Adaptive Neural Network Subspace Method for Solving Partial Differential Equations with High Accuracy
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917855736365056 |
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| author | Lin, Zhongshuo Wang, Yifan Xie, Hehu |
| author_facet | Lin, Zhongshuo Wang, Yifan Xie, Hehu |
| contents | Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopted to build the basis of the finite dimensional subspace. Then the discrete solution is obtained by using the subspace approximation. Especially, based on the subspace approximation, a posteriori error estimator can be derivated by the hypercircle technique. This a posteriori error estimator can act as the loss function for adaptively refining the parameters of neural network. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02586 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Adaptive Neural Network Subspace Method for Solving Partial Differential Equations with High Accuracy Lin, Zhongshuo Wang, Yifan Xie, Hehu Numerical Analysis 65N30, 65N25, 65L15, 65B99 Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopted to build the basis of the finite dimensional subspace. Then the discrete solution is obtained by using the subspace approximation. Especially, based on the subspace approximation, a posteriori error estimator can be derivated by the hypercircle technique. This a posteriori error estimator can act as the loss function for adaptively refining the parameters of neural network. |
| title | Adaptive Neural Network Subspace Method for Solving Partial Differential Equations with High Accuracy |
| topic | Numerical Analysis 65N30, 65N25, 65L15, 65B99 |
| url | https://arxiv.org/abs/2412.02586 |