Physics-Informed Neural Network based on Non-overlapping Domain Decomposition Methods for Two-Dimensional Magnetostatic Field Problems

Abstract

PINN is a method for training neural networks by incorporating the error for an initial boundary value problem of partial differential equations into the loss function, and many studies have been reported. In order to improve the accuracy of PINN, it is desirable to increase the size of the training data set and to use a distribution with low discrepancy sequences. However, it is difficult to divide the point set while maintaining the characteristics of the distribution in the case of parallel processing of large training data. Therefore, this paper focuses on non-overlapping domain decomposition methods, which are known as a parallel numerical method for the finite element method. Especially, in addition to the classical Dirichlet-Neumann, Neumann-Neumann, and Dirichlet-Dirichlet algorithms, an iterative DDM algorithm based on the conjugate gradient method is developed for PINN. In addition, this paper applies the proposed method to a two-dimensional magnetostatic field problem and demonstrates numerical examples.