The time evolution of physical systems is described by differential equations, which depend on abstract quantities like energy and force. Traditionally, these quantities are derived as functionals based on observables such as positions and velocities. Discovering these governing symbolic laws is the key to comprehending the interactions in nature. Here, we present a Hamiltonian graph neural network (HGNN), a physics-enforced GNN that learns the dynamics of systems directly from their trajectory. We demonstrate the performance of HGNN on n-springs, n-pendulums, gravitational systems, and binary Lennard Jones systems; HGNN learns the dynamics in excellent agreement with the ground truth from small amounts of data. We also evaluate the ability of HGNN to generalize to larger system sizes, and to hybrid spring-pendulum system that is a combination of two original systems (spring and pendulum) on which the models are trained independently. Finally, employing symbolic regression on the learned HGNN, we infer the underlying equations relating the energy functionals, even for complex systems such as the binary Lennard-Jones liquid. Our framework facilitates the interpretable discovery of interaction laws directly from physical system trajectories. Furthermore, this approach can be extended to other systems with topology-dependent dynamics, such as cells, polydisperse gels, or deformable bodies.

翻译：物理系统的时间演化由微分方程描述，这些方程依赖于能量和力等抽象量。传统上，这些量是基于位置和速度等可观测量推导出的泛函。发现这些主导性符号定律是理解自然界相互作用的关键。本文提出了一种哈密顿图神经网络（HGNN），这是一种物理强化的图神经网络，能够直接从系统轨迹中学习动力学。我们在n-弹簧、n-摆、引力系统以及二元伦纳德-琼斯系统上展示了HGNN的性能；HGNN能够从少量数据中学习到与真实情况高度一致的动力学。我们还评估了HGNN泛化到更大系统规模以及混合弹簧-摆系统（由两个独立训练的原始系统——弹簧和摆——组合而成）的能力。最后，通过对学习到的HGNN进行符号回归，我们推导出了与能量泛函相关的基本方程，甚至适用于二元伦纳德-琼斯液体等复杂系统。我们的框架促进了直接从物理系统轨迹中可解释地发现相互作用定律。此外，该方法可推广至其他具有拓扑依赖动力学的系统，例如细胞、多分散凝胶或可变形体。