Learning complex multi-agent system dynamics from data is crucial across many domains, such as in physical simulations and material modeling. Extended from purely data-driven approaches, existing physics-informed approaches such as Hamiltonian Neural Network strictly follow energy conservation law to introduce inductive bias, making their learning more sample efficiently. However, many real-world systems do not strictly conserve energy, such as spring systems with frictions. Recognizing this, we turn our attention to a broader physical principle: Time-Reversal Symmetry, which depicts that the dynamics of a system shall remain invariant when traversed back over time. It still helps to preserve energies for conservative systems and in the meanwhile, serves as a strong inductive bias for non-conservative, reversible systems. To inject such inductive bias, in this paper, we propose a simple-yet-effective self-supervised regularization term as a soft constraint that aligns the forward and backward trajectories predicted by a continuous graph neural network-based ordinary differential equation (GraphODE). It effectively imposes time-reversal symmetry to enable more accurate model predictions across a wider range of dynamical systems under classical mechanics. In addition, we further provide theoretical analysis to show that our regularization essentially minimizes higher-order Taylor expansion terms during the ODE integration steps, which enables our model to be more noise-tolerant and even applicable to irreversible systems. Experimental results on a variety of physical systems demonstrate the effectiveness of our proposed method. Particularly, it achieves an MSE improvement of 11.5 % on a challenging chaotic triple-pendulum systems.

翻译：从数据中学习复杂多智能体系统动力学对于物理模拟和材料建模等诸多领域至关重要。相较于纯数据驱动方法，现有物理信息融合方法（如哈密顿神经网络）通过严格遵循能量守恒定律引入归纳偏置，从而提升样本学习效率。然而，现实系统（如含摩擦弹簧系统）并非严格能量守恒。基于此，我们聚焦于更普适的物理原理：逆时对称性——该原理表明系统动力学在时间回溯时应保持不变。该原理既能维持保守系统的能量守恒，又可作为非保守可逆系统的强归纳偏置。为注入此归纳偏置，本文提出一种简单有效的自监督正则化项作为软约束，通过对齐基于连续图神经网络常微分方程（GraphODE）预测的正向与反向轨迹，有效施加逆时对称性，从而在经典力学框架下提升更广泛动力系统的预测精度。此外，我们通过理论分析证明，该正则化本质上最小化了ODE积分步骤中的高阶泰勒展开项，使模型具备更强的抗噪能力，甚至可适用于不可逆系统。在多种物理系统上的实验结果验证了所提方法的有效性，其中在具有挑战性的混沌三摆系统上实现了11.5%的均方误差改进。