Projection Methods for Operator Learning and Universal Approximation
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917342726848512 |
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| author | Zappala, Emanuele |
| author_facet | Zappala, Emanuele |
| contents | We obtain a new universal approximation theorem for continuous (possibly nonlinear) operators on arbitrary Banach spaces using the Leray-Schauder mapping. Moreover, we introduce and study a method for operator learning in Banach spaces $L^p$ of functions with multiple variables, based on orthogonal projections on polynomial bases. We derive a universal approximation result for operators where we learn a linear projection and a finite dimensional mapping under some additional assumptions. For the case of $p=2$, we give some sufficient conditions for the approximation results to hold. This article serves as the theoretical framework for a deep learning methodology in operator learning. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12264 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projection Methods for Operator Learning and Universal Approximation Zappala, Emanuele Numerical Analysis Artificial Intelligence Machine Learning We obtain a new universal approximation theorem for continuous (possibly nonlinear) operators on arbitrary Banach spaces using the Leray-Schauder mapping. Moreover, we introduce and study a method for operator learning in Banach spaces $L^p$ of functions with multiple variables, based on orthogonal projections on polynomial bases. We derive a universal approximation result for operators where we learn a linear projection and a finite dimensional mapping under some additional assumptions. For the case of $p=2$, we give some sufficient conditions for the approximation results to hold. This article serves as the theoretical framework for a deep learning methodology in operator learning. |
| title | Projection Methods for Operator Learning and Universal Approximation |
| topic | Numerical Analysis Artificial Intelligence Machine Learning |
| url | https://arxiv.org/abs/2406.12264 |