Adaptive neural network basis methods for partial differential equations with low-regular solutions
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arXiv
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| Format: | Preprint |
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2024
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| author | Huang, Jianguo Wu, Haohao Zhou, Tao |
| author_facet | Huang, Jianguo Wu, Haohao Zhou, Tao |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_01998 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Adaptive neural network basis methods for partial differential equations with low-regular solutions Huang, Jianguo Wu, Haohao Zhou, Tao Numerical Analysis 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. |
| title | Adaptive neural network basis methods for partial differential equations with low-regular solutions |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2411.01998 |