2025/06/11 by Long Yuan, Yuan, Long, Menghui Wu +3 · 1 citation
Computer Science · Engineering · Physics and Astronomy · #65N30 #65N55 #68T07 #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Model Reduction and Neural Networks #Neural Networks and Applications #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2506.09309
openalex publication_date 2025/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.