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 $\Omega$ 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 $\Omega_{k}$ ($1\le k\le K$). Secondly, the low-regular solutions on different subdomains \(\Omega_{k}\)~($1\le k\le K$) 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.

翻译：本文旨在设计一种自适应神经网络基函数方法，用于数值求解具有低正则解的二/三维二阶半线性偏微分方程。该方法通过结合一类浅层神经网络及其多尺度类比所生成的基函数、自适应方法中的残差策略以及非重叠区域分解法而获得。首先，基于解残差，我们将整体区域 $\Omega$ 划分为 $K+1$ 个非重叠子区域，分别记为 $\{\Omega_k\}_{k=0}^K$，其中精确解在子区域 $\Omega_{0}$ 上光滑，在子区域 $\Omega_{k}$ ($1\le k\le K$) 上具有低正则性。其次，不同子区域 \(\Omega_{k}\) ($1\le k\le K$) 上的低正则解通过不同尺度的神经网络进行逼近，而子区域 \(\Omega_0\) 上的光滑解则通过初始化的神经网络进行逼近。再次，我们通过直接求解线性最小二乘问题或采用高斯-牛顿法求解非线性最小二乘问题来确定待定系数。所提方法可自然地推广至多层级情形。最后，我们将此自适应方法应用于多个二/三维峰值问题，以展示其高效的计算性能。