Adaptive Neural Network Basis Methods for Partial Differential Equations with Low-Regular Solutions (Communications in Computational Physics, 2025)
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 $Ω$ is iteratively decomposed and eventually partitioned into $K+1$ non-overlapping subdomains, denoted respectively as $\{Ω_k\}_{k=0}^K$, where the exact solution is smooth on subdomain $Ω_0$ and low-regular on subdomain $Ω_k$ $(1≤ k≤ K)$. Secondly, the low-regular solutions on different subdomains $Ω_k$ $(1≤ k≤ K)$ 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.