−2 dx  �  dt ≤  ≤  � T  0 ||∇uN(t)||2p/(p−2)  Lp(U)  ��  U |∇uN(t)|2dx  � p−4  p−2 dt ≤  ≤ C  � T  0 ||∇uN(t)||2p/(p−2)  Lp(U)  dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='18)  Of course we also assume p ̸= 4, or the thesis would follow trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' So assume  2 < p < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Applying again Gagliardo-Nirenberg, Hölder and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='17):  10  � T  0 ||∇uN(t)||4  L4(U)dt ≤  � T  0 ||∇uN(t)||p  Lp(U)||uN(t)||4−p  H2(U)dt ≤  ≤  �� T  0 ||∇uN||  2p  p−2  Lp(U)  � p−2  p  �� T  0 ||uN||2  H2(U)  � 4−p  2  ≲  �� T  0 ||uN||2  H2(U)  � 4−p  2  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='19)  and we can conclude since (4 − p)/2 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Notice the assumption n = 2 is needed in order to have the necessary  Gagliardo-Niremberg estimates in the previous Proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Also, notice how the  homogeneity of degree 4 is preserved both in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='18) and in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='19), where to conclude  we uniformly bound some quantities depending on uN, namely (  �  U |∇uN(t)|2dx)  p−4  p−2  and  �� T  0 ||∇uN||  2p  p−2  Lp(U)  � p−2  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  The estimates obtained in the previous Proposition actually yield uniform esti-  mates of uN and zN in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U)) thanks to the classical fact that ||u||L2(U) +  ||∆u||L2(U) is an equivalent norm for H2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' To recapitulate, we have (up to a  subsequence we will not rename):  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  (uN, zN) weakly- ∗ converging in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H1(U))  (uN, zN) weakly converging in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U))  (∂tuN, ∂tzN) weakly converging in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U))  (uN, zN) converging in C(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U))  (uN, zN) converging in the strong topology in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H1(U))  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='20)  where the compact embeddings in C(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U)) and L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H1(U)) are ob-  tained by applying the Aubin-Lions lemma (see [3], [13], [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' We are now ready  to prove the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Existence and uniqueness of strong solutions  Let U ⊂ R2 be a bounded Lipshitz domain and let (u0, z0) ∈ [H1(U)]2 with  0 ≤ z0 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Then there exists a unique strong solution (u, z) of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Let (u, z) be the weak limit of (uN, zN) in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' H2(U)), let’s see how  the pair is a solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' This will be sufficient to prove the thesis thanks to  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Let ψ ∈ VM = Span{e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' , eM} be a test function for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='6) with  N > M, so it holds:  �  U uN(t)ψ  �  ��  �  1  =  �  U πN[u0]ψ −  � t  0  �  U(ηǫ + φ(zN)2)∇uN∇ψ  �  ��  �  2  −  � t  0  �  U(uN − gN)ψ  �  U zN(t)ψ  �  ��  �  3  =  �  U πN[z0]ψ − 2ǫ  � t  0  �  U ∇zN∇ψ −  � t  0  �  U φ′(zN)φ(zN)|∇uN|2ψ  �  ��  �  4  +  � t  0  �  U  1 − zN  2ǫ  ψ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='21)  11  and we want to show we can pass to the limit in every highlighted term, since  for the others it’s trivial by weak convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  As for 1 and 3 , we can pass to the limit thanks to the compactness in  C(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  For 2 , we have (by dominated convergence) strong convergence in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U))  of (ηǫ + φ(zN)2), and weak convergence of ∇uN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' So their product weakly  converges and we can pass to the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  We already saw in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='4 how ∇uN is uniformly bounded in L4(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L4(U)),  which is the same as saying |∇uN|2 is uniformly bounded in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Then, up to taking another subsequence, |∇uN|2 ⇀ |∇u|2 in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Since φ′(zN)φ(zN) → φ′(z)φ(z) in the strong L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' L2(U)) topology by  dominated convergence, their product weakly converges and we can pass to  the limit in 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  It only remains to prove that (u, z) satisfy the homogeneous Neumann bound-  ary conditions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  To do that we first have to make sense of ∂nu for any u ∈ H2(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' We define  ∂n : H2(U) → H1/2(∂U)  as  ∂nu(ψ) =  �  U ∆uΨdx +  �  U ∇u∇Ψdx,  where ψ ∈ H1/2(∂U) and Ψ ∈ H1(U) is an extension of ψ to the whole U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' In  particular Ψ will be chosen according to the trace extension operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Ψ = Eψ,  where E is defined as:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Trace extension operator, see [17]  Given a bounded, Lipshitz domain Ω ⊂ Rn and 1 < p < +∞, there exists a  linear and bounded trace extension operator  E : W 1− 1  p ,p(∂Ω) → W 1,p(Ω)  such that Tr(Eu) = u for every u ∈ W 1− 1  p(∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  The operator ∂n just defined is continuous, indeed using ||Eψ||H1(U) ≤ C||ψ||H1/2(U):  ||∂nu||H1/2(∂U) =  sup  ψ∈H1/2(∂U)  �  ∂nu(ψ)  ||ψ||H1/2(∂U)  �  ≤  ≤ C  sup  ψ∈H1/2(∂U)  �  1  ||Eψ||H1(U)  �  U ∆uEψdx +  �  U ∇u∇Eψdx  �  ≤  ≤ C(||∆u||L2(U) + ||∇u||L2(U)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Moreover it is known that W 1−1/p,p(∂U) compactly embeds into Lp(∂U) (see  [7]), so  12  ∂n : H2(U) �→ L2(∂U) is weak-strong continuous,  meaning it sends weakly converging sequences in strong converging ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' We  have to prove that ∂nu(t) = ∂nz(t) = 0 for almost every t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Since the argument is  the same we’ll only show that the boundary conditions hold for u, moreover for  simplicity we assume that u(t) ∈ H2(U) for every t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' By weak-strong continuity of  ∂n we have that for every t ∈ [0, T], modulo a subsequence (which depends on t):  uN(t)  H2(U)  −−−⇀ u(t) =⇒ ∂nuN(t)  L2(∂U)  −−−−→ ∂nu(t) as N → +∞,  but since ∂nuN(t) ≡ 0 for every N we have the thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  References  [1] Luigi Ambrosio, Nicola Fusco, and Diego Pallara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Functions of bounded vari-  ation and free discontinuity problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Courier Corporation, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
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+page_content=' Ennio De Giorgi, page 654, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
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+page_content=' Hitchhiker’s  guide to the fractional sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
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+page_content=' Partial differential equations, volume 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
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+page_content=' Analysis of gradient flow of a regular-  ized mumford-shah functional for image segmentation and image inpaint-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
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+page_content='  Revisiting brittle fracture as an en-  ergy minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Journal of the Mechanics and Physics of Solids,  46(8):1319–1342, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [11] Thierry Gallouet and Alexis Monier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' On the regularity of solutions to elliptic  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' (7), 19(4):471–488, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [12] David Gilbarg, Neil S Trudinger, David Gilbarg, and NS Trudinger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Elliptic  partial differential equations of second order, volume 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Springer, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [13] Jacques-Louis Lions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  Quelques méthodes de résolution de problemes aux  limites non linéaires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [14] Jacques Louis Lions and Enrico Magenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Non-homogeneous boundary value  problems and applications: Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' 1, volume 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Springer Science & Business  Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [15] Norman G Meyers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' An Lp-estimate for the gradient of solutions of second  order elliptic divergence equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Annali della Scuola Normale Superiore di  Pisa-Classe di Scienze, 17(3):189–206, 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [16] David Bryant Mumford and Jayant Shah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Optimal approximations by piece-  wise smooth functions and associated variational problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Communications  on pure and applied mathematics, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [17] Jindrich Necas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Les méthodes directes en théorie des équations elliptiques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [18] Jacques Simon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Compact sets in the space Lp(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Annali di Matematica  pura ed applicata, 146(1):65–96, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  [19] Michael Struwe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Geometric evolution problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content=' Nonlinear partial differential  equations in differential geometry, 2:257–339, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}
+page_content='  14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdFRT4oBgHgl3EQfyTjG/content/2301.13645v1.pdf'}

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