# Physics-Informed Regularization for Domain-Agnostic Dynamical System Modeling

arXiv: https://arxiv.org/abs/2410.06366 · HTML: https://mlsanity.com/paper/2410.06366/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 56 references that are in the map.
- First posted on arXiv 8 Oct 2024.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Heterogeneous Graph Transformer](https://mlsanity.com/paper/2003.01332/) (2020)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [iGibson 2.0: Object-Centric Simulation for Robot Learning of Everyday Household Tasks](https://mlsanity.com/paper/2108.03272/) (2021)
- [Graph Neural Ordinary Differential Equations](https://mlsanity.com/paper/1911.07532/) (2019)
- [Hamiltonian Graph Networks with ODE Integrators](https://mlsanity.com/paper/1909.12790/) (2019)
- [Graph Normalizing Flows](https://mlsanity.com/paper/1905.13177/) (2019)
- [Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning](https://mlsanity.com/paper/2002.08860/) (2020)
- [Reversible Architectures for Arbitrarily Deep Residual Neural Networks](https://mlsanity.com/paper/1709.03698/) (2017)
- [Learning Continuous System Dynamics from Irregularly-Sampled Partial Observations](https://mlsanity.com/paper/2011.03880/) (2020)
- [Identifying nonlinear dynamical systems via generative recurrent neural networks with applications to fMRI](https://mlsanity.com/paper/1902.07186/) (2019)
- [Neural Dynamics on Complex Networks](https://mlsanity.com/paper/1908.06491/) (2019)
- [Equivariant Graph Neural Operator for Modeling 3D Dynamics](https://mlsanity.com/paper/2401.11037/) (2024)
- [A probabilistic model for the numerical solution of initial value problems](https://mlsanity.com/paper/1610.05261/) (2016)
- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [Generalizing Graph ODE for Learning Complex System Dynamics across Environments](https://mlsanity.com/paper/2307.04287/) (2023)
- [Time-Reversal Symmetric ODE Network](https://mlsanity.com/paper/2007.11362/) (2020)
- [CF-GODE: Continuous-Time Causal Inference for Multi-Agent Dynamical Systems](https://mlsanity.com/paper/2306.11216/) (2023)
- [Scalable Graph Networks for Particle Simulations](https://mlsanity.com/paper/2010.06948/) (2020)
- [Causal Graph ODE: Continuous Treatment Effect Modeling in Multi-agent Dynamical Systems](https://mlsanity.com/paper/2403.00178/) (2024)
- [ModLaNets: Learning Generalisable Dynamics via Modularity and Physical Inductive Bias](https://mlsanity.com/paper/2206.12325/) (2022)

## Built on by (most cited first, 9 of 9)

- [Graph ODEs and Beyond: A Comprehensive Survey on Integrating Differential Equations with Graph Neural Networks](https://mlsanity.com/paper/2503.23167/) (2025)
- [Architecture-Aware Learning Curve Extrapolation via Graph Ordinary Differential Equation](https://mlsanity.com/paper/2412.15554/) (2024)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [Towards Generalizable PDE Dynamics Forecasting via Physics-Guided Invariant Learning](https://mlsanity.com/paper/2509.24332/) (2025)
- [PAINET: A Principled Efficient Transformer for 3D Dynamics Modeling](https://mlsanity.com/paper/2510.04233/) (2025)
- [Predicting Time Series of Networked Dynamical Systems without Knowing Topology](https://mlsanity.com/paper/2412.18734/) (2024)
- [GeoAda: Efficiently Finetune Geometric Diffusion Models with Equivariant Adapters](https://mlsanity.com/paper/2507.02085/) (2025)
- [Generalizing Dynamics Modeling More Easily from Representation Perspective](https://mlsanity.com/paper/2603.22655/) (2026)
- [Time-varying Interaction Graph ODE for Dynamic Graph Representation Learning](https://mlsanity.com/paper/2604.24811/) (2026)

## Abstract (arXiv)

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.