SympGNNs: Symplectic Graph Neural Networks for identifiying high-dimensional Hamiltonian systems and node classification
This addresses the challenge of scaling Hamiltonian system learning to high-dimensional many-body systems for physics and machine learning applications, with incremental improvements in graph neural networks.
The paper tackles the problem of learning high-dimensional Hamiltonian systems, where existing models like SympNets struggle, by introducing Symplectic Graph Neural Networks (SympGNNs), which effectively handle system identification in examples like a 40-particle coupled Harmonic oscillator and a 2000-particle molecular dynamics simulation, and achieve competitive accuracy in node classification tasks.
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.