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Volume 14, Issue 3
Recovering the Source Term in Elliptic Equation via Deep Learning: Method and Convergence Analysis

Chenguang Duan, Yuling Jiao, Jerry Zhijian Yang & Pingwen Zhang

East Asian J. Appl. Math., 14 (2024), pp. 460-489.

Published online: 2024-05

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  • Abstract

In this paper, we present a deep learning approach to tackle elliptic inverse source problems. Our method combines Tikhonov regularization with physics-informed neural networks, utilizing separate neural networks to approximate the source term and solution. Firstly, we construct a population loss and derive stability estimates. Furthermore, we conduct a convergence analysis of the empirical risk minimization estimator. This analysis yields a prior rule for selecting regularization parameters, determining the number of observations, and choosing the size of neural networks. Finally, we validate our proposed method through numerical experiments. These experiments also demonstrate the remarkable robustness of our approach against data noise, even at high levels of up to 50%.

  • AMS Subject Headings

65N15, 65N20, 65N21

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{EAJAM-14-460, author = {Duan , ChenguangJiao , YulingYang , Jerry Zhijian and Zhang , Pingwen}, title = {Recovering the Source Term in Elliptic Equation via Deep Learning: Method and Convergence Analysis}, journal = {East Asian Journal on Applied Mathematics}, year = {2024}, volume = {14}, number = {3}, pages = {460--489}, abstract = {

In this paper, we present a deep learning approach to tackle elliptic inverse source problems. Our method combines Tikhonov regularization with physics-informed neural networks, utilizing separate neural networks to approximate the source term and solution. Firstly, we construct a population loss and derive stability estimates. Furthermore, we conduct a convergence analysis of the empirical risk minimization estimator. This analysis yields a prior rule for selecting regularization parameters, determining the number of observations, and choosing the size of neural networks. Finally, we validate our proposed method through numerical experiments. These experiments also demonstrate the remarkable robustness of our approach against data noise, even at high levels of up to 50%.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.2023-271.290324}, url = {http://global-sci.org/intro/article_detail/eajam/23157.html} }
TY - JOUR T1 - Recovering the Source Term in Elliptic Equation via Deep Learning: Method and Convergence Analysis AU - Duan , Chenguang AU - Jiao , Yuling AU - Yang , Jerry Zhijian AU - Zhang , Pingwen JO - East Asian Journal on Applied Mathematics VL - 3 SP - 460 EP - 489 PY - 2024 DA - 2024/05 SN - 14 DO - http://doi.org/10.4208/eajam.2023-271.290324 UR - https://global-sci.org/intro/article_detail/eajam/23157.html KW - Inverse source problem, deep neural network, stability estimate, convergence rate. AB -

In this paper, we present a deep learning approach to tackle elliptic inverse source problems. Our method combines Tikhonov regularization with physics-informed neural networks, utilizing separate neural networks to approximate the source term and solution. Firstly, we construct a population loss and derive stability estimates. Furthermore, we conduct a convergence analysis of the empirical risk minimization estimator. This analysis yields a prior rule for selecting regularization parameters, determining the number of observations, and choosing the size of neural networks. Finally, we validate our proposed method through numerical experiments. These experiments also demonstrate the remarkable robustness of our approach against data noise, even at high levels of up to 50%.

Duan , ChenguangJiao , YulingYang , Jerry Zhijian and Zhang , Pingwen. (2024). Recovering the Source Term in Elliptic Equation via Deep Learning: Method and Convergence Analysis. East Asian Journal on Applied Mathematics. 14 (3). 460-489. doi:10.4208/eajam.2023-271.290324
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