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 $Ω$ 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 $Ω_{k}$ ($1\le k\le K$). 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.

Figures (14)

• Figure 1: A schematic diagram from $K=0$ to $K=2$.
• Figure 2: The schematic diagrams of the domain for example \ref{['sec:linear-2d-peak']}.
• Figure 3: The $\operatorname{err}_{L_{2}}$ with the number of scaled neural network basis functions.
• Figure 4: The exact solution (Left), the predicted solution (Middle) and absolute error (Right) when $M^{*}=1000$.
• Figure 5: The schematic diagrams of the domain for example \ref{['sec:nonlinear-peaks']}.

Theorems & Definitions (1)

• Remark 1