Solving the Helmholtz Equation by Sparse Fundamental Solution Neural Network

Abstract

A sparse fundamental solution neural network (SFSNN) for solving the Helmholtz equation with constant coefficients and relatively large wave numbers $ k $ is proposed. The method combines the strengths of fundamental solution techniques and neural networks by employing a radial basis function neural network, where fundamental solution functions serve as activation functions. Since these functions inherently satisfy the homogeneous Helmholtz equation, SFSNN only requires boundary sampling, significantly accelerating training. To enhance sparsity and generalization, an $\ell_1$ regularization term of the weights is introduced into the loss function, reformulating the weight optimization as a least absolute shrinkage and selection operator (Lasso) problem. This not only reduces the number of basis functions but also improves the network’s generalization capability. Numerical experiments validate the method’s effectiveness for high-wavenumber isotropic Helmholtz equations in two dimensions and three dimensions. The results reveal that when the analytical solution is a linear combination of fundamental solutions, SFSNN accurately identifies their centers. Otherwise, the number of required basis functions scales as $ N = O(k^{(\tau(d-1)))} $, where $\tau < 1$ and $ d $ is the problem dimension. Moreover, SFSNN has been successfully extended to non-homogeneous and semi-infinite Helmholtz equations, achieving high accuracy. Codes of the examples in this paper are available at https://github.com/wangzhiwensuda/SFSNN-Helmholtz-problem.