Structure-informed operator learning for parabolic Partial Differential Equations
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909389526401024 |
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| author | Benth, Fred Espen Detering, Nils Galimberti, Luca |
| author_facet | Benth, Fred Espen Detering, Nils Galimberti, Luca |
| contents | In this paper, we present a framework for learning the solution map of a backward parabolic Cauchy problem. The solution depends continuously but nonlinearly on the final data, source, and force terms, all residing in Banach spaces of functions. We utilize Fréchet space neural networks (Benth et al. (2023)) to address this operator learning problem. Our approach provides an alternative to Deep Operator Networks (DeepONets), using basis functions to span the relevant function spaces rather than relying on finite-dimensional approximations through censoring. With this method, structural information encoded in the basis coefficients is leveraged in the learning process. This results in a neural network designed to learn the mapping between infinite-dimensional function spaces. Our numerical proof-of-concept demonstrates the effectiveness of our method, highlighting some advantages over DeepONets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_09511 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Structure-informed operator learning for parabolic Partial Differential Equations Benth, Fred Espen Detering, Nils Galimberti, Luca Analysis of PDEs Numerical Analysis Probability In this paper, we present a framework for learning the solution map of a backward parabolic Cauchy problem. The solution depends continuously but nonlinearly on the final data, source, and force terms, all residing in Banach spaces of functions. We utilize Fréchet space neural networks (Benth et al. (2023)) to address this operator learning problem. Our approach provides an alternative to Deep Operator Networks (DeepONets), using basis functions to span the relevant function spaces rather than relying on finite-dimensional approximations through censoring. With this method, structural information encoded in the basis coefficients is leveraged in the learning process. This results in a neural network designed to learn the mapping between infinite-dimensional function spaces. Our numerical proof-of-concept demonstrates the effectiveness of our method, highlighting some advantages over DeepONets. |
| title | Structure-informed operator learning for parabolic Partial Differential Equations |
| topic | Analysis of PDEs Numerical Analysis Probability |
| url | https://arxiv.org/abs/2411.09511 |