Tensor Neural Network Based Machine Learning Method for Elliptic Multiscale Problems
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910381464616960 |
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| author | Lin, Zhongshuo Liu, Haochen Xie, Hehu |
| author_facet | Lin, Zhongshuo Liu, Haochen Xie, Hehu |
| contents | In this paper, we introduce a type of tensor neural network based machine learning method to solve elliptic multiscale problems. Based on the special structure, we can do the direct and highly accurate high dimensional integrations for the tensor neural network functions without Monte Carlo process. Here, with the help of homogenization techniques, the multiscale problem is first transformed to the high dimensional limit problem with reasonable accuracy. Then, based on the tensor neural network, we design a type of machine learning method to solve the derived high dimensional limit problem. The proposed method in this paper brings a new way to design numerical methods for computing more general multiscale problems with high accuracy. Several numerical examples are also provided to validate the accuracy of the proposed numerical methods. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_16380 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tensor Neural Network Based Machine Learning Method for Elliptic Multiscale Problems Lin, Zhongshuo Liu, Haochen Xie, Hehu Numerical Analysis 35B27, 65M05, 65L15, 68T07 In this paper, we introduce a type of tensor neural network based machine learning method to solve elliptic multiscale problems. Based on the special structure, we can do the direct and highly accurate high dimensional integrations for the tensor neural network functions without Monte Carlo process. Here, with the help of homogenization techniques, the multiscale problem is first transformed to the high dimensional limit problem with reasonable accuracy. Then, based on the tensor neural network, we design a type of machine learning method to solve the derived high dimensional limit problem. The proposed method in this paper brings a new way to design numerical methods for computing more general multiscale problems with high accuracy. Several numerical examples are also provided to validate the accuracy of the proposed numerical methods. |
| title | Tensor Neural Network Based Machine Learning Method for Elliptic Multiscale Problems |
| topic | Numerical Analysis 35B27, 65M05, 65L15, 68T07 |
| url | https://arxiv.org/abs/2403.16380 |