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

in Neural networks : the official journal of the International Neural Network Society by Alan John Varghese, Zhen Zhang, George Em Karniadakis

TLDR

  • SympGNN, a new neural network model, can effectively learn Hamiltonian systems and perform node classification in high-dimensional many-body systems, resolving the struggles of existing models.
  • The model combines symplectic maps with permutation equivariance and is demonstrated on two physical examples, achieving comparable accuracy to state-of-the-art models.
  • SympGNN also overcomes oversmoothing and heterophily problems, paving the way for its application in various fields.

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 combine 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.

Overview

  • This study introduces a new neural network model, SympGNN, designed to learn Hamiltonian systems and perform node classification.
  • SympGNN combines symplectic maps with permutation equivariance, a property of graph neural networks, to effectively handle system identification in high-dimensional Hamiltonian systems.
  • The study proposes two variants of SympGNN: G-SympGNN and LA-SympGNN, arising from different parameterizations of the kinetic and potential energy.

Comparative Analysis & Findings

  • The authors 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.
  • The results show that SympGNN can accurately learn the correct dynamics for high-dimensional Hamiltonian systems and perform node classification comparably with state-of-the-art models.
  • SympGNN is also able to overcome the oversmoothing and heterophily problems, two key challenges in the field of graph neural networks.

Implications and Future Directions

  • The development of SympGNN has significant implications for the study of complex systems, including many-body systems and molecular dynamics simulations.
  • Future work could involve applying SympGNN to other physical systems and exploring its potential applications in fields such as chemistry, materials science, and machine learning.
  • The authors suggest that SympGNN could also be used to solve other challenging problems in physics and chemistry, such as large-scale simulations of quantum many-body systems.
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