Existing neural network models to learn Hamiltonian systems, such as SympNets, although accurate in low-dimensions, struggle to learn the correct dynamics for high-dimensional many-body systems. Herein, we introduce Symplectic Graph Neural Networks (SympGNNs) that can effectively handle system identification in high-dimensional Hamiltonian systems, as well as node classification. SympGNNs combines symplectic maps with permutation equivariance, a property of graph neural networks. Specifically, we propose two variants of SympGNNs: i) G-SympGNN and ii) LA-SympGNN, arising from different parameterizations of the kinetic and potential energy. We demonstrate the capabilities of SympGNN on two physical examples: a 40-particle coupled Harmonic oscillator, and a 2000-particle molecular dynamics simulation in a two-dimensional Lennard-Jones potential. Furthermore, we demonstrate the performance of SympGNN in the node classification task, achieving accuracy comparable to the state-of-the-art. We also empirically show that SympGNN can overcome the oversmoothing and heterophily problems, two key challenges in the field of graph neural networks.

本研究解决了现有神经网络模型在高维多体哈密顿系统中学习动态的困难，提出了Symplectic Graph Neural Networks（SympGNNs），有效结合了辛映射与图神经网络的置换等变性。通过两种变体（G-SympGNN与LA-SympGNN）的实验，表明SympGNN在高维系统识别与节点分类中表现出色，且能克服图神经网络中的两个主要挑战：过平滑与异质性。

SympGNN：用于识别高维哈密顿系统和节点分类的哈密顿图神经网络