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

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YOU? · · 2025 · Open Access · · DOI: https://doi.org/10.4208/cicp.oa-2024-0310

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.

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Type: article
Landing Page: https://doi.org/10.4208/cicp.oa-2024-0310
PDF: https://global-sci.org/index.php/cicp/article/download/23353/36227
OA Status: bronze
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DOI: https://doi.org/10.4208/cicp.oa-2024-0310

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

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Publication year: 2025

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Publication date: 2025-11-28

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Authors: Jianguo Huang, Haohao Wu, Tao Zhou

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Landing page: https://doi.org/10.4208/cicp.oa-2024-0310

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