Let data talk: data-regularized operator learning theory for inverse problems
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911806266540032 |
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| author | Chen, Ke Wang, Chunmei Yang, Haizhao |
| author_facet | Chen, Ke Wang, Chunmei Yang, Haizhao |
| contents | Regularization plays a pivotal role in integrating prior information into inverse problems. While many deep learning methods have been proposed to solve inverse problems, determining where to apply regularization remains a crucial consideration. Typical methods regularize neural networks via architecture, wherein neural network functions parametrize the parameter of interest or the regularization term. We introduce a novel approach, denoted as the "data-regularized operator learning" (DaROL) method, designed to address PDE inverse problems. The DaROL method trains a neural network on data, regularized through common techniques such as Tikhonov variational methods and Bayesian inference. The DaROL method offers flexibility across different frameworks, faster inverse problem-solving, and a simpler structure that separates regularization and neural network training. We demonstrate that training a neural network on the regularized data is equivalent to supervised learning for a regularized inverse map. Furthermore, we provide sufficient conditions for the smoothness of such a regularized inverse map and estimate the learning error in terms of neural network size and the number of training samples. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_09854 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Let data talk: data-regularized operator learning theory for inverse problems Chen, Ke Wang, Chunmei Yang, Haizhao Numerical Analysis 65N21 G.1.8 Regularization plays a pivotal role in integrating prior information into inverse problems. While many deep learning methods have been proposed to solve inverse problems, determining where to apply regularization remains a crucial consideration. Typical methods regularize neural networks via architecture, wherein neural network functions parametrize the parameter of interest or the regularization term. We introduce a novel approach, denoted as the "data-regularized operator learning" (DaROL) method, designed to address PDE inverse problems. The DaROL method trains a neural network on data, regularized through common techniques such as Tikhonov variational methods and Bayesian inference. The DaROL method offers flexibility across different frameworks, faster inverse problem-solving, and a simpler structure that separates regularization and neural network training. We demonstrate that training a neural network on the regularized data is equivalent to supervised learning for a regularized inverse map. Furthermore, we provide sufficient conditions for the smoothness of such a regularized inverse map and estimate the learning error in terms of neural network size and the number of training samples. |
| title | Let data talk: data-regularized operator learning theory for inverse problems |
| topic | Numerical Analysis 65N21 G.1.8 |
| url | https://arxiv.org/abs/2310.09854 |