SympGNNs: Symplectic Graph Neural Networks for identifiying high-dimensional Hamiltonian systems and node classification

TL;DR

SympGNNs are novel symplectic graph neural networks designed to effectively identify high-dimensional Hamiltonian systems and perform node classification, overcoming limitations of previous models in complex physical systems.

Contribution

Introduction of SympGNNs with two variants that combine symplectic maps and permutation equivariance for high-dimensional system identification and node classification.

Findings

Successfully modeled a 40-particle harmonic oscillator.

Accurately classified nodes in a 2000-particle molecular system.

Overcame oversmoothing and heterophily issues in graph neural networks.

Abstract

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.