Sparse-regularized high-frequency enhanced neural network for solving high-frequency problems

High-frequency problems frequently arise in various scientific and engineering applications. In this paper, we propose a high-frequency enhanced neural network (HFNN) to solve high-frequency partial differential equations. The basic idea of HFNN is to decompose the numerical solution into high-frequency and low-frequency components, and employ specific neural networks to handle these components separately by embedding high-frequency functions into the network. To further enhance the performance of the HFNN, we introduce a sparse-regularized high-frequency enhanced neural network (SR-HFNN) algorithm. The SR-HFNN algorithm employs a two-stage training strategy, where the first stage mainly learns to remove irrelevant frequency information through sparse regularization. By leveraging the power of deep neural networks and sparse learning, our proposed SR-HFNN algorithm demonstrates superior performance in solving high- frequency partial differential equations and inverse problems. The numerical results validate the fast convergence and high approximation accuracy of the SR-HFNN algorithm for high-frequency partial differential equations.