Adaptive neural network basis methods for partial differential equations with low-regular solutions

Abstract

This paper aims to devise an adaptive neural network basis method for numerically solving a second-order semilinear partial differential equation (PDE) with low-regular solutions in two/three dimensions. The method is obtained by combining basis functions from a class of shallow neural networks and the resulting multi-scale analogues, a residual strategy in adaptive methods and the non-overlapping domain decomposition method. At the beginning, in view of the solution residual, we partition the total domain $\hat{I}\text{c◯}$ into $K+1$ non-overlapping subdomains, denoted respectively as $\{\hat{I}{\text{c◯}}_k{\}}_{k=0}^K$, where the exact solution is smooth on subdomain $\hat{I}{\text{c◯}}_0$ and low-regular on subdomain $\hat{I}{\text{c◯}}_k$ ($1\le k\le K$). Secondly, the low-regular solutions on different subdomains \(Ω_{k}\)~($1\le k\le K$) are approximated by neural networks with different scales, while the smooth solution on subdomain \(Ω_0\) is approximated by the initialized neural network. Thirdly, we determine the undetermined coefficients by solving the linear least squares problems directly or the nonlinear least squares problem via the Gauss-Newton method. The proposed method can be extended to multi-level case naturally. Finally, we use this adaptive method for several peak problems in two/three dimensions to show its high-efficient computational performance.