An unconditionally convergent discontinuous Galerkin neural network method activated by plane waves for Helmholtz equation and Maxwell's equations

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 unconditional only if nonlinear initial parameters for the quasi-maximizers are selected by suitably distributed propagation directions, and established without the assumption on the boundedness of the neural network parameters. Numerical experiments confirm the effectiveness of the proposed method.