Physics-Informed Regularization for Domain-Agnostic Dynamical System Modeling

We sincerely thank you for your valuable comments and suggestions for improving our paper! We would like to make the following claims and hope they can address your concerns.

W1. About the use case for Vanilla neural ODE

Thank you for the valuable suggestion! Our TRS loss can be coupled with other models as well, due to its simplicity and flexibility. In our experiments for single-agent systems, TREAT actually becomes neuralODE + TRS loss, as there is no graph structure considered if there’s only one agent (also mentioned in Appendix D.2 page 25 line 673). As shown in Table 1, TREAT consistently outperforms other baselines across diverse single-agent datasets, showing its strong generalization ability and effectiveness. Due to its simplicity, we believe other models, such as NeuralODE and flow-based methods, can also benefit from TRS loss.

W2. Theorem 3.1

Thanks for pointing this out. We will add an explanation between the original space y and the embedding space z. Our theorem aims to highlight the differences between with and without TRS loss. Both scenarios involve an encoder and a decoder. However, since y and z correspond to each other through the encoder and decoder, the errors introduced by the encoder and decoder do not affect the validity of Theorem 3.1.

Q1. Is TRS a universal property of all classic ODEs?

A deterministic ODE can indeed be solved numerically from the endpoint to trace back to the initial points. However, the definition of time-reversal symmetry (TRS) in physics requires that the results of solving the system forward and backward in time must be consistent. This does not always hold for any given ODE function. In Appendix B (page 18, line 523 ~ page 19, line 540), we provide examples of reversible and irreversible systems. These examples illustrate their ODE functions and verify the systems’ energy conservation and time-reversal symmetry. For example, spring systems with frictions do not satisfy TRS, and are also not energy-conservative.

Q2. Explanation of Figure 4 for short-term predictions

Thanks for pointing it out! We first observe that on damped springs and forced springs, TREAT offers comparable results, and on the pendulum, TREAT significantly outperforms others.

As for the simple spring with shorter prediction lengths, we assume the potential reasons as follows:

1. For TRS-ODEN and HODEN, which do not have encoder and decoder structures, we compute each object’s initial state via linear spline interpolation if it is missing. Since Simple Spring is a trivial system, it is less sensitive to the initial state and easier to learn. Also, TRS may be less important for making short-term predictions than long-term predictions as suggested by Figure 4. Therefore, in the short term, forcing TRS and having a more complex encoder structure (TREAT) may add additional burdens to the model learning process, and simple models like TRS-ODEN and HODEN can already work well. However, we can observe in the long run, TREAT significantly achieves better results.

2. TRS-ODEN and HODEN are two single-agent baselines that do not consider the interactions among agents, i.e., the graph structure. As the dynamics of the Simple Spring are not very complicated, within a short prediction length, the interactions between objects may not have a significant impact on the trajectories. This aligns with observations in existing literature [1].

[1] Zijie Huang, Yizhou Sun and Wei Wang. "Coupled Graph ODE for Learning Interacting System Dynamics." KDD 2021.

Thank you very much for your valuable feedback!

Regarding your minor concern, we made the following clarification:

The definition of TRS is $\frac{d R\circ x}{dt} = -F(R\circ x)$ as shown in page 4 line 111-120. For Hamiltonian systems, the state x contains both position q and velocity p, i.e. $x = (q,p)$, and the reversing operator would work as $R\circ (q,p) = (q,-p)$ according to [1]. This is can simply be understood as: if a pendulum swings from A to B (with no frictions and no external forces, etc), we can just revert its velocity in the opposite direction, and let it swing back from B to A, the two trajectories would be the same. In this case, only velocity is reversed and position at B is kept the same.

Now let's check for spring systems with frictions, would it satisfy $\frac{d R\circ x}{dt} = -F(R\circ x)$. As shown in Appendix B.3, the deterministic ODE functions for such systems are defined as $\frac{dq}{dt} = \frac{p}{m}$, $\frac{dp}{dt} = -kq-\gamma\frac{p}{m}$, where $m$ is the mass. Since $R\circ (q,p) = (q,-p)$, going back to the TRS definition $\frac{d R\circ x}{dt} = -F(R\circ x)$, we need to check for position $q$, does the first ODE function follow $\frac{dq(-t)}{d(-t)} = -F(q(-t), p(-t))$, which implies $\frac{dq(t)}{dt} = -F(q(t), -p(t))$. Similarly, for velocity $p$, we need to check does the second ODE function follows $\frac{dp(-t)}{d(-t)} = -F(q(-t), p(-t))$, which implies $\frac{dp(t)}{dt} = F(q(t), -p(t))$.

We can easily verify that the first ODE function holds ($\frac{dq(t)}{dt} = -F(q(t), -p(t))$), while the second ODE function fails ($\frac{dp(t)}{dt} = F(q(t), -p(t))$). Therefore, spring systems with frictions do not satisfy TRS.

Please kindly let us know if there are any questions further, for which we're happy to provide additional information and clarifications. We truly appreciate the time and effort you’ve invested in reviewing our work!

[1] Lamb, J. S., & Roberts, J. A. (1998). Time-reversal symmetry in dynamical systems: a survey. Physica D: Nonlinear Phenomena, 112(1-2), 1-39.

We're grateful for your support and helpful feedback! Please kindly find our response below to the questions raised in the review and let us know if you have any further questions.

W1: Reference about TRS-ODEN.

Thank you for your advice! We would like to discuss more about TRS-ODEN in the introduction part in the revised version. However, we would like to clarify a few points: We do not position our paper as an improved version of how to enforce TRS as in TRS-ODEN paper. Instead, we would like to find a general physics-informed prior for domain-agnostic dynamical system modeling. To this end, we found that TRS is a great property to achieve that. We would like to emphasize our major contribution is to find the numerical benefits of TRS regularization, i.e. minimizing higher-order Taylor expansion terms during ODE integration steps. This makes the proposed TRS loss widely applicable to a wide range of dynamical systems, regardless of their physical properties (i.e. even irreversible systems). Therefore, we did not see TRS-ODEN as a direct competitor. Nonetheless, in sections where details are needed, we discussed more details about TRS-ODEN (first appeared in Sec.2). However, we will make sure to clarify this further in the introduction part to make it clearer.

Q1: Cases where TRS is violated

Thanks for your question! Indeed, some systems may not strictly obey the TRS due to situations such as time-varying external forces, internal friction, and underlying stochastic dynamics. For example, spring systems with frictions do not adhere to TRS. We illustrate examples for reversible and irreversible systems in Appendix B (page 18, line 523 ~ page 19, line 540). Therefore, a desired model shall be able to flexibly inject time-reversal symmetry as a soft constraint, so as to cover a wider range of real-world dynamical systems. Note that from the numerical aspect, we also theoretically prove that the TRS loss effectively minimizes higher-order Taylor expansion terms during ODE integration, offering a general numerical advantage for improving modeling accuracy across a wide array of systems, regardless of their physical properties (even for irreversible systems). Therefore, TREAT achieves high-precision modeling from both aspects as depicted in Figure 1(a).

Q2-1: Can TRS decrease numerical errors across different solvers?

To show TRS can in general help to reduce numerical errors across different choices of solvers, we show the performance of the Euler method and Runge-Kutta (RK4) methods, which have different trade-off between precision and time. As suggested by the results below (also shown in Appendix E.1 on page 26, line 726-735), TREAT consistently outperforms LGODE, our strongest baseline, across different solvers and datasets. We also notice that the improvement ratio $\frac{LGODE - TREAT}{LGODE}$ is larger when using RK4 compared to Euler. This suggests that our TRS loss minimizes higher-order Taylor expansion terms, thus compensating for numerical errors introduced by ODE solvers.

Q2-2: Evaluation metric

Thank you for your valuable feedback. Since our work focuses on deterministic trajectory prediction, calculating the MSE between the predicted trajectory and the ground truth trajectory is the most direct and effective measure. Therefore, we followed the experimental settings of existing work [1] [2].

[1]Huang, Z., et al. "Learning continuous system dynamics from irregularly-sampled partial observations." Advances in Neural Information Processing Systems 2020

[2] Kipf, Thomas, et al. "Neural relational inference for interacting systems." International conference on machine learning. PMLR, 2018.

Q3: Code

Thank you for your feedback. We apologize for not providing sufficient docs and comments for the code. We have refined our code structure and documentation in the anonymous GitHub repo.

Q4: Discussion about limitations

Thank you for your valuable suggestion. We have included the limitations section in Appendix H (page 27, lines 776-779). We apologize for any inconvenience this may have caused. In the revised version, we promise to include it in the main part of the paper.

We thank you for your insightful comments on improving our paper. However, we believe there’s some misunderstanding and would like to make the following claims to address your concerns.

W1: The difference to the final implementations.

We would like to mention that both methods are approximations towards TRS. In Appendix A.4 (page16), we show the difference of the final implementations and their visualizations in Figure 8 (assuming one integration step).

The TRS loss in TREAT (ours) follows Lemma 2.1: $R \circ \Phi_t \circ R \circ \Phi_t = \text{I}$ It means: we start from $\boldsymbol{\hat{y}}_i ^{\text{fwd}}(0)$ and first move forward one step, reaching the state $\boldsymbol{\hat{y}}_i ^{\text{fwd}}(1)$. Then we reverse it and move forward one step in the opposite direction, getting $\boldsymbol{\hat{y}}_i ^{\text{rev}}(0)$. Finally, we reverse it and it shall restore back to the same state. That is ideally it should be the same as $\boldsymbol{\hat{y}}_i ^{\text{fwd}}(0)$.

The second reverse loss in TRS-ODEN follows Eqn 5 as $R \circ \Phi_t = \Phi_{-t}\circ R$. It means we first reverse the initial state and move forward one step in the opposite direction to reach $\boldsymbol{\hat{y}}_i ^{\text{rev2}}(-1)$. We then perform a symmetric operation to reach $\boldsymbol{\hat{y}}_i ^{\text{rev2}}(1)$, which should align with the forward one $\boldsymbol{\hat{y}}_i ^{\text{fwd}}(1)$.

The key differences are illustrated in Figure 8. Our method forces the starting point of backward trajectories to be the same as the end point of the forward one. In TRS-ODEN, the backward trajectories start from the same point as the forward one. We analytically show why our implementation is better based on Figure 8 in Appendix A.4.

W2: Why our implementation is better.

As shown in Appendix A.4 (page 16 line 494-501), we analytically show our implementation based on Lemma 2.1 to approximate TRS has a lower maximum error compared to TRS-ODEN, supported by empirical experiments in Sec. 4.2.

We here use one integration step for illustration purpose. Specifically, if we assume the two reconstruction losses are of the same value $a$, and the two TRS losses have reached the same value $b$, we show that the maximum error between the reversal and ground truth trajectory for each agent, made by TREAT is smaller. In TREAT, the maximum error is $max(a,b)$, whereas TRS-ODEN is $a+b$.

Finally, we would like to summarize our contribution again: we do not position our paper as an improved version of how to enforce TRS as in the TRS-ODEN paper. Instead, we would like to find a general physics-informed prior for domain-agnostic dynamical system modeling. To this end, we found that TRS is a great property to achieve that. Our major contribution is a physics-inspired regularizer that works well beyond physics domains, due to its numerical properties. Specifically, our TRS loss minimizes higher-order Taylor expansion terms during ODE integration steps. This makes the proposed TRS loss widely applicable to a wide range of dynamical systems, regardless of their physical properties.

Q1.

Thanks for your question. The reversing operator R is an involution by definition [1], i.e. $R\circ R = I$. We will clarify this in Lemma 2.1 based on your valuable suggestion.

For example: Consider reversing the position $q(t)$, we have $R\circ q(t) = q(t)$ [1]. Therefore we can get $R\circ R\circ q(t)=R \circ q(t) = q(t)$,

As for velocity $p(t)$, we have $R\circ p(t) = -p(t)$ [1]. Therefore we can also get $R\circ R\circ p(t)=R\circ (-p(t))= p(t)$

[1] Lamb, J. S., & Roberts, J. A. (1998). Time-reversal symmetry in dynamical systems: a survey. Physica D: Nonlinear Phenomena, 112(1-2), 1-39.

Q2.

The definition of $R$ is illustrated in section 2.2 (page 4 line 111-128). It is a reversing operator defined as $R: {x} \mapsto R \circ {x}, R \circ {x} (t) = {x}(-t)$ where $x$ is the observational state of the system. For different systems, $x$ can contain different state variables, so this definition is universal to all systems. For example, if $x = (q,p)$ where $q$ is the location and $p$ is the velocity, $R\circ x = (q,-p)$ [1].

[1] Lamb, J. S., & Roberts, J. A. (1998). Time-reversal symmetry in dynamical systems: a survey. Physica D: Nonlinear Phenomena, 112(1-2), 1-39.

Q3.

As shown in Figure 2 and Figure 8, $z^{\text{rev}}(t’_0)=R \circ z^{\text{fwd}}(t_K)=z^{\text{fwd}}(t_K)$. Combining this with Eqn 8, we obtain $y^{\text{rev}}(t’_j)$ and finally, we derive Eqn 9.

Q4.

Yes. Theorem 3.1 does not restrict the form of the ODE function $g$. The ODE function $g$ can be any NN in practice.

Q5.

Thanks for pointing this out! The definition of $\mathcal{L}_{reverse2}$ is proposed in TRS-ODEN, and detailed in our Appendix A.4. We will take your advice and add it to the main text for clarification.

We sincerely appreciate your insightful comments and valuable advice on improving our paper. We highly appreciate your recognition of the novelty and significance of our work. Regarding your questions, we detailed our responses below.

W1: Empirical results on additional real-world examples.

We appreciate your suggestions to add more real-world examples. In our experiments, we have 9 diverse datasets spanning across 1.) single-agent, multi-agent systems; 2.) simulated and real-world systems; and 3.) systems with different physical priors. Most of them are simulated datasets, as real-world datasets do not conform neatly to specific physical laws, making it challenging to classify them as either conservative or reversible straightforwardly and to compare them with existing physics-informed baselines. Notably, on one real-world human motion dataset (walking object), TREAT outperforms other baselines significantly, showing case its strong generalization ability and numerical benefits.

To address your concern, we additionally added a new human motion dataset (dancing object). The results below suggest that TREAT consistently outperforms other baselines on the new real-world dataset, showing its effectiveness.

W2: review on other energy-preserving/time-reversal neural ODE solvers.

We appreciate your suggestion to incorporate more related work on energy-preserving and time-reversal neural ODE solvers. In response, we have drafted the following additions to our revised manuscript.

Various methods have been developed to maintain the total energy of dynamic systems over time [1][2][3][4]. However, strictly energy-conservative approaches can be unrealistic for non-isolated systems that interact with their environments. Conversely, methods that allow for both energy-conserving and dissipative models, as well as reversible and irreversible systems, offer more flexibility [5][6][7][8]. These methods, however, require prior knowledge of the system's properties. For example, [7] necessitates determining the appropriate bracket operator for different systems. Our model, in contrast, introduces a unified approach that learns both the trajectories and the aforementioned properties dynamically, without needing to assign a specific property beforehand.

[1] Greydanus, Samuel, Misko Dzamba, and Jason Yosinski. "Hamiltonian neural networks." Advances in neural information processing systems 32 (2019).

[2] Gruver et al, Deconstructing the inductive biases of Hamiltonian neural networks, ICLR 2022.

[3]Han, Chen-Di, et al. "Adaptable Hamiltonian neural networks." Physical Review Research 3.2 (2021): 023156.

[4] Mattheakis, Marios, et al. "Hamiltonian neural networks for solving equations of motion." Physical Review E 105.6 (2022): 065305.

[5] Zhong et al, Dissipative symoden: Encoding Hamiltonian dynamics with dissipation and control into deep learning, et al, ICLR 2020 workshop.

[6] Morrison, Philip J. "A paradigm for joined Hamiltonian and dissipative systems." Physica D: Nonlinear Phenomena 18.1-3 (1986): 410-419.

[7] Gruber, et al, Reversible and irreversible bracket-based dynamics for deep graph neural networks, NeurIPS 2023.

[8] Huh et al, Time-reversal symmetric ode network, NeurIPS 2020.

Q1. Can the TRS loss be coupled with other models?

Our TRS loss can be coupled with other models as well, due to its simplicity and flexibility. In our experiments for single-agent systems, TREAT actually becomes neuralODE + TRS loss, as there is no graph structure considered if there’s only one agent. As shown in Table 1, TREAT consistently outperforms other baselines across diverse single-agent datasets, showing its strong generalization ability and effectiveness. Due to its simplicity, we believe other models, such as NeuralODE and flow-based methods, can also benefit from TRS loss.