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Physics-Informed Regularization for Domain-Agnostic Dynamical System Modeling
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Proposes TREAT, a GraphODE-based model regularized by a time-reversal symmetry loss that aligns forward and reverse latent trajectories, with a theorem claiming it minimizes higher-order ODE integration error.
Problem
Learning physical dynamical systems from data with neural ODEs is prone to violating physical properties and accumulating errors, and existing physics-informed models (e.g., HNNs) are tied to specific physical priors such as energy conservation.
Approach
Adds a self-supervised time-reversal symmetry loss that aligns forward predicted trajectories with reverse trajectories integrated backward from the forward endpoint with negated dynamics; the authors argue this reduces higher-order Taylor error terms, which helps irrespective of whether the system is conservative or irreversible.
Key results
- TREAT achieves the lowest MSE on all reported datasets in the multi-agent and single-agent tables, e.g., 1.2527e-2 on Pendulum vs 1.4156e-2 for LG-ODE.
- The abstract reports an 11.5% MSE improvement on the chaotic triple-pendulum scenario.
- Ablation shows the adopted reverse-loss formulation beats the gt-rev and rev2 (TRS-ODE style) variants on most datasets.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add the forward/reverse trajectory TRS regularizer | GraphODE (TREAT) · Simple Spring, Forced Spring, Damped Spring, Pendulum, Human Motion, and Attractor | MSE (10^-2) | Helped |
| Add the forward/reverse trajectory TRS regularizer | GraphODE (TREAT) · Simple Spring, Forced Spring, and Damped Spring, single-agent | MSE (10^-2) | Helped |
| Use ground-truth-to-reverse-trajectory loss instead of forward/reverse alignment (ablation) | TREAT · Simple Spring, multi-agent | MSE (10^-2): 1.1178 -> 1.1313 | Hurt |
| Use ground-truth-to-reverse-trajectory loss instead of forward/reverse alignment (ablation) | TREAT · Forced Spring, multi-agent | MSE (10^-2): 1.4525 -> 1.5254 | Hurt |
| Use ground-truth-to-reverse-trajectory loss instead of forward/reverse alignment (ablation) | TREAT · Damped Spring, multi-agent | MSE (10^-2): 0.5944 -> 0.6171 | Hurt |
| Use ground-truth-to-reverse-trajectory loss instead of forward/reverse alignment (ablation) | TREAT · Pendulum, multi-agent | MSE (10^-2): 1.2527 -> 1.6158 | Hurt |
| Use ground-truth-to-reverse-trajectory loss instead of forward/reverse alignment (ablation) | TREAT · Human Motion | MSE (10^-2): 0.2192 -> 0.2495 | Hurt |
| Use ground-truth-to-reverse-trajectory loss instead of forward/reverse alignment (ablation) | TREAT · Single-agent Simple Spring, Forced Spring, Damped Spring, and Attractor | MSE (10^-2) | Mixed |
| Use TRS-ODEN-style reversal loss instead of forward/reverse alignment (ablation) | TREAT · Simple Spring, Forced Spring, Damped Spring, Pendulum, Human Motion, and Attractor | MSE (10^-2) | Hurt |
| Use TRS-ODEN-style reversal loss instead of forward/reverse alignment (ablation) | TREAT · Single-agent Simple Spring, Forced Spring, and Damped Spring | MSE (10^-2) | Hurt |
Surprising findings
- The authors claim TRS loss helps even for irreversible systems such as Damped Spring, where TREAT outperforms baselines.
- HODEN, which enforces energy conservation, performs poorly on several systems where it is misapplied, e.g., Pendulum MSE around 741 vs 1.25 for TREAT.
Limitations
- Only incorporates temporal-domain inductive bias; spatial symmetries such as translation and rotation equivariance are not addressed.
- Reverse-loss computation roughly doubles the ODE integration cost per step, per the remark on computational overhead.
- Baselines HODEN and TRS-ODEN require initial states approximated via linear spline interpolation, which hurts their performance on chaotic systems.
Experimental setup
Datasets: Simple Spring Forced Spring Damped Spring Pendulum Human Motion Attractor
Benchmarks: Simple Spring Forced Spring Damped Spring Pendulum Human Motion Attractor
Metrics: MSE
Baselines: LatentODE HODEN TRS-ODEN TRS-ODEN_GNN LG-ODE
Builds on: GraphODE LG-ODE Hamiltonian Neural Networks TRS-ODE
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