Adaptive Neural Network Basis Methods for Partial Differential Equations with Low-Regular Solutions

This paper aims to devise an adaptive neural network basis method for numerically solving a second-order semi-linear partial differential equation 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. Firstly, based on the solution residual, the domain $\Omega$ is iteratively decomposed and eventually partitioned into $K+1$ non-overlapping subdomains, denoted respectively as $\{\Omega_k\}_{k=0}^K$, where the exact solution is smooth on subdomain $\Omega_0$ and low-regular on subdomain $\Omega_k$ $(1\leq k\leq K)$. Secondly, the low-regular solutions on different subdomains $\Omega_k$ $(1\leq k\leq K)$ are approximated by neural networks with different scales, while the smooth solution on subdomain $\Omega_0$ 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.