Physics-Informed Regularization for Domain-Agnostic Dynamical System Modeling
Authors: Zijie Huang
Wanjia Zhao
Jingdong Gao
Ziniu Hu
Xiao Luo
Yadi Cao
Yuanzhou Chen
Yizhou Sun
Wei Wang
Venue: NeurIPS 2024 poster (2024)
Keywords: Physics-informed Neural Networks;  Dynamical Systems ; Graph Neural Networks
Primary area: machine_learning_for_physical_sciences
TL;DR: We present TREAT, a novel framework that achieves high-precision modeling for a wide6 range of dynamical systems from the numerical aspect, by enforcing Time-Reversal7 Symmetry (TRS) via a novel regularization term.

Abstract: Learning complex physical dynamics purely from data is challenging due to the intrinsic properties of systems to be satisfied. Incorporating physics-informed priors, such as in Hamiltonian Neural Networks (HNNs), achieves high-precision modeling for energy-conservative systems. However, real-world systems often deviate from strict energy conservation and follow different physical priors. To address this, we present a framework that achieves high-precision modeling for a wide range of dynamical systems from the numerical aspect, by enforcing Time-Reversal Symmetry (TRS) via a novel regularization term. It helps preserve energies for conservative systems while serving as a strong inductive bias for non-conservative, reversible systems. While TRS is a domain-specific physical prior, we present the first theoretical proof that TRS loss can universally improve modeling accuracy by minimizing higher-order Taylor terms in ODE integration, which is numerically beneficial to various systems regardless of their properties, even for irreversible systems. By integrating the TRS loss within neural ordinary differential equation models, the proposed model TREAT demonstrates superior performance on diverse physical systems. It achieves a significant 11.5% MSE improvement in a challenging chaotic triple-pendulum scenario, underscoring TREAT’s broad applicability and effectiveness.

Physics-Informed Regularization for Domain-Agnostic Dynamical System Modeling
Zijie Huang
Wanjia Zhao
Jingdong Gao' Ziniu Hu
Xiao Luo'
Yadi Cao
Yuanzhou Chen' Yizhou Sun
Wei Wang'
University of California Los Angeles, Sttanford University
CCaiiorrna Institute of Technology https://treatoock.
Abstract
Learning complex physical dynamics purely from data is challenging due to the intrinsic properties of systems to be satisfied. Incorporating innssce-informed priors, such as Hamiltonian Neural Networks (HNNs), achieves begh-precision modeling for energy energy-conservate conservvtive systems. However, real-world systems often deviate from strict energy conservation and follow different physical priors. To ad- dress this, we present framework that achieves high-precision modeling for wide range of dynamical systems from the numerical aspect, by enforeing Symmetry (TRS) via novel regularization term. It helps preserve energies for conservative systems while serving as strong inductive bias for non-conservative, reversible systems. While TRS domain specifiee physical prior, we present the first theoretical proof that TRS loss can universally improve modeling accuracy by minimizing higher-order Taylor terms in ODE integration, which is numerically beneficial to various systems regardless of their properties, even for irreversible systems. By integrating the TRS loss within neural ordinary differential equation diverse
models, the proposed model TREAT demonstrates superior performance physical systems. achieves significant 11.5% MSE improvement in a challeng- ing chaotic triple-pendulum scenario. underscoring TREAT's s broad applicability and effectiveness. Code and further details are available at hhee
Introduction
Dynamical systems, spanning applications from physical simulations (Kipf et al., 2018 Wang et al.
2020 Lu et al.) al.||220.. 2022 Huang et al. 2023 Luo et al. [202a Xu et al.) 2024 Luo et al. 20024) to robotic control (Li al.] 2022 Ni and Qureshi 2022). are challenging model due intricate dynamic patterns and potential interactions under multi-agent settings. Traditional numerical simulators require extensive domain knowledge for design, which sometimes unknown Saancez-GGnzzlee et al. 2020), and can consume significant computational resources (Wang et al. 2024). Therefore, directly learning dynamics from the observational data becomes an attractive alternative.
Existing deep learning approaches
Gonzalez et al. 2020 Plaff et al.) 2021 Han et al.
2022ab usually learn fixed-step transition function to predict system dynamics from timestamp / to timestamp +++ and rollout trajectories recursively. The transition function can have different
inductive biases. such as Graph Neural Networks (GNNs) (Pfaff et al., 2020 Martinkus al.
2021 Lam et al. 2023 Cao et al., 2023) for capturing pair-wise interactions among agents through message passing. Most recently, neural ordinary differential equations (Neural ODEs) (Chen et al.
"Equal contribution, Corresponding to Zijie Huang <zijichuang waa.hhaann@es.ulla.dd....
Wanjia Zhao <wanji-
azh @ es wos.sttanfrdddduu
i Work done as a visiting student at UCLA
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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High Digh--Procisio Modeling Dynamical Systems
Injecting Physical
Reducing Error Accumulation over Integration Steps
TRS Loss
dP dr o(A=")+..
Energy- ConggyConsevvatiee
dR(a) - (RRG)))
A2
Ag
Numerical Erros
dt (g.p)
A
Reversal
Ag
Dynamical Systems under
Identical pasitions
RG) cq. -p)
method
Classical Mechunics
forward
Integnation steps
(b.1) Physical Prices
(b.2) Time- Reversal Symmetry+
buckward
(b.3) Accumalation
Figure 1: (a) High-precision modeling for dynamical systems; (b.1) Classification of classical
mechanical systems based on (Tolman 1938 Lamb and Roberts (998);(b2) Tim-Reversal Symmetry
(Ilustration:(b.3) Error accumulation in numerical solvers
2018 Rubanova al. 2019) have emerged potent solution for modeling system dynamics in a
continuous manner, which offer superior prediction accuracy over discrete models in the long-range. and can handle systems with partial observations. In particular, GraphODEs (Huang al. 2020 Luo
et al. 2023b; Zang and Wang. 2020; Jiang et al. [2023 Luo et al., 2023ch extend NeuralODEs to
model interacting (multi-agent) dynamical systems, where agents co-evolve and form trajectories jointly.
However, the complexity of dynamical systems necessitates large amounts of data. Models trained on limited data risk violating fundamental physical principles such as energy conservation. A promising strategy improve modeling accuracy involves incorporating physical inductive biases (Raissi et al. 2019 Crammer et al. 2020). Existing models like Hamiltonian Neural Networks (HNNs) (Greydanus) al. 2019: Sanchez- Gonzalez et al., 2019) strictly enforce energy conservation, yielding more accurate predictions for conservatvee systems. However, not all real-world systems strictly adhere to energy conservation, and they may adhere to various physical priors. Other methods that model both energy- -ereyy--nneviinggg and dissipative systems, as well as reversible systems, offer more flexibility (Zhong et al.] 22200] Gruber al., (022)) Nevertheless, they often rely on prior knowledge of the system and are also limited to systems with corresponding physical priors. Such system diversity largely limits the usage of existing models which are designed for specific physical prior.
To address this, we present a framework that achieves high-precision modeling for a wide range of dynamical systems from the numerical aspect, by enforcing Time-Reversal Symmetry (TRS) via a novel regularization term. Specifically, TRS posits that a system's 's dynamics should remain invariant when time is reversed (Lamb and Roberts 1998). To incorporate TRS, we propose simflc-yet-fffective self-supervised regularization term that acts as a soft constraint. This term forward and backward trajectories predicted by a neural network and we use GraphODE as the backbone. We 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 wide array of systems, regardless of their physical properties. It forces the model to capture fine- ane-ggrined physical properties such as jerk (the derivatives of accelerations) and provides more regularization for long-term prediction. We also justify our TRS design choice, showing case its superior performance both analytically and empirically. We name the model as TREAT Reversal Symmetry ODE).
Note that TRS itself is a physical prior, that is broader than energy conservation as depicted in Figure b.. It covers classical congy-conservative systems such as Newtonian mechanics, and also non-conservative, reversible systems like Stokes flow (Pozrikidis) 2001), commonly encountered in microfluidics (Kim and Karrila, 2013) Cao and Li 2018](Ca et al.) 2019). Therefore, TRS loss achieves
modeling from both the physical aspect, and the numerical aspect as shown in Figure a), making it aamain-agnostt and widely applicable to various dynamical systems. We systematically conduct experiments across diverse datasets spanning across I.) single-agent, multi-agent systems; 2.) simulated and real-world systems; and 3.) systems with different physical

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priors. TREAT consistently outperforms state-of of-the-attteee art baselines, affirming its effectiveness and versatility across various dynamic scenarios.
Our primary contributions can be summarized as follows:
We introduce TREAT, powerful framework that achieves high-precision modeling for
wide range of systems from the numerical aspect, by enforcing Time-Reversal Symmetry (TRS) via regularization term.
We establish the first theoretical proof that the time-reversal symmetry loss could in general help learn more fine-grained and long- long-context system dynamics from the numerical aspect, regardless of systems' physical properties (even irreversible systems). This bridges the specific physical implication and the general numerical benefits of the physical prior -TRS. We present empirical evidence of TREAT's state-of-the-ar performance in a variety of systems over 9 datasets, including real-world & simulated systems, etc. It yields significant MSE improvement of 1 1155% on the challenging chaotic triple-pent riple-pendulum system.
2 Preliminaries and Related Work
We represent dynamical system graph 9 = (V, where V denotes the node of N agent and & denotes the set of edges representing their physical interactions. For simplicity, we assumed to be static over time. Single-agent dynamical system is special case where the graph only has one node. In the following, we use the multi-agent setting by default to illustrate our model. We denote X (t) E Rxxd as the feature matrix at timestamp t for all agents, with d as the feature dimension. Model input consists of trajectories of feature matrices over M historical timestamps X (l-MM:11
X ((_1)} and G. The timestamps i_ l-M can have non- non-uniform intervals and take any continuous values. Our goal is to learn neural simulator fo()): : (f-t-M::1)..C 9 - (ff::)). which predicts node dynamies Y (t) the future on timestamps 0= <.. .... l = T sampled within [0, 77. We use (t) denote the targeted dynamic vector of agent at time f. In some cases when we are only predicting system feature trajectories, (-) = X (-).
(X(I-M)
2.1 NeuralODE for Dynamical Systems
NeuralODEs (Chen et al.) 2018 Rubanova et al. 2019) are a family of continuous models that define the evolution of dynamical systems by ordinary differential equations (ODEs). The state
evolution can be described as: (t) := daa(() (z) (t), =2(t) ..- ---2w(t),, where 2:) E Rd denotes the latent state variable for agent at timestamp t. The ODE function y parameterized by neural network such as Multi-Layer Perception (MLP). which is automatically learned from data. GraphODEs (Poli al., 2019 [Hunn et al.) 2020 Luo et al.. 2023b Wenn al. [202]] Huang et al.) 2024) are special cases of NeuralODEs, where g a Graph Neural Network (GNN) to capture the continuous interaction among agents.
GraphODEs have been shown to achieve superior performance, especially in long-range predictions and can handle data irregularity issues. They usually follow the procderrprceessor-ddeodde archi- tecture, where an encoder first computes the latent initial states z (to), ... .. (fo) for all agents simultaneously based their historical observations as in Eqn [
z (lo), 22(1)) NN (1a) = SENC (I-M:
Then the GNN-based ODE predicts the latent trajectories starting from the learned initial states. The latent state (() can be computed at any desired time using a numerical solver such as Runge- Kuttais (Schober et al. 2019) as:
2, (t) ODE-Solver (xx 2x(t + J
zN dt.
(2)
Finally, a decoder extracts the predicted dynamics (t) based on the latent states 2, (t) for any (t) = fDEc(z,())).
timestamp t:
Following Kipf et al. 2018 we use "agents" to denote "objects" in dynamical systems, which is different from "intelligent agent" in AL
3

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However, vanilla GraphODEs can violate physical properties of a system, resulting in unrealistic predictions. We therefore propose inject physics-informed regularization term to make more accurate predictions.
2.2 Time-Reversal Symmetry (TRS)
Consider a dynamical system de- scribed in the form dæ(()) F(x(tt), where æ=t) Ω is the ob- served states such as positions. The system is said to follow the Time- Reversal Symmetry
dt
f (ba)
forward trajectory
there exists a
R +by i dy
(6)
reversing operator R Ω + Ω such that (Lamb and Roberts 1998):
Latent reverse trajectory
CG
d(R R o x())) - FF (4)
Figure 2: Illustration of time-reversal symmetry based on Lemma total length of the trajectory f, to T. t is the time index in the reverse trajectory, which points to the same time kg k in the forward trajectory.
dt
where denote the action of func-
tional R on the function æ.
Intuitively, we can assume xt)) is the position of flying ball and the con- ventional reversing operator is defined as R H R R (t) x=-t). This implies when <t)) is a forward trajectory position with initial position (0),, x(-t) is then a position in the time- reversal trajectory, where x--() is calculated using the same function F, but with the integration flying ball: at
time reversed, i.e. dl +> d(-t). Eqn shows how to create the reverse trajectory each position, the velocity (i.e., the derivative of position with respect to time) should be the opposite. In neural networks, we usually model trajectories in the latent space via z Saanhezz-Gonzalee et al., 2020), which can be decoded back to real observation state i.c. positions. Therefore, apply the reversal operator for z.
Now we introduce a time evolution operator Φ, such that o, z(t) z(t + T) for arbitrary t. t E IR satisfies &r, ΦT. + where o denotes composition. The time evolution operator helps us to move forward (when r > 0) or backward (when < 0) through time, thus forming a trajectory. Based on (Lamb and Roberts 1998), in terms of the evolution operator, Equ p jipplies:
R 6e = o_ o R = o, R
(5)
which means that moving forward steps and then turning to the opposite direction is equivalent
to firstly turning to the opposite direction and then moving backwards t step. Eqn 5 has been
widely used describe time-reversal symmetry in existing literature (Huh et al. 2020 Valperga
et al. 2022). Nevertheless, we propose the following lemma, which is more intuitive to understand and straightforward to guide the design of our time-reversal regularizer.
Lemma 2.1. Eqn is equivalent to Ro 6; o R o
where I denotes identity mapping.
Lemma 2.1 means if we move t steps forward, then turn to the opposite direction, and then move forward for more steps, shall restore back the same state. This is illustrated in Figure where the reverse trajectory should be the same as the forward trajectory It can be understood as rewinding a video to the very beginning. The proof of Lemma 2.1 is Appendix A.2
Method: TREAT
We present novel framework TREAT that achieves high-precision modeling for wide range of systems from the numerical aspect, by enforcing Lime-Reversal Symmetry (TRS) via regularization
'Tie--rveesal symmetry is a property of physical systems, which requires the forward and reverse trajectories be generated by the same mechanism F(-). It differs from reversibility of neural networks (Chang al. [2018]
Liu et al. (2019), which is property machine learning models and ensures the recovery of input from output via reversed operator f- (-). We highlight the detailed discussions in Appendix
We explain Figure with implementation in Appendix A.1

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Processor: Learnable ODE function
ODE Solve (g. (Z, ... "N (to)), (fo kx)
Encoder
Decoder
21(0)
(a(
20
(to) 4 (t)
Latent dynamics d7
oond (+) fosc (t)
(t,) (t 1).)
nud (1R-1) (ew)
0
stud(to)
Forward trajectory y!dd
g oy (ooc
0
y teN 1
(tx) (t))
22
02
51th
(ti-1)
Reverse trajectory St
(C))
t 142-tt-2t11
Leeverse Σ
t tK-1fx
Input 9
Time-Reversal Symmetry Constraint
Output Y
Figure 3: Overall framework of TREAT. O2, O2. Og are connected agents. follows the encoder- processor-decoder architecture introduced in Sec 2.1 A novel TRS loss is incorporated improve modeling accuracy across systems from the numerical aspect, regardless of their physical properties.
term. It improves modeling accuracy regardless of systems' physical properties. We first introduce our architecture design, followed by theoretical analysis to explain its numerical benefits.
TREAT uses GraphODE (Huang et al. 2020) the backbone and flexibly incorporates TRS as regularization term based on Lemma 2.1 This term aligns model forward and reverse trajectories. In practice, our model predicts the forward trajectories at a series of timestamps {tx k= ground truth The reverse trajectories are also the
K
observations are discrete, where to < 6 < < l same series of K timestamps so to be aligned with the forward one, which we denote as { ) k=o satisfying 0 = % < <... < txx = T. It's important to note that the values of the time variable t in the reverse trajectories do not represent real time, but serve as indexes of reverse trajectories. This leads to the relation 1'x- = T Lg. which means the reverse trajectories at timestamp 1'x - correspond the forward trajectories time 4.- For example, l = T 1 = 0. indicates I6 and (x are both pointing to the same real time T, which is the ending point of the forward trajectory as shown in Figure
3] Based on Lemma 2.1 the difference of the two trajectories at any observed time should be small, i.c. =_wd((tt s no (2x (k)+)). This serves the guideline for our regularizer design. The weight of the regularizer also adjustable to adapt different systems. The overall framework is depicted in Figure 3
3.1 Tive-Reversal Symmetry Loss and Training
Forward Trajectory Prediction and Reconstruction Loss. For multi-agent systems, we utilize the GNN operator described (Kipf al.) 2018) as our ODE function g(.), which drives the system to move forward and output the forward trajectories for latent states 2fdd(t) (t) at each continuous time t [0, I] and each agent We then employ Multilayer Perceptron (MLP) decoder to predict output trajectories g(dd(t) based on the latent states, We summarize the whole procedure as:
2( dz( g( 2222// (1), 2h) (1)d (1),... 2nd (1));
(6)
z fud((t) = (I 9); gfad(() = fDEc(= (t))
5

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To train the model, we use the reconstruction loss that minimizes the L2 distance between predicted forward trajectories (ff ( k= and the ground truth trajectories (tx(tt)))R as:
N
K
yi(ls) - G. g (ta (tw))
Lpeed
1=1
Reverse Trajectory Prediction and Regularization Loss. We design novel time- tine-reversal symme- try loss as soft constraint to flexibly regulate systems' behavior based on Lemma 2.1 Specifically, we first compute the latent reverse trajectories ="t by starting from the ending state of the forward one, traversed back over time. We then employ the decoder output dynamic trajectories y²t (t).
(t) dz (t)
-9(21" (t), 22 (t), .. (t),
dt
(8)
z/ (1)) = (Lx), (1) = SDEC (2, (z.
Next, based Lemma 2.1 if the system follows Time-Reversal Symmetry, the forward and backward trajectories shall be exactly overlap. We thus design the reversal loss by minimizing the L2 distances between model forward and backward trajectories decoded from the latent trajectories:
Lreverse
(4x) 0. (K
WR
Finally, we jointly train TREAT weighted combination of the two losses:
L = Lpred + aCreverse
(x(tx) g!" a
(fx" - 0f
(10)
where a is positive coefficient balance the two losses based on different targeted systems.
The computational time of Lreverse is of the same scale as the reconstruction loss Lpred- As the computation process of the reversal loss is to first use the ODE solver to generate the reverse trajectories, which has the same computational overhead as computing the forward trajectories, and then compute the L2 distances.
Remark.
3.2 Theoretical Analysis of Time-Reversal Symmetry Loss
We next theoretically show that the time-reversal symmetry loss numerically helps to improve prediction accuracy in general, regardless of systems' physical properties. Specifically, show that minimizes higher-order Taylor expansion terms during the ODE integration steps.
Theorem 3.1. Let At denote the integration step size in an ODE solver and T be the prediction length. The reconstruction loss Lored defined in Eqn 7 is 3 AP). The time-reversal loss Lreverse defined in Eqn is O Att).
We prove Theorem 3.1 in Appendix A.3 From Theorem we can see two nice properties of
our proposed time-reversal loss: 1) Regarding the relationship to At, Creverse is optimizing high- order term A which forces the model to predict fine-grained physical properties such as jerk (the derivatives of accelerations). In comparison, the reconstruction loss optimizes A/2, which mainly guides the model to predict the locations/velocities accurately. Therefore, the combined loss enables model to be more noise-tolerable; 2) Regarding the relationship to T, Creverse is more sensitive to total sequence length SI"), thus it provides more regularization for long-context prediction, a key challenge for dynamic modeling.
TRS Loss Design Choice. We define Creverse the distance between model forward trajectories and backward trajectories. Based on the definition of TRS in Sec. 2.2 there are other implementation choices. One prior work TRS-ODE (Huh et al.) 2020) designed TRS loss based on Eqn 5 where reverse trajectory shares the same starting point as the forward one. However, we show that our implementation based on Lemma 2.T to approximate time-reversal symmetry has a lower maximum error compared to their implementation below, supported by empirical experiments in Sec. 4.2

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Lemma 3.2. Let Lreverse be the TRS implementation of TREAT based on Lemma 2.1 Creverse2 be the one in (Huh et al. 2020) based on Eq When the reconstruction loss defined in Eqn of both methods are equal, and the two TRS losses are equal, i.e. Lreverse Leeversel. the maximum error between the reversal and ground truth trajectory for each agent, i.e. Max Errorr rew max (19.(1x) - ". (12 1,2.... N, made by TREAT is smaller.
We prove Lemma 3.2 in Appendix A.4. Another implementation is to minimize the distances
between model backward trajectories and ground truth trajectories. When both forward and backward trajectories are close to ground-truth, they are implicitly symmetric. The major drawback is that at the early stage of learning when the forward is far away from ground truth (Cpred), such implicit regularization does not force time-reversal symmetry, but introduces more noise.
4 Experiments
Datasets. We conduct systematic evaluations over five multi-agent systems including three 5-body spring systems (Kipf et al.] 2018), complex chaotic pendulum system and a real-world motion capture dataset (Carnegie Mellon University| 2003); and four single-agent systems including three spring systems (with only one node) and a chaotic strange attractors system (Huh et al. 2020).
The settings of spring systems include: 1) conservative, i.e. interactions with the environments, we call it Simple Spring; 2) non-conservative with frictions, we call it Damped Spring: 3) non- conservative with periodic external forces, we call it Forced Spring. The Pendulum system contains three connected sticks in a 2D plane. It is highly sensitive to initial states, with minor disturbances leading significantly different trajectories (Shinbrot et al.. 1992 [Arejcewicz et al. 2008). The real- world motion capture dataset (Carnegie Mellon University (2003) describes the wal walking trajectories of a person, each tracking a single joint. We call it Human Motion. The strange attractor consists of symmetric attractor/repellor force pairs and is chaotic (Sproti 2015). It is also highly sensitive to the initial states (Koppe et al. 2019). We call it Attractor.
Towards physical properties, Simple Spring and Pendulum are conservative and reversible; Force Spring and Attractor reversible but won-conservative; Damped Spring are irreversible and non- conservative. For Human Motion, it does not adhere to specific physical laws since it is a real-world dataset. Details of the datasets and generation pipelines can be found inAppendix C
Task Setup. We conduct evaluation by splitting trajectories into two halves: (M), [Mx++1 [x] where timestamps can be irregular. We condition the first half of observations to make predictions for the second half as in (Rubanova et al. 2019). For spring datasets and Pendulum, we generate iiregular-sampeed trajectories and set the training samples to be 20,000 and testing samples to be 5,000 respectively. For Attractor, We generate 1,000 and 50 trajectories for training and testing respectively following Huuh et al. (2020). 10% of training samples are used as validation sets and the maximum trajectory prediction length is 60. Details can be found in Appendix C
Baselines. We compare TREAT against three baseline types: 1) pure data-driven approaches including LG-ODE (Huang et al. 2020) and LatentODE (Rubanova et al. 2019), where the first one is a multi- agent approach considering pair-wise interactions, and the second one is single-agent approach that predicts each trajectory independently: 2) emergy-preservee HODEN (Greydanus et al. 2019); and 3) time-reversal TRS- (Huh et al. 2020).
The latter two are single-agent approaches and require initial states as given input. To handle missing initial states in our dataset, we approximate the initial states for the two methods via linear spline interpolation (Endre Süli 2003). In addition, we substitute the ODE network in TRS-ODEN with GNN (Kipf et al. (2018) as TRS- TRS-ODENGNN. which serves as a new multi-agent approach for fair comparison. HODEN cannot be easily extended to the multi-agent setting as replacing the ODE function with a GNN can violate energy conservation of the original HODEN. For running LGODE and TREAT on single-agent datasets, we only include self-loop edges in the graph G = (V,E), which makes the ODE function g simple MLP. Implementation details can be found in Appendix D.2
7

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Table Evaluation results on MSE (10-2) Best results are bold numbers and second- results
in underline numbers. Human Motion is real-world dataset and all others are simulated datasets.
Multi-Agent Systems
Single-Agent Systems
Simple Spring
Forced Spring
Damped Spring
Human Motion
Simple Spring
Forced Spring
Damped Spring
Datuset
Pendulum
Attractor
La
5,2622
5.0277
3,3419
2,6894
2.9061
5,7957
0.4563
1,3012
0.58394
HODEN
3.0039
4.0668
8.7950
741.2296
1.9855
3.2119
4.004
1,5675
54,2912
TRS-ODEN
3.6785
4.4465
1.7595
741.4988
0,5400
3.0271
0,4056
1,5667
2,2683
TRS- ODENONN
1,4115
2.1102
0.5951
596.0319
0.2609
/
/
/
LG-ODE
1.7429
1,8929
0.9718
1,4156
0.7610
1.6156
0.1465
1.1223
0. 0.6942
TREAT
1.1178
1,4525
0.5944
1,2527
0.2192
1.6026
0,0960
1.0750
0.5581
(-bllttion of our method with different implementation of Lreserse
TREAT ge--ee
1,1313
1.5254
0.6171
1.6158
0.2495
1.6190
0.1104
1.1205
0.6364
TREAT
1.6786
1.9786
0.9692
1.5631
0.8785
1.6901
0.0983
1.0952
0.7286
4.1
Main Results
Table 1 shows the prediction performance on both multi-agent systems and single-agent systems measured by mean squared error (MSE). We can see that TREAT consistently surpasses other models, highlighting its generalizability and the efficacy of the proposed TRS loss.
For multi-agent systems, approaches that consider interactions among agents (LG-ODE, TRS- ODENGNN. TREAT) consistently outperform single-agent baselines (LatentODE, HODEN, TRS- ODEN), and TREAT achieves the best performance across datasets.
The chaotic nature the Pendulum system and the Attractor system, with their sensitivity to initial states 0 poses extreme challenges for dynamic modeling. This leads to highly unstable predictions for models like HODEN and TRS-ODEN, as they estimate initial states via inaccurate linear spline interpolation (Endre Süli 2003). In contrast, LatentODE, LG-ODE, and TREAT employ advanced encoders that infer latent states from observed data and demonstrate superior accuracy. Among them, TREAT achieves the most accurate predictions, further showing its robust generalization capabilities.
We observe that misapplied inductive biases can degrade results, which limits the usage of physics- informed methods that are designed for individual physical prior such as HODEN. HODEN only excels on energy-conee conreycvnsevvttivee systems, such as Simple Spring compared with LatentODE and TRS- ODEN in the multi-agent setting. Its performance drop dramatically on Force Spring, Damped Spring, and Attractrr. Note that HODEN naively forces each agent to be energy corryy-conservative, instead of the whole system. Therefore, it performs poorly than LG-ODE, TREAT in the multi-agent settings.
For the Human Motion dataset, characterized by its dynamic ambiguity as it does not adhere to specific physical laws, we cannot directly determine whether it is conservative or time-reversal. For such system with an unknown nature, TREAT outperforms other purely data-driven methods significantly, showcasing its strong numerical benefits in improving prediction accuracy across diverse system types. This is also shown by its superior performance on Damped Spring, which is irreversible.
LE-ORF
LGGOF
LG-DDE TREAT
0.02
0.03
TIS-OGEN
TREAT
HODEN
0.04
0.05
HODEN
TREAT
TRS-CIEEN
0.00
0.02
TIR--DEN HODEN
MIE
TItS-DOON
1G-00F
TREAT
HO01h
0.03
0.02
20
40
30
60
20
30
60
50
60
20
30
40
60
Prediction Lengths
Prediction Lengtns
Lengths
Lengthe
(a) Sumple Spring
(b) Damped Spring
(e) Forced Spging
(d) Pendulum
Figure 4: Varying prediction lengths across multi-agent datasets (Pendulum MSE is in log values).
Video show Pendulum is highly sensitive to initial states.
8

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0.015
0.009
0.218
0.018
0.014
TREAT LB ODE
TREAT L6 00E
0.008
000214
0.016
0.013
0.007
0.01
5.02
6.02
6.1
0.02
0.012
500
1000
Simple Spring
(b) Dumped
Forced Spring
Pendulum
Figure 5: Varying a values across multi-agent datasets.
Ablation and Sensitivity Analysis
Ablation on implementation of Creverse. We conduct two ablation by changing the implementation
of Creverse discussed in Sec. 3.2 1) TREAT C... Etrre-g-rev which computes the reversal loss as the L2 distance between ground truth trajectories to model backward trajectories; 2) TREAT C.ey which implements the TRS loss based on 5las in TRS-ODEN (Huh al. 2020). From the last block of Table 1 we can clearly see that our implementation achieves the best performance against the two.
Evaluation across prediction lengths. We vary the maximum prediction lengths from 20 60 and report model performance as shown in Figure 4 As the prediction step increases, TREAT consistently maintains optimal prediction performance, while other baselines exhibit significant error accumulations. The performance gap between TREAT and baselines widens when making long-range predictions, highlighting the superior predictive capability of TREAT.
Evaluation across different a. We vary the values of the coefficient a defined in Eqn 10 which balances the reconstruction loss and the TRS loss. 3 being neither too high nor too low. This is because when a is too small, the model tends to neglect the TRS physical bias, resulting in error accumulations. Conversely, when becomes large, the model can emphasize TRS at the cost of accuracy. Nonetheless, across different values, TREAT consistently surpasses the purely data-driven LG-ODE, showcasing its superiority and flexibility in modeling diverse dynamical systems.
that the optimal values
We study TREAT's sensitivity towards solver choice and observation ratios in Appendix E. and Appendix E.2 respectively.
Courrd Truh
TRRAT
L.G-OO
FOOGFN
Daangy
3,5
3.50
1.0
3.00
0.6
0.6
5.0
0.4
2.2.5
-0.5
-0.5
330
02
20
0A
0.8
-22
0.2
375
02
0.3
0.0
191 Uarped Speen
2.88
L$ ODE
TREAT
380
0.6-
HODEN
322
6.4
3.0
330
Fo.2
3.79
375
-64
-33
0.
02
0.4
-01
-2.2
0.
0.6
1.5
0.8
Sysunn Erooe
1.0
ce.
0.2-
0.0
1292
-04
0.4
0.0
-05
0.5
2.5
3.5
0.5
0.0
Figure 6: Visualization for 5-body spring systems (trajectory starts from light dark colors).

---

KTente
Xlaantz
X1e
TIEEAT bevern Late
3.0
TREAT
Lo o0c Tevrrall Lole
TREAT Severos leet
2.0
TREAT
TT teve Ligi
1.00
HRAA
TREAT LG E6-00E
LG ODE
LG ODE
LG tanerral
0.6
9.4
20
40
60
25
40
10
20
30
50
100 Epochs Pendulum
150 200
Simple Spring
Damppd
Ferced Spring
Figure 7: TRS loss visualization across multi-agent datasets (scales of two y-axes are different).
4.3 Visualizations
Trajectory Visualizations. Model predictions and ground truth are visualized in Figure 6 As HODEN single-agent baseline that individually forces every agent's energy to be constant over time which is not valid, the predicted trajectories is having the largest errors and systems' total energy not conserved for all datasets. The purely data-driven LG-ODE exhibits unrealistic energy patterns, as seen in the energy spikes in Simple Spring and Force Spring. In contrast, TREAT, incorporating reversal loss, generates realistic energy trends, and consistently produces trajectories closest to the ground truth, showing its superior performance.
Reversal Loss Visualizations To illustrate the issue of energy explosion from the purely data-driven LG-ODE, we visualize the TRS loss over training epochs from LG-ODF and TREAT in Figure 7 As results suggest, LG-ODE has increased TRS loss over training epochs, meaning it violating the time-reversal symmetry sharply, in contrast to TREAT which has decreased reversal loss over epochs.
5
Conclusions
We propose TREAT, deep learningframework that achieves high-precision modeling for wide range of dynamical systems by injecting time-reversal symmetry as an inductive bias. TREAT features novel regularization term to softly enforce time-reversal symmetry by aligning predicted forward and reverse trajectories from a GraphODE model. Notably, we theoretically prove that the regularization term effectively minimizes higher-order Taylor expansion terms during the ODE integration, which serves as a general numerical benefit widely applicable to various systems (even irreversible systems) regardless of their physical properties. Empirical evaluations on different kinds of datasets illustrate TREAT's superior efficacy in accurately capturing real-world system dynamics.
6 Limitations
Currently, TREAT only incorporates inductive bias from the temporal aspect, while there are many important properties in the spatial aspect such translation and rotation equivariance (Satorras et al.) 2021 Han et al., 20226] Xu et al. (022). Future endeavors that combine biases from both temporal and spatial dimensions could unveil new frontier in dynamical systems modeling.
Acknowledgement
This work was partially supported by NSF22002 NSF 2106859, 23F2501 NSF 221 1551, NSF 1937599, NSF 2119643, NSF 2303037, NSF 2312501, DARPA 1RR0012290100/HRROO 1260656, HR0011 12490370, NIH U54HG012517, NIH U24DK097771, NASA, SRC JUMP 2.0 Center, Amazon Research Awards, and Snapchat Gifts.
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14

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A Theoretical Analysis
A.1 Implementation of the Time-Reversal Symmetry Loss
Algorithm The implementation of Crenerse
Require: latent initial states <to) the ODE function g(-); number of agents N: 1: for each i N do
2: Compute the latent forward trajectory at timestamps {th, K
Fiud (tx) ODE-Solver(g, [2f =x" (to).... fie (to)), tx). Reach the final state (tx((tt).
3: The initial state of the reverse trajectory is defined as 2. ";" (1) x; =tx( and the dynamics
the system which is the ODE function g(-) is also reversed as -9(-)-
4: Compute the latent reverse trajectory at timestamps {t/ 11 k=
21Th (t() ODE-SODE ODE-Solver(s) (xT"(()) 2 (t/) (t,))... (t)).. ()).
5:
gf (fx) (2," D.thh (t%) = fDEC (z. (1)))
6: end for
(tx - D. (t'x ||
7: Creverse = EN Ex= K
A.2 Proof of Lemma!
Proof. The definition of time-reversil symmetry is given by:
R d. Q_ R o, R Here, R is an involution operator, which means Ro o R = I. First, we apply the time evolution operator both sides of Eqn 11
(11)
Φe o R th = 2e o o0 R
(12)
Simplifying, obtain:
Φ: Ro 0. R
(13)
Next, we apply the involution operator both sides of the equation:
Ro or R d, Ro
(14)
Since R R we finally arrive at:
Ro 0i R di =1
(15)
which means the trajectories can overlap when evolving backward from the final state.
A.3 Proof of Theorem 3.1
Let At denote the integration step size in ODE solver and T be the prediction length. The time stamps of the ODE solver are (ty) where f5+1 tj = At for = 0, ... , > 1). Next suppose during the forward evolution, the updates go through states 2nd
pl))
for j 0, ... T, where (0x((1,) is position, ( is momentum, while during the reverse (tz) = (
for j = O, ... T, in reverse order. The ground truth trajectory is (t,) = (t,), (t)(t) for ......T.
evolution they go through states (0))(1)))
For the sake of brevity in the ensuing proof, we denote 28 (t₂) by zff =)))) by 2fnnd and = by 2it and we will use Mathematical Induction to prove the theorem.
A.3.1 Reconstruction Loss (Cprea) Analysis.
First, we bound the forward loss E/= ||z - 2 I1. Since method models the momentum and position of the system, write the following Taylor expansion of the forward process, where
15

---

for any 0 s
9j+1 = qj
fwd + (p) /mm + (p) /2m) +((F)),
(16a)
fwd = P/j P fwd Al + 0( PJ+1
(16b)
Pj+1 = Pj + (
(16c)
and for the ground truth process, we also have from Taylor expansion that
2t 9j+1 gj + (pf /m) (p) At O(At)),
(17a)
p Pÿ + +D, D O(At))
(17b)
p P + (At)).
(17c)
With these, we aim to prove that for any = 0, ... T. the following hold :
l fwd 1/l12 1 Cfvd x22 At?,
(18a)
lipk swd p 12 5 Cdwd kAt,
(18b)
where CTwd and C2 are constants.
Base Case k = O: Based on the initialization rules, is obvious that ||
0 00'1/ = 0 and
[pp p ll2 0, thus (8ab and (18b) both hold for k 0.
Inductive Hypothesis: Assume [1aa and 18b) hold for j. which means:
|g fwil
0 112
Chud J² j² Ar?;
(19a)
1pp lod pf S Cfwd jt
(19b)
Inductive Proof: We need to prove (8a) and 18bb hold for h = +
First, using (lcc) and (17c), we have
1 pf 1l1 O(At) pfud P ll + O(Ú + 1)Ar) O(1),
(20)
where we iterate through j.j 1,... in the second equality. Then using (17b and (6bb) we get for j that
fwd
1
31
= (p fud + + fid At) (p) + DJ At) + +
< l1p fudd 5 + ||0) 13 Al+ Al+
< j + (()) AL,
where the first inequality uses the triangle inequality, and in the second inequality we (99)) as well as (20). We can see there exists Cfwd such that the final expression above is upper bounded by Ofwed (j At. with which the claim holds for +
Next for (1a)) using
17a and (6a). we get for any that
of 1
(g (p, /m]At /2mmmmmmm))))
(Øz + (m/5/m)Att (13)) (25)22m)P))) O(AP))2
of
1/222 m
_
||p
rad 1b
1 |1p!5mm 15 11, + +(Ar))
2m
Crud
O(1) At?
where the first inequality uses the triangle inequality, and in the second inequality (1a)) and 19bb as well as (20). Thus with an appropriate Cfudd we have the final expression above is upper bounded by Chied G 1)2 and the claim holds for j + 1.
Since both the base case and the inductive step have been proven, by the principle of mathematical induction, [8al and (18b) holds for all l 0, T.
16

---

With this, we finish the forward proof by plugging (8a) and
(8b) into the loss function:
Σ || 2
p Rwd
p
2
twd
oj 112
jm
T (Cfru)
ΣEP + (C2mu) 2
5
O
A.3.2 Reversal Loss (Creverse) Analysis.
Next we analyze the reversal loss Σj= [|[(2)))) expansion residual terms for more in-depth analysis.
2|nd1 For this, we need to refine the Taylor
First reconsider the forward process. Since the process is generated from the learned network, may assume that for some constants CJ, C22 and C3, the states satisfy the following for any S j < T:
q
fwdd
fwd 4t+1
(P5+1/ (D)+1/m)Ar)) /mmmm/ + (P)+1 Amm rem, fwd,3
fud
(2
P) fwdd
Pj+1
fwd Al + rem fwd,2
(21b)
p
Pf11
rem
fwd,1
(21c)
where the remaining terms Taylor expansions for the reverse process:
rem fwd,é
|| S At for i = 1,2, 3. Similarly, we have approximate
9 eov Jov + (Df+1/ / + (D)f1/2m))) + +emm, rov.3
(22a)
p3" pj+1 pay At rem, 2.
(22b)
D." =pft1 + rem; rev.
(22c)
where
[rem]" rev.á
112 S At for i = 1,2,3.
We will prove via induction that for k = T,T 1, ... ,,
[ 4%
fwd 2l I C3 (T - At,
(23a)
[[((p)))
Pk fwd < C (T' At)
(23b)
[[((P)))
Pk
fwd
S C (T A)
(23c)
where CToV, CITy and Cg are constants.
The entire proof process is analogous to the previous analysis of Reconstruction Loss.
Base Case A T:: Since the reverse process is initialized by the forward process variables at k T.
fwdd is obvious that 4 (2cc) all hold for k = 0.
47t l|2 = PT
fwdd
rev
Pr
PT
|| 0. Thus (3a). (3b) and
Inductive Hypothesis: Assume the inequalities (3b). (23a) and (3c hold for j + 1, which means:
[[(()H+1) / 112 < C3 (T (j + 1))
(24a)
11 < C2 (T (j + A1? [R(1)H1)
twd
(24b)
[R(p)
p lll S C7 (T (j + 1))
(24c)
Inductive Proof: We need to prove (3b
23ah and (3cc holds for = j.
First, for 23c
using
21c and 22c
we get for any j that
R(P)"))
||(2T==1 + rem, rev, - (p)+1 + rem, Ford1 >/1
||R( p/y ||, + ||rem rev.1 l12 + ||rem Fwd. ll2
CT (T - -j-- 1--144 1) + 2c,At,
17

---

where the first inequality uses the triangle inequality, and the second inequality plugs in (24c). taking CT 2c,, the above is upped bounded by C7 (T jt,t, and (23b) holds for y.
Thus
Second, for (2b), using 2 and 22b). we get
[ p. 11 -11 - (pf71 + 13511 A rem, (e ") (p)41 p At rem; remm) la
< B(p521) p + [((5+1) D5+ 11 A ||rem,
112 + |rem) fod. 112
C (I -j 1)2 + CT
1) + 20g At..
where the first inequality uses the triangle inequality, and in the second inequality we use (24a) and (24)). Thus taking C2 = max (CTT/ /2, 202). we have the final expression above is upper bounded by C2 (T Al?, and so the claim holds for j.
Finally, for (24a), we use (2 and (2a) get
[ q 2
[1fff (pF+1 / /m) At + /2m) + rem, je.....)) (gft1 (p/ddd /mm /mmmmmmmttt + (prmd /2mm At
rrem" fvd.3) l12
[((]tt)) - 1j+tl11
fidd
112
1
[[
Pj+ill2 11. At + 2m
fwdd
P CT (T j 1) + 20g A+3.
-
fwd 11 At + |rem,
;
rev.3
|rmm
fwdd3 ||
2 (T j 1)² + 2m
03 (T - 1)at
m
where the first inequality uses the triangle inequality, and in the second inequality we 24
24b
/6m, 2c3]. we have the final expression above is and (24c). Thus taking C = max { C2 /3m,, CT0//m, upper bounded by Ci (T Ats, and the claim holds for j-
Since both the base case and the inductive step have been proven, by the principle of mathematical induction, (33)), (23a) and (23c) hold for all k T,T I,... O.
With this we finish the proof by plugging 23bb and (23a) into the loss function:
r
T
T
[R(2
nev
2]
fud
12
=
[R(p))
- p)) fid 112
RR
- 9f find
112
1=0
5=0
1=
(25)
$ (Cror)
(T [))))aar* (Cgov) '07 (T [T)))Aar A
j=0
j=0
A(T)
A.4 Proof of Lemma 3.2
TREAT: R o & o R Φ =1
TRS-ODEN: R $: = +-+ o (
(()))(())) (1)
(0)
( B (
st (0) xx()
x((1)
year m100 y/"
Eprnt (1) -y((1) I1; =
Eyred [1))pe(()) - y1(1) 16 :
Lyswe (1(0) (o I =b
Emowrnes = || (1) sp 11 =
(-1)
Ground truth Trajectory
MaxError TREAT max(a,b) MaxError TRS-ODEN +
Forward Trajectory
Reverse Trajectory
Figure 8: Comparison between two reversal loss implementation
18

---

We expect an ideal model to align both the predicted forward and reverse trajectories with the ground truth. As shown Figure we integrate one step from the initial state g(vd(() (0) (which is the same as (0)) and reach the state (1).
The first reverse loss implementation (ours) follows Lemma 2. as R 2 o Ro 2, = I. which means when evolve forward and reach the state (1( we reverse into (-1) R(g, (1) and go back to reach y!ah (0), then reverse it to get ((j("")) (0)), which ideally should be the same as (0))
The second reverse loss implementation follows Ean reverse the initial state as (0) (0),, then evolve the reverse trajectory in the opposite direction to reach g,-2 (-1). and then perform symmetric operation to reach g,a2(()) with the forward trajectory.
4 = t- o R. which means we first
(1), aligning
We assume the two reconstruction losses Lpred (1) yı (1)1lB
the same. For the
rime-reversal losses, we also assume they have reached the same value b:
Lreverse
|R(D (0)) (0)) + ([(00 (-1)) (1) = [ " (0)) D. ( (0) b,
Creverse2 10: (0) D: 2 + 21 (1) (1)1( 1 (1) y ( (1)11 := b,
(wd
As shown in Figure 8 where we illustrate the worst case scenario Mar Error gt new
max [1:; D + TREAT and TRS-ODEN, we can see that in our implementation the worst error is the maximum of two loss, while the TRS-ODEN's implementation has the risk of accumulating the error together, making the worst error being the sum of both:
Max ErroPTAAT = max R(D (0)) - (:(0))l: R - yx(1) = max a, b),
Mox MozErrorTRS.ODEN max [100:"²00
(1) -x(())
max { (-1)) (1)
= ||0,0 g. (1)]1 + |0 (1) y(1))22 = a +
(26)
So it is obvious that MarErTOTTEAT made by TREAT is smaller., which means our model achieves a smaller error the maximum distance between the reversal and ground truth trajectory.
B Example of varying dynamical systems
We illustrate the energy conservation and time reversal of the three n-body spring systems used in our experiments. We use the Hamiltonian formalism of systems under classical mechanics describe their dynamics and verify their energy conservation and time-reversibility characteristics.
The scalar function that describes a system's 's motion is called the Hamiltonian, H. and is typically equal to the total energy of the system, that is, the potential energy plus the kinetic energy (North) 202T) It describes the phase space equations of motion by following two first-order ODEs called Hamilton's 's equations:
dq HH(q. p) dp 0H(q. p)
(27)
dt Op dt
Oq
where q E R" .P E R", and H: R2a + IR are positions, momenta, and Hamiltonian of the system.
Under this formalism, energy conservative is defined by dHH/dt O, and the time-reversal symmetry is defined by H(g. p,t,,,t) p)t) H(q. p,, -t) Lamb and Roberts 1998).
B.1 Conservative and reversible systems.
A simple example is the isolated n-dydy spring system, which can be described by:
dqp pi
dt
dp; Σ -kqi--pp
(28)
dt
JEN.
where q = (qı 121.qq.... q.. qN)) a set of positions of each object, , p = (P1, Pa, " PNn) is a set of momenta of each object, m, is mass of each object, à is spring constant.
19

---

The Hamilton's 's equations are:
HH(q. p)
dq: Pi
0pi dt
(29)
0H(q. p)
dpi
Σ k(q - qj),
dqi dt
JEN:
Hence, we can obtain the Hamiltonian through the integration of the above equation.
N
1
H(q.p) p)
k(q: -t)²
(30)
2m;
2
i=1
JEN;
Verify the systems' energy conservation
d'H(q.p)
N
N
N
Σ
k( - -1)))== =0, 0,
(31)
dt
dt 2m,
dt
t=1
JEN,
So it is conservative.
Verify the systems' time-reversal symmetry We do the transformation R: (q. + (q. PP -t).
N
2
N
N
H(q.p) =
Pi
k(qi -q)²
2m14
2
i=1
JEN;
(32)
N
p2)
N
H(q. -p)
2m14
+
2
k(q - 90)
1=1 JEN,
It is obvious H(q. p) = H(q. -p), is reversible
B.2 Non-conservative and reversible systems.
simple example n-body spring system with periodical external force, which can be described by:
dqi Pi
dt m
(33)
dp: k(q: qj) - ki cos wl,
dt JEN.
The Hamilton's equations are:
0H(q. p) dqi Pi
Op: dt
m
(H(q.)) p)
dpi Σ JEN.
(34)
k(q:
qi) kn coswl,
0q: dt
Hence, we can obtain the Hamiltonian through the integration of the above equation:
N
N
N
H(q.
pi
k(q
q1)2+
a k coswt,
(35)
2m,
2
1=1
i=
t=1
Verify the systems' energy conservation
dH.(p.))
1
N
N
N
k(q
4i)
+ dt.
ni k cos coswl) wl)
dt.
dt
2m
JEN;
=0
g:hn cos woswt)
(36)
dt
wgiki sin o
20

---

So it is non-conservative.
Verify the systems' time-reversal symmetry We do the transformation R (q.,,)) + (q. -p- -1).
2
N
Pi
H(q.p)
k(q
+
qi 1 cos wt,
2m14
i=1
i=1
JEN.
1=1
(37)
N
?
N
N
N
H(q. =
+
k(q
- -u)) +
Σ
qi * h1 cos (-t),,
2m
2
2
i=1
JEN;
1=1
It is obvious H(q, p.t) H(q. -p.t). it is reversible
B.3
Non-conservative and irreversible systems.
simple example is n-body spring system with frictions proportional its velocity,y is the coefficient of friction, which can be described by:
dq: P
dt
m
(38)
dp; koq:
- Pi
dt
The Hamilton's 's equations are:
8k(q.)) p) dq: Pi
0p: dt m
(q. dpi = Σ
(39)
k(q: -q1 +
dqu dt JEN,
Hence, we can obtain the Hamiltonian through the integration of the above equation:
H(q p)
N
Pr
2
N
N
k(q:
qi) 2 +
7 p.
dl,
(40)
2m,
2
m11
0
m
i=1
JEN,
Verify the systems' energy conservation
dH(p.p)
1
pi
N
k(q
-1)))
+
1
N
Pi
2
dt)
dl
dl
2m
2
2
dl
m
Jn
m
JEN,
-0
+ di
Σ
dt)
(41)
171
1=1
7 Pi
m m
So it is non-conservative.
Verify the systems' time-reversal symmetry We do the transformation R (q.p.t) + (q. -p, -t).
H(q.p)=
N
Pi
2
N
k(q
N
Pi
dl.
2m,
2
2
J0
i=1
i=1
JEN,
(=1
(42)
H(q. -p) =
N
(-pi)2
N
N
k(q
-1)²
++
N
(-t)
pi dl d(-t). t)
2m;
2
2
Jo
1=1
1=1
JEN;
-=1
It is obvious H(q.p,t) H(q. -p,t). t).
it is inreversibee
21

---

C Dataset
In our experiments, all datasets are synthesized from ground-truth physical law via sumulation. We generate five simulated datasets; three n-body spring systems under damping, periodic, or no external force. one chaotic tripe pendulum dataset with three sequentially connected stiff sticks that form and chaotic strange attractor. We name the first three as Sipmle Spring, Forced Spring. and Damped Spring respectively. For multi-agent systems, all n-body spring systems contain interacting balls, with varying connectivities. Each Pendulum system contains 3 connected stiff sticks. For single-agent systems, all spring systems contain only one ball. For the chaotic single Attractor, we follow the setting of (Huh et al., 2020).
For the n-body spring system, we randomly sample whether a pair of objects are connected, and model their interaction via forces defined by Hooke's law. In the Damped spring, the objects have an additional friction force that is opposite to their moving direction and whose magnitude is proportional to their speed. In the Forced spring. all objects have the same external force that changes direction periodically. We show in Figure [a). the energy variation in both of the Damped spring and Forced spring is significant. For the chaotic triple Pendulum the equations governing the motion are inherently nonlinear. Although this system is deterministic, it is also highly sensitive to the initial condition and numerical errors (Shinbrot et al., 1992, Awrejcewicz et al., 2008, Sttachoiikk and Okada, 2006). This property is often referred to the "butterfly effect". as depicted in Figure 9 Unlike for n-body spring systems, where the forces and equations of motion can be easily articulated, for the Pendulum, the explicit forces cannot be directly defined, and the motion of objects can only be described through Lagrangian formulations (North] 202T), making the modeling highly complex and raising challenges for accurate learning. We simulate the trajectories by using Euler's 's method for
Original mitial condition: 9g 1e-3 pertachation:
10-
le2 perturbation:
02
Joint 0
10
100 200
300
400
500
50U
70U
Time siaps
Figure 9: Illustration show the pendulum is highly-sensitive to initial states
n-body spring systems and using the 4th order Runge- Kutta (RK4) method for the Pendulum and Attractor.. For all spring systems and Pendulum, We integrate with a fixed step size and subsample every 100 steps. For training, we use total of 6000 forward steps. To generate irregularly sampled partial observations, we follow (Huang et al. (220) and sample the number of observations from a uniform distribution U(40, 52) and draw the observations uniformly for each object. For testing, we additionally sample 40 observations following the same procedure from PDE steps [6000, 12000], besides generating observations from steps [1, 6000]. The above sampling procedure is conducted independently for each object. We generate 20k training samples and 5k testing samples for each dataset. For Attractor, we integrate total of 600 forward steps for training and subsample every 10 steps. For testing, we additionally sample 40 observations from step [600, 1200].The The irregularly sampled partial observations generation the same as above. We generate 1000 training samples and 50 testing samples following (Huh et al. 2020). Therefore, for all datasets, condition length is
22

---

60 steps and prediction length is 40s steps. The features (position/velocity) are normalized to the maximum absolute value of I across training and testing datasets.
We also compute the Maximum Lyapunov Exponent (MLE) to assess the chaos level of the systems, using the formula:
|6(1)l1 ||8(0)]
X = maz In
We set fixed initial values for each dataset and generate 10 trajectories by perturbing the initial values with random noise (0, 0.0001). We calculate the Maximum Lyapunov Exponent (MLE) between any two trajectories. Finally, compute the average and std of MLE from all pairs to gauge the chaotic behavior of each dataset. The data is presented in the table below:
Table 2: MLE of different Multi-agent Systems
Dataset
Simple Spring
Forced Spring
Dumped Spring
Pendulum
MLE(in 60 steps)
0.4031 + 0.3944
1.0087 1.0577 0.6307 + 0.7065 34.1832 + 30.1446
From the table, it's evident that the order of MLE values is: Pendulum > three Spring datasets. This observation is consistent with the evaluation results based MSE presented in previous responses in Table 3 which indicates that as the prediction length(steps size) increases, there is a more significant performance degradation of all models Pendulum dataset.
In the following subsections, we show the dynamical equations of each dataset in detail.
C.1
Spring Systems
C.1.1 Simple Spring
The dynamical equations of simple spring are as follows:
dq: Pi
dt
(43)
dp: Σ
k(q: qj)
dt
JEN.
where where (q1: 92: ................................ set of momenta of each object. We set the mass of each object m 1, the spring constant, = 0.1.
9N) is set of positions of each object, p (P1. (P1.P2... P2....
.PN) is
C.1.2 Damped Spring
The dynamical equations of damped spring are as follows:
dqi m
dt
dp: Σ kbqq: 9i) -
Pi
(44)
dt JEN;
m
where where q (qı ....... qN) is set of positions of each object, (P1, P2. .p..:..,N)) is
set of momenta of each object, We set the mass of each object m = 1, the spring constanti = 0.1. the coefficient of friction 7 10.
C.1.3 Forced Spring
The dynamical equations of forced spring system are as follows:
dqi P
dt m
(45)
dpi Σ
-k(qi
qi)
ki coswt,
di JEN.
23

---

where where = (q1. Q2: 9N) is set of positions of each object, (P1. Pa. PNN is of momenta of each object. We the mass of each object 1, the spring constantk 0.1. the external strength k 10 and the frequency of variation =
We simulate the positions and momentums of three spring systems by using Euler methods follows:
qi(4 1) qi(() + dqi Pt(t + 1) pt(t)
dt
(46)
dp:
At
dt
where da, and dp: were defined as above for each datasets, and At = 0.001 is the integration steps. di
di
C.2 Chaotic Pendulum
In this section, we demonstrate how to derive the dynamics equations for a chaotic triple pendulum using the Lagrangian formalism.
The moment of inertia each stick about the centroid is
1 ml?
1
(47)
12
The position of the center of gravity of each stick is as follows:
T1 sin 01, y1 = cos 01
2
T2 I(sin 02 sin 02), y/2 = =1(os000 cos 02)
(48)
1
1
T3 [(sin 0 sin 82 sin 0s), ys -I(cos 01 + cos 02 2
cos 0y)
The change in the center of gravity of each stick is:
i 2 cas 01 01. ý1 sin 01
12 = l(cos 01 01 + 2
cos 02 62). $2 M(sin 01 sin 02 (2)
(49)
1
1
(ccos 01 01 cos 02 cos Os 0s), ús l(sin 01 01 sin 02 02 + sin Og 0s))
2
The Lagrangian of this triple pendulum system is:
=7
m(a? 2 + 2622 xis" + vh² + 1/2 vss") + 100 + +02² + -mg(Un + U/2 + Us)
(50)
ml(90200 cos(O) 02) 30.07 1 cos (0, 03) + 30,0gl cas (02 0s) + 1-707 + 4031 + 031
15g cos (0: + 99 cos (02) 3g cos (0s))
The Lagrangian equation defined as follows:
0c
Oc
(51)
00
00
and we also have:
0C OT
p
00
00
(52)
dC
OC
di ai
00
where p is the Angular Momentum. can list the equations for each of the three sticks separately:
ØL
aL
P1 =
001
p1
001
ØC
dL
P2
122
(53)
002
802
Oc
Ac
P3 00g
P 00g
24

---

Finally, we have
01 6(9p) cosl (2(0) P3)) 22pp cool (1) - 0b) 9p2 cosl 01 1000------000000 200 20:)-+1ppp cool (1 0g) cosl -0,
mi? [8] cosl cos(2(6; 65 ) 9 rosl 2 0b)
45 cool 2 [5]] 1 169
0₂ 6(27p; cos(0, 02) 9pt cos(
0g 20, ) 9pp cost 2 11 0g) p7 cos 20, 02 0g) +57ps cas 02) p7p
ml( (81 ( 2 U2 ) cos 11 0a + 45 cost (200, 03)) 169)
0g 6/21p; cos(0i -7pp cus (0 -26, -28,40.) +0,) 27p2 co0 -8, 0.) +57pz cos (8, 10 +.)+ cos(2(0)
m coo (20 B2)) 2 -100000111--111111111)10) ((42-
pi - (30201 sin (0) 02) + 0.0g1 sin (01 02) + 5g sin (01)
P1 ml (-30 fyl sin (01 82) fg0g1 sin (02 0z) + 3g sin (02)
pr ml (0: f11 sin (0: 0z) f24g1 sin (02 0s) g sin (0s)
(54)
We simulate the angular of the three sticks by using the Runge-Kutta 4th Order Method as follows:
(0) O(t, 0()) At
At 0(t 2 (t) + A0'(t) At
At
(t = 0(t + At
2 (t) + A0) At
2
(55)
At 6(t + At, (t) (0) At
AO - (A00 (1) + (0 + A0
(1) + (1)
( 1) - (t) + AL(
where Ó was defined as above, and A 0.0001 is the integration steps.
C,3
Chaotic Strange Attractor
The dynamical equations of this reversible strange attractor follows:
dr dt + yz, dy -I2, dt
(56)
dz
y²² + 22yz,
di
x. y. x E R
The above equations can presented as (d(t), ((t), z(t)) Dynamic(z(t), x()).
We simulate K(t)) = (a(t), v(t), ztt) by using the Runge Kutta 4th Order Method as follows:
AK, (t) = D (ymmmi(((())) (K * At
AKy(t) = (y)mmc((((t) + AK1(t) (t)
At
2
AKa(() = D (mamic(((() + 4K2(t)
Al
2
(57)
(Ka = D Dynamic (K(() + AK₃ At 1
(K - (AK + AK2(t) AKs(t) + AK)())
6
K(I K(I) AK(!)
We sampling =(40) randomly from uniform distribution [1, while fixing x=(o) y(()) = 0. We set the trajectory lengths of both training and test dataset to 600, with regular time-step size At 0.03 and the sample frequency of 10. We add Gaussian noise 0.05m, *N(O,, 1) to training trajectories.
25

---

C.4
Human Motion
For the real-world motion capture Mataset(Carnegg Mellon University 2003). we focus on the walking sequences of subject 35. Each sample in this dataset is represented by 31 trajectories, each corresponding the movement of single joint. For each joint, we first randomly sample the number of observations from a uniform distribution U(30, 42) and then sample uniformly from the first 50 frames for training and validation trajectories. For testing, we additionally sampled 40 observations from frames [51,90 99).We .We split different walking sequences into training (15 trials) and test sets (7 trials). For each walking sequence, we further split it into several non-overlapping small sequences with maximum length 50 for training, and maximum length 100 for testing. In this way, we generate total 220 training samples and 27 testing samples. We normalize all features (position/velocity to maximum absolute value of I across training and testing datasets.
D
Model Details
In the following we introduce in details how we implement our model and each baseline.
D.I Initial State Encoder
For multi-agent systems, the initial state encoder computes the latent node initial states (4) for all agents simultaneously considering their mutual interaction. Specifically, it first fuses all observations into a temporal graph and conducts dynamic node representation through a spatial-temporal GNN as in (Huang et al. 2020
n3es to h+1 (())
Σ
I a01))1000 W.h
((MDEN))))
(110-10) (Wahh i(t))
W W 1-1 ((0)
h111 i(C))
h h/1 ((e)) + TE(t' -t)
(58)
Vd
At
A
TE(At)2. sin 100002:/4
TE(At)2141 cos
100002:/4
where || denotes concatenation; (-)) is a non lonerr activation function; d is the dimension of node embeddings. The node representation is computed as a weighted summation over its neighbors plus residual connection where the attention score is a transformer-baset (Vaswani et al. 2017)
dot-product of node representations by the use of value, key, query projection matrices W.. W k. W
Here h (e) is the representation of agent j at time in the 1-th layer. i(t') is the general index for neighbors connected by temporal edges (where t' A t) and spatial edges (where
and * j). The temporal encoding (Hu et al. 2020) is added to a neighborhood node representation in order to distinguish its message delivered via spatial and temporal edges. Then, we stack layers to get the final representation for each observation node: h! = h Finally, we employ self-attention mechanism to generate the sequence representation u, for each agent as their latent initial states:
hiht) a, = -aah ((> K
1 K
( a (
E
W
(59)
where a; is the average of observation representations with a nonlinear transformation Wa and hg = h + TE((). K is the number of observations for each trajectory. Compared with recurrent models such as RNN, LSTM (Sepp and Jürgen, 1997). it offers better parallelization for accelerating training speed and in the meanwhile alleviates the vanishing/cxploding gradient problem brought by long sequences. For single-agent Systems, there only left the self-attention mechanism component.
Given the latent initial states, the dynamics of the whole system are determined by the ODE function 9 which parametrize a GNN in (Huang al.] 2020) for Multi- Agget Systems to capture the continuous interaction among agents. For single-agent systems, we only include self-loop edges in the graph g (V,E), which makes the ODE function y a simple ML..
We then employ Multilayer Perceptron (MIP) as a decoder to predict the trajectories (.()) from the latent states (t).
26

---

(t), (2((), ... NN (t) ODEsolver(a, (=1(to), =2(to) =N (t0, 1 -..))))
(60)
(t) fdee (z)(t)
D.2 Implementation Details
TREAT
For multi-agent systems, our implementation of TREAT follows GraphODE pipeline. We implement the initial state encoder using 2-layer GNN with hidden dimension of 64 across all datasets. We use ReLU for nonlinear activation. For the sequence self-attention module, we set the output dimension to 128. The encoder's output dimension is set to 16, and add 64 additional dimensions initialized with all zeros to the latent states (t) to stabilize the training processes as in (Huang et al. 2021). The GNN ODE function is implemented with single-layer GNN from (Kipf et al.) 2018) with hidden dimension 128. For single-agent systems, we only include self-loop edges in the graph = (V,E), which makes the ODE function g simple MLP. To compute trajectories,
the Runge-Kuttar method from torchdiffeq python package s(Chen et al. (021] as the ODE solver and a one-layer MLP as the decoder.
We implement our model in pytorch. Encoder, generative model, and the decoder parameters are jointly optimized with Adam W optimizer 0.0001 for spring datasets and 0.00001 for Pendulum. The batch size for all datasets is set to 512.
(I (Loshchilov and Hutter] (219)) using learning rate of
TREAT =g-eey and TREAT Crey=ret share the same architecture and hyparameters as TREAT, with different implementations of the loss function. T. instead of comparing forward and reverse trajectories, we look at the L2 distance between the ground truth and reverse trajectories when computing the reversal loss.
For TREAT Cree neev2 we implement the reversal loss following (Huh et al. 2020) with one difference: we do not apply the reverse operation to the momentum portion of the initial state to the ODE function. This is because the initial hidden state is an output of the encoder that mixes position and momentum information. Note that we also remove the additional dimensions to the latent state that TREAT has. To reproduce our model's results, we provide our code implementation link here
LatentODE
We implement the Latent ODE sequence to sequence model as specified in (Rubanova et al.) 2019). We use a 4-layer ODE function in the recognition ODE, and 2-layer ODE function in the generative
ODE. The recognition and generative ODEs use Euler and Dopri5 as solvers (Chen et al.. 2021), respectively. The number of units per layer is 1000 in the ODE functions and 50 in GRU update networks. The dimension of the recognition model set to 100. The model is trained with learning rate of 0.001 with an exponential decay rate of 0.999 across different experiments. Note that since latentODE a single-agent model, we compute the trajectory of each object independently when applying to multi-agent systems.
HODEN
To adapt HODEN, which requires full initial states all objects, to systems with partial observations, compute each object's 's initial state via linear spline interpolation if it is missing. Following the setup in (Huh et al., 2020), we have two 2-layer linear networks with Tanh activation in between as ODE functions, in order model both positions and momenta. Each network has 1000-unit layer followed by single-unit layer. The model is trained with a learning rate of 0.00001 using a cosine single-agent model, we compute the trajectory of each object independently when applying it to multi-agent systems.
Shheduler.HODER
TRS-ODEN
Similar HODEN, we compute each object's initial state via linear spline interpolation if it is missing. As in (Huh et al. 2020), we use a 2-layer linear network with Tanh activation in between as the ODE functions, and the Leapfrog method for solving ODEs. The network has 1000 hidden units and is trained with learning rate of 0.00001 using a cosine scheduler. TRS-ODEN is single-agent model, we compute the trajectory of each object independently when applying to multi-agent systems.
27

---

TRS-ODENGNN
For TRSODENGNN, we substitute the ODE function in TRS-ODEN with GraphODE network. The GraphODE generative model is implemented with a single-layer GNN with hidden dimension 128. As in HODEN and TRS-ODEN, we compute each object's missing initial state via línear spline interpolation and the Leapfrog method for solving ODE. For all datasets, we use 0.5 as the coefficient for the reversal loss in (Huh al. 2020), and 0.0002 the learning rate under cosine scheduling.
LGODE
Our implementation follows (Huang et al. (020) except we remove the Variational Autoencoder (VAE) from the initial state encoder. Instead of using the output from the encoder GNN as the mean and std of the VAE, we directly use it as the latent initial state. That is, the initial states are deterministic instead of being sampled from a distribution. We use the same architecture as in TREAT and train the model using Adam W optimizer with a learning rate of 0.0001 across all datasets.
E
Additional Experiments
E.1
Comparison of different solvers
We next show our model's sensitivity regarding solvers with different precisions. Specifically, we compare against Euler and Runge-Kuttaa (RK4) where the latter is higher-precision solver. We show the comparison against LGODE and TREAT in Table
We can firstly observe that TREAT consistently outperforms LGODE, which is our strongest baseline across different solvers and datasets, indicating the effectiveness of the proposed time-reversal symmetry loss. Secondly, the improvement ratios get larger when using RK4 over Euler. This can be understood as our reversal loss is minimizing higher-order Tayler expansion terms (Theoreom 3.1 thus compensating numerical errors brought by ODE solvers.
compute the improvement ratio as LGODE-TREDT LGODE We can see that
Table 3: Evaluation results on MSE (10 3) over different solvers for multi-agent systems.
Dataset
Simple Spring Euler
Forced Spring
Dumped Spring Euler
Pendulum
Solvers
RK4
Euler
RK4
RK4
Euler
RK4
LGODE TREAT
1.8443
1.7429
2.0462
1.8929
1.1686
0.9718
1.4634
1.4156
1.4864
1.1178
1.6058
1.4525
0.8070
0.5944
1.3093
1.2527
% Improvement
19.4057
35.8655
21.5228
23.2659
30.9430
38.8352
10.5303
11.5075
E,2
Evaluation across observation ratios.
For LG-ODE and TREAT, the encoder computes the initial states from observed trajectories. To show models' sensitivity towards data sparsity, we randomly mask out 40% and 80% historical observations and compare model performance. As shown in Table when changing the ratios from 80% to 40%. we observe that TREAT has a smaller performance drop compared with LG-ODE, especially on the more complex Pendulum dataset (LG-ODE decreases 22.04% while TREAT decreases 1.62%). This indicates that TREAT is less sensitive toward data sparsity.
Table 4: Results of varying observation ratios on MSE (10-2) of multi-agent datasets.
Dataset
Simple Spring
Forced Spring
Damped Spring
Pendulum
Observation Ratios
0.8
0.4
0.8
0.4
0.8
0.4
0.8
0.4
LG-ODE TREAT
1.7054
1.6889
1.7554
2,0370
0.9305
1.0217
1.4314
1.7469
1.1176
1.1429
1.3611
1.5109
0.6920
0.6964
1.2309
1.2110
F
Discussion about Reversible Neural Networks
In literature, there is another line of research about building reversible neural networks (NNs). For example, (Chang et al. 2018) formulates three architectures for reversible neural networks to address
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the stability issue and achieve arbitrary deep lengths, motivated by dynamical system modeling. al. 2019) employs normalizing flow to create generative model of graph structures. They all propose novel architectures to construct reversible NN where intermediate states across layer depths do need be stored, thus improving memory efficiency.
Liu
However, we'd like to clarify that reversible NNs (RevNet) do not resolve the time-reversal symmetry problem that we're studying. The core of RevNet is that input can be recovered from output via reversible operation (which is another operator), similar as any linear operator W (-) have a reversed projector W (-) In the contrary. what we want to study is that the same operator can be used for both forward and backward prediction over time, and keep the trajectory the same. That being said, generate the forward and backward trajectories, we are using the same g(-), instead of g (-) respectively.
In summary, though both reversible NN and time-reversal symmetry share similar insights and intuition, they're talking about different things: reversible NNs make every operator g()) having g (-), while time-reeersible assume the trajectory get from ş/wd g(z) and gbod -g() to be closer. Making 9 to be reversible cannot make the system to be time-reversible.
G Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. TREAT is trained upon physical simulation data (e.g.. spring and pendulum) and implemented by public libraries in PyTorch. During the modeling, we neither introduces any social/ethical bias nor amplify any bias in the data. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
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Reviews:
[{"review_id": "uSrp9AlRsn", "reviewer": "Reviewer_hqsc", "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.", "questions": "Does the method work well with models other than GraphODE?", "limitations": "The authors briefly touched upon the possibility of incorporating properties in the spatial aspect such as translation and rotation equivariance.", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 4, "strengths": "* The idea of imposing time-reversal symmetry via a regularization term is novel and well-motivated.\n* The implementation of the regularization term is simple yet the effect is visible.\n* Overall the article is well written, and technical details are explained clearly.\n* 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.\n* The article lacks a more comprehensive review on other  energy-preserving / time-reversal neural ODE solvers.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n### **W1: Empirical results on additional real-world examples.**\nWe 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. \n\n\nTo 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. \n\n|model | TREAT | LGODE | TRS_ODEN |HODEN|LatentODE|\n|--|--|--|--|--|--|\n|MSE|$\\underline{2.5420}$|2.7270|3.6885|4.4342|23.0157|\n\n### **W2: review on other energy-preserving/time-reversal neural ODE solvers.**\n\nWe 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.\n\nVarious 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.\n\n\n[1] Greydanus, Samuel, Misko Dzamba, and Jason Yosinski. \"Hamiltonian neural networks.\" Advances in neural information processing systems 32 (2019).\n\n[2] Gruver et al, Deconstructing the inductive biases of Hamiltonian neural networks, ICLR 2022.\n\n[3]Han, Chen-Di, et al. \"Adaptable Hamiltonian neural networks.\" Physical Review Research 3.2 (2021): 023156.\n\n[4] Mattheakis, Marios, et al. \"Hamiltonian neural networks for solving equations of motion.\" Physical Review E 105.6 (2022): 065305.\n\n[5] Zhong et al, Dissipative symoden: Encoding Hamiltonian dynamics with dissipation and control into deep learning, et al, ICLR 2020 workshop.\n\n[6] Morrison, Philip J. \"A paradigm for joined Hamiltonian and dissipative systems.\" Physica D: Nonlinear Phenomena 18.1-3 (1986): 410-419.\n\n[7] Gruber, et al, Reversible and irreversible bracket-based dynamics for deep graph neural networks, NeurIPS 2023.\n\n[8] Huh et al, Time-reversal symmetric ode network, NeurIPS 2020.\n\n### **Q1. Can the TRS loss be coupled with other models?**\nOur 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.\nDue to its simplicity, we believe other models, such as NeuralODE and flow-based methods, can also benefit from TRS loss.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dYZO9wi5N0", "reviewer": "Reviewer_SfDm", "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.", "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？\n\n2. The definition of R does not appear in the paper. Can the method proposed in this paper hold for any R?\n\n3. How is regularization term 9 obtained based on Lemma 2, as R did not appear in the final regularization term 9？\n\n4. Does Theorem 3.1 hold for any untrained NN? \n\n5. The definition of $L_{reverse2}$ does not appear in the paper.  Can the condition $L_{reverse2}=L_{reverse}$ be satisfied?", "limitations": "Yes", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "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.\nAlso, 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:\n\n1.  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. \n\n2. 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.\n\n3. Some descriptions in the text are not clear enough, please refer to the question section below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n### **W1: The difference to the final implementations.**\nWe 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).\n\nThe TRS loss in TREAT (ours) follows Lemma 2.1: $R \\circ \\Phi_t \\circ R \\circ \\Phi_t = \\text{I} $\nIt 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 $\\boldsymbol{\\hat{y}}_i ^{\\text{fwd}}(0)$.\n\nThe 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 $\\boldsymbol{\\hat{y}}_i ^{\\text{fwd}}(1)$.\n\nThe 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.\nWe analytically show why our implementation is better based on Figure 8 in Appendix A.4.\n\n### **W2: Why our implementation is better.**\nAs 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.\n\nWe here use one integration step for illustration purpose. Specifically, if we assume the two reconstruction losses \nare 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 $a+b$.\n\nFinally, 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. \n\n### **Q1.**\nThanks 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.\n\nFor example:\nConsider 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) $,\n\nAs for velocity $p(t)$,  we have $R\\circ p(t) = -p(t)$ [1]. Therefore we can also get $R\\circ R\\circ p(t)=R\\circ (-p(t))= p(t)$\n\n[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.\n\n### **Q2.**\nThe 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].\n\n[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.\n\n### **Q3.** \nAs shown in Figure 2 and Figure 8, $z^{\\text{rev}}(t’_0)=R \\circ z^{\\text{fwd}}(t_K)=z^{\\text{fwd}}(t_K)$. \nCombining this with Eqn 8, we obtain $ y^{\\text{rev}}(t’_j)$\nand finally, we derive Eqn 9.\n\n### **Q4.** \nYes. Theorem 3.1 does not restrict the form of the ODE function $g$. The ODE function $g$ can be any NN in practice.\n\n### **Q5.** \nThanks 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.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "vDTGRTv1JS", "reviewer": "Reviewer_D3dA", "summary": "The paper proposes a novel Time-Reversal Symmetry (TRS) graph neural ODE, where the TRS is introduced as a soft regularisation term.", "questions": "Are there cases where the time-reversal symmetry expectation is violated?\n\nA 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?\n\nThe 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.\n\nThe work will benefit from additional proofreading.", "limitations": "The authors do not provide a separate Limitations section, and do not discuss limitations of the work in the Conclusions section.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "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. \n\n### **W1: Reference about TRS-ODEN.** \nThank 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:\nWe 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.\n\n### **Q1: Cases where TRS is violated**\nThanks 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).\n\n### **Q2-1: Can TRS decrease numerical errors across different solvers?**\nTo 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.\nWe 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.\n\n| Dataset &nbsp;&nbsp; &nbsp;&nbsp;  &nbsp; &nbsp; &nbsp; &nbsp; | Simple Spring   &nbsp;  &nbsp;&nbsp;&nbsp;| Forced Spring &nbsp;&nbsp;&nbsp;| Damped Spring  &nbsp;&nbsp;&nbsp;     | Pendulum                  | \n|---------------|-----------------------|---------------|---------------------|--------------------|\n\nSolvers| &nbsp; Euler   |  &nbsp;  RK4   | &nbsp; Euler   |  &nbsp;  RK4   | &nbsp; Euler  |  &nbsp;  RK4  | &nbsp; Euler   |  &nbsp;  RK4      \n:--|:--|:--|:--|:----|:---|:-----|:---------|:-----------\nLGODE         | 1.8443  | 1.7429      | 2.0462  | 1.8929      | 1.1686  | 0.9718      | 1.4634  | 1.4156   \nTREAT         | **1.4864** | **1.1178** | **1.6058** | **1.4525** | **0.8070** | **0.5944** | **1.3093** | **1.2527** \n\\% Improvement | 19.4057 | 35.8655     | 21.5228 | 23.2659     | 30.9430 | 38.8352     | 10.5303 | 11.5075  \n\n### **Q2-2: Evaluation metric**\nThank 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]. \n\n[1]Huang, Z., et al.  \"Learning continuous system dynamics from irregularly-sampled partial observations.\" Advances in Neural Information Processing Systems 2020\n\n[2] Kipf, Thomas, et al. \"Neural relational inference for interacting systems.\" International conference on machine learning. PMLR, 2018.\n\n### **Q3: Code**\nThank 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.\n\n### **Q4: Discussion about limitations**\nThank 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.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "aAEHaOieYP", "reviewer": "Reviewer_edxC", "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.", "questions": "1.\tFigure 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?\n\n2.\tIn figure 4a, do you have any insights why the TREAT does not do well compared with others with shorter prediction length.", "limitations": "n/a", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "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.\n\nThe paper provides theories that TRS minimizes higher-order Taylor terms, enhancing modeling accuracy.\n\nThe paper conducted extensive experiments on nine datasets show that TREAT outperforms other neuralODE models.", "weaknesses": "1.\tThe 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). \n\n2.\tThe 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.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "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.\n\n\n### **W1. About the use case for Vanilla neural ODE**\n\nThank you for the valuable suggestion! \nOur 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.\nDue to its simplicity, we believe other models, such as NeuralODE and flow-based methods, can also benefit from TRS loss.\n\n### **W2. Theorem 3.1**\nThanks 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.\n\n### **Q1. Is TRS a universal property of all classic ODEs?**\nA 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.\n\n### **Q2. Explanation of Figure 4 for short-term predictions**\nThanks 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.\n\nAs for the simple spring with shorter prediction lengths, we assume the potential reasons as follows:\n\n1. 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. \n\n2. 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].\n\n[1] Zijie Huang, Yizhou Sun and Wei Wang. \"Coupled Graph ODE for Learning Interacting System Dynamics.\" KDD 2021.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper introduces Time-Reversal Symmetry (TRS) as a regularization term to improve NODEs. The TRS regularization is designed to align forward and backward trajectories predicted by a NODE, promoting energy conservation and stability in the system. The paper is well-motivated, and the proposed method is both innovative and relevant, with theoretical justifications provided to support the benefits of TRS in improving modeling accuracy. The approach is validated with a number of experiments across nine diverse datasets, demonstrating good performance compared to baseline models. In summary, the paper is fun to read and stimulating. 

Given the strengths of the paper (i.e., the novel contribution, nice theoretical foundations, and experimental validation) and the very positive reviews, I recommend accepting this paper. However, I urge the authors to carefully revise the manuscript when preparing the camera ready version. Specifically, I suggest to expand the empirical scope and providing clearer theoretical explanations, as well as to address all the discussed issues during the rebuttal phase.
Decision: Accept (poster)