Surrogate Modeling of Particle Methods Based on Graph Neural Networks

Abstract

For the purpose of accelerating numerical analysis, we propose a surrogate model for a particle method based on IsoGCNs, which are variants of graph neural networks (GNNs). In mechanical phenomena, when transformations such as rotations, translations, and reflections are applied to the input, the output takes on the same transformations. This property is called equivariance. IsoGCNs improve learning efficiency by introducing the equivariance into GNNs, and have been shown to be useful for finite element methods. This study shows an IsoGCN formulation for the particle method by considering relationships between particles as a graph. The proposed method is verified by comparing the errors and computation time of numerical and surrogate analyses of the three-dimensional heat conduction equation. In this paper, the problem of estimating the temperature distribution after one second from the initial temperature distribution is considered. As a result, the surrogate analysis requires more computation time than the numerical analysis with a time step size of one second. However, the accuracy is comparable, and it confirms that it can learn three-dimensional heat conduction equations.