Solving High-dimensional Parametric Elliptic Equation Using Tensor Neural Network
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909089830797312 |
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| author | Chen, Hongtao Fu, Rui Wang, Yifan Xie, Hehu |
| author_facet | Chen, Hongtao Fu, Rui Wang, Yifan Xie, Hehu |
| contents | In this paper, we introduce a tensor neural network based machine learning method for solving the elliptic partial differential equations with random coefficients in a bounded physical domain. With the help of tensor product structure, we can transform the high-dimensional integrations of tensor neural network functions to one-dimensional integrations which can be computed with the classical quadrature schemes with high accuracy. The complexity of its calculation can be reduced from the exponential scale to a polynomial scale. The corresponding machine learning method is designed for solving high-dimensional parametric elliptic equations. Some numerical examples are provided to validate the accuracy and efficiency of the proposed algorithms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00040 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solving High-dimensional Parametric Elliptic Equation Using Tensor Neural Network Chen, Hongtao Fu, Rui Wang, Yifan Xie, Hehu Numerical Analysis 35B27, 60H15, 60H35, 68T07 In this paper, we introduce a tensor neural network based machine learning method for solving the elliptic partial differential equations with random coefficients in a bounded physical domain. With the help of tensor product structure, we can transform the high-dimensional integrations of tensor neural network functions to one-dimensional integrations which can be computed with the classical quadrature schemes with high accuracy. The complexity of its calculation can be reduced from the exponential scale to a polynomial scale. The corresponding machine learning method is designed for solving high-dimensional parametric elliptic equations. Some numerical examples are provided to validate the accuracy and efficiency of the proposed algorithms. |
| title | Solving High-dimensional Parametric Elliptic Equation Using Tensor Neural Network |
| topic | Numerical Analysis 35B27, 60H15, 60H35, 68T07 |
| url | https://arxiv.org/abs/2402.00040 |