A discontinuous Galerkin plane wave neural network method for Helmholtz equation and Maxwell's equations

TL;DR

This paper introduces a novel discontinuous Galerkin plane wave neural network method for efficiently solving Helmholtz and Maxwell's equations, combining neural networks with adaptive finite element techniques.

Contribution

It develops an innovative neural network-based basis construction within a discontinuous Galerkin framework, with proven convergence and adaptive refinement for wave equations.

Findings

Numerical experiments demonstrate the method's effectiveness.

Convergence of the method is theoretically established.

Adaptive basis construction improves solution accuracy.

Abstract

In this paper we propose a discontinuous Galerkin plane wave neural network (DGPWNN) method for approximately solving Helmholtz equation and Maxwell's equations. In this method, we define an elliptic-type variational problem as in the plane wave least square method with $h-$refinement and introduce the adaptive construction of recursively augmented discontinuous Galerkin subspaces whose basis functions are realizations of element-wise neural network functions with $hp-$refinement, where the activation function is chosen as a complex-valued exponential function like the plane wave function. A sequence of basis functions approaching the unit residuals are recursively generated by iteratively solving quasi-maximization problems associated with the underlying residual functionals and the intersection of the closed unit ball and discontinuous plane wave neural network spaces. The convergence results of the DGPWNN method are established without the assumption on the boundedness of the neural network parameters. Numerical experiments confirm the effectiveness of the proposed method.