Physics-Informed Regularization for Domain-Agnostic Dynamical System Modeling

Summary

The paper introduces a framework named TREAT (Time-Reversal Symmetry ODE) aimed at improving dynamical system modeling through a physics-informed approach. It incorporates Time-Reversal Symmetry (TRS) as a regularization term to enhance model precision. This approach is shown to preserve energy in conservative systems and provide strong inductive bias for non-conservative, reversible systems. Theoretical proofs are provided to demonstrate the numerical benefitd∂. The framework's effectiveness is validated through experiments on nine diverse datasets, showing improvement over existing models.

Strengths

The paper proposed the TRS as a regularization term for dynamical system modeling. this regularization intuitively makes sense, by augmenting the reverse trajectory as additional data for training. The paper provides theories that TRS minimizes higher-order Taylor terms, enhancing modeling accuracy. The paper conducted extensive experiments on nine datasets show that TREAT outperforms other neuralODE models.

Weaknesses

1. The paper only uses graphODE as backbone, from my understanding the TRS regularization can also fit with other neuralODE methods, it would be interesting to add experiments to show its compatibility (e.g. vanilla neuralODE). 2. The theoretical analysis in theorem 3.1 bounds the error in the embedding space z, however, the loss function in eqn 10 describes the error from original space y. it is worth mentioning the difference due to encoder/decoder error.

Questions

1. Figure 1 shows the time reserve symmetry system is inclusive in the general classical mechanics. I’m confused regarding the TRS definition in physics and the numerical TRS used in this paper. In my understanding, any dynamical system with deterministic ode description can reverse trace the initial points by solving it from the end point numerically? therefore, the method should be generally applicable? 2. In figure 4a, do you have any insights why the TREAT does not do well compared with others with shorter prediction length.

Rating

7

Confidence

3

Soundness

3

Presentation

3

Contribution

3

Limitations

n/a

Summary

The paper proposes a novel Time-Reversal Symmetry (TRS) graph neural ODE, where the TRS is introduced as a soft regularisation term.

Strengths

The paper presents a novel method to combat numerical errors for graph neural ODEs.

Weaknesses

The paper does not introduce early enough that the time-reversal symmetry (TRS) method has been applied to other DL methods such as the TRS-ODEN.

Questions

Are there cases where the time-reversal symmetry expectation is violated? A comparison to a numerical solution could serve as an additional benchmark to show how TRS decreases the numerical errors and improves the prediction. Also, why considering only MSE as a performance metric? The code is anonymised, but it really needs cleaning and proper documentation to the very least docstrings and comments; I have not tested running the code. The work will benefit from additional proofreading.

Rating

6

Confidence

4

Soundness

3

Presentation

2

Contribution

3

Limitations

The authors do not provide a separate Limitations section, and do not discuss limitations of the work in the Conclusions section.

Summary

This paper proposes a regularization term to enforce Time-Reversal Symmetry (TRS) for modeling dynamical systems. The method helps preserve energies for conservative systems while serving as a strong inductive bias for non-conservative reversible systems. They also prove that TRS loss can universally improve modeling accuracy by minimizing higher-order Taylor terms in ODE integration. Numerical examples show that integrating the TRS loss within neural ordinary differential equation models demonstrates superior performance on diverse physical systems.

Strengths

This paper focuses on learning complex physical dynamics from data and incorporating physics priors into learning models, which is an important area. This paper proposes a regularization term to enforce Time-Reversal Symmetry (TRS), which is an important physics prior. Also, the paper numerically shows that integrating the TRS loss within neural ordinary differential equation models demonstrates superior performance on diverse physical systems.

Weaknesses

While the paper on time-reversal symmetry demonstrates several strengths, there are also some potential weaknesses that could be addressed: 1. Time-Reversal Symmetric ODE Network has been proposed (https://arxiv.org/pdf/2007.11362). This article is also cited in this paper. However, this paper only points out that two articles use different properties (Equation 5 and Lemma 2.1) to sign the regularization term, the difference in the final regularization term is not fully explained. 2. The numerical experiments also demonstrated that the method proposed in this paper outperforms existing time-reversal symmetry regularization (https://arxiv.org/pdf/2007.11362). However, the reason why this method is better in this paper is not explained in detail. 3. Some descriptions in the text are not clear enough, please refer to the question section below.

Questions

1. In the proof of Lemma 2.1, it is required that R be an involution operator. However, such assumption is not stated in Lemma 2.1. Is there any evidence to prove that this assumption is reasonable？ 2. The definition of R does not appear in the paper. Can the method proposed in this paper hold for any R? 3. How is regularization term 9 obtained based on Lemma 2, as R did not appear in the final regularization term 9？ 4. Does Theorem 3.1 hold for any untrained NN? 5. The definition of $L_{reverse2}$ does not appear in the paper. Can the condition $L_{reverse2}=L_{reverse}$ be satisfied?

Rating

6

Confidence

3

Soundness

3

Presentation

3

Contribution

3

Limitations

Yes

Summary

This paper proposes a method to enhance neural ordinary differential equations (ODEs) by enforcing approximate Time-Reversal Symmetry. A self-supervised regularization term is introduced to align forward and backward trajectories predicted by a neural network, promoting energy conservation and stability in the system. Experiments are conducted on 9 datasets, comprising both real-world and simulated systems. The proposed approach outperforms other baseline methods, demonstrating its effectiveness in improving neural ODEs.

Strengths

* The idea of imposing time-reversal symmetry via a regularization term is novel and well-motivated. * The implementation of the regularization term is simple yet the effect is visible. * Overall the article is well written, and technical details are explained clearly. * The results show significant improvement over other baseline models.

Weaknesses

* It would be more interesting to see if the model works for other real-world examples/experiments. * The article lacks a more comprehensive review on other energy-preserving / time-reversal neural ODE solvers.

Questions

Does the method work well with models other than GraphODE?

Rating

8

Confidence

4

Soundness

3

Presentation

4

Contribution

4

Limitations

The authors briefly touched upon the possibility of incorporating properties in the spatial aspect such as translation and rotation equivariance.

General Response to all Reviewers

We sincerely thank all reviewers for their valuable time and insightful feedbacks! We acknowledge the positive comments such as "Novel approach and well-motivated " (Reviewer hqsc, D3dA), “Simple-yet-effective approach and superior performance” (Reviewer hqsc, SfDm,edxC), "Well-written" (Reviewer hqsc), "Theoretical Analysis” (Reviewer edxC). We have also responded to all the questions point by point and will incorporate these suggestions into our revision to enhance our paper.