Adaptive neural network basis methods for partial differential equations with low-regular solutions Article Swipe
Jianguo Huang
,
Haohao Wu
,
Tao Zhou
·
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2411.01998
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2411.01998
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.
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- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2411.01998
- https://arxiv.org/pdf/2411.01998
- OA Status
- green
- Cited By
- 1
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4404353335
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https://openalex.org/W4404353335Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2411.01998Digital Object Identifier
- Title
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Adaptive neural network basis methods for partial differential equations with low-regular solutionsWork title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
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2024Year of publication
- Publication date
-
2024-11-04Full publication date if available
- Authors
-
Jianguo Huang, Haohao Wu, Tao ZhouList of authors in order
- Landing page
-
https://arxiv.org/abs/2411.01998Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/2411.01998Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://arxiv.org/pdf/2411.01998Direct OA link when available
- Concepts
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Basis (linear algebra), Artificial neural network, Mathematics, Partial differential equation, Applied mathematics, Differential equation, Computer science, Mathematical analysis, Artificial intelligence, GeometryTop concepts (fields/topics) attached by OpenAlex
- Cited by
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1Total citation count in OpenAlex
- Citations by year (recent)
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2025: 1Per-year citation counts (last 5 years)
- Related works (count)
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10Other works algorithmically related by OpenAlex
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