Error analysis for hybrid finite element/neural network discretizations
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866908847685238784 |
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| author | Kapustsin, Uladzislau Kaya, Utku Pfefferer, Johannes Richter, Thomas |
| author_facet | Kapustsin, Uladzislau Kaya, Utku Pfefferer, Johannes Richter, Thomas |
| contents | We describe and analyze a hybrid finite element/neural network method for predicting solutions of partial differential equations. The methodology is designed for obtaining fine scale fluctuations from neural networks in a local manner. The network is capable of locally correcting a coarse finite element solution towards a fine solution taking the source term and the coarse approximation as input. Key observation is the dependency between quality of predictions and the size of training set which consists of different source terms and corresponding fine & coarse solutions. We provide the a priori error analysis of the method together with the stability analysis of the neural network. The numerical experiments confirm the capability of the network predicting fine finite element solutions. We also illustrate the generalization of the method to problems where test and training domains differ from each other. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11271 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Error analysis for hybrid finite element/neural network discretizations Kapustsin, Uladzislau Kaya, Utku Pfefferer, Johannes Richter, Thomas Numerical Analysis 65Y10, 65N12, 65N30 We describe and analyze a hybrid finite element/neural network method for predicting solutions of partial differential equations. The methodology is designed for obtaining fine scale fluctuations from neural networks in a local manner. The network is capable of locally correcting a coarse finite element solution towards a fine solution taking the source term and the coarse approximation as input. Key observation is the dependency between quality of predictions and the size of training set which consists of different source terms and corresponding fine & coarse solutions. We provide the a priori error analysis of the method together with the stability analysis of the neural network. The numerical experiments confirm the capability of the network predicting fine finite element solutions. We also illustrate the generalization of the method to problems where test and training domains differ from each other. |
| title | Error analysis for hybrid finite element/neural network discretizations |
| topic | Numerical Analysis 65Y10, 65N12, 65N30 |
| url | https://arxiv.org/abs/2310.11271 |