Improved randomized neural network methods with boundary processing for solving elliptic equations
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916337186504704 |
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| author | Zhou, Huifang Sheng, Zhiqiang |
| author_facet | Zhou, Huifang Sheng, Zhiqiang |
| contents | We present two improved randomized neural network methods, namely RNN-Scaling and RNN-Boundary-Processing (RNN-BP) methods, for solving elliptic equations such as the Poisson equation and the biharmonic equation. The RNN-Scaling method modifies the optimization objective by increasing the weight of boundary equations, resulting in a more accurate approximation. We propose the boundary processing techniques on the rectangular domain that enforce the RNN method to satisfy the non-homogeneous Dirichlet and clamped boundary conditions exactly. We further prove that the RNN-BP method is exact for some solutions with specific forms and validate it numerically. Numerical experiments demonstrate that the RNN-BP method is the most accurate among the three methods, the error is reduced by 6 orders of magnitude for some tests. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_18457 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved randomized neural network methods with boundary processing for solving elliptic equations Zhou, Huifang Sheng, Zhiqiang Numerical Analysis We present two improved randomized neural network methods, namely RNN-Scaling and RNN-Boundary-Processing (RNN-BP) methods, for solving elliptic equations such as the Poisson equation and the biharmonic equation. The RNN-Scaling method modifies the optimization objective by increasing the weight of boundary equations, resulting in a more accurate approximation. We propose the boundary processing techniques on the rectangular domain that enforce the RNN method to satisfy the non-homogeneous Dirichlet and clamped boundary conditions exactly. We further prove that the RNN-BP method is exact for some solutions with specific forms and validate it numerically. Numerical experiments demonstrate that the RNN-BP method is the most accurate among the three methods, the error is reduced by 6 orders of magnitude for some tests. |
| title | Improved randomized neural network methods with boundary processing for solving elliptic equations |
| topic | Numerical Analysis |
| url | https://arxiv.org/abs/2407.18457 |