Year: 2024
Author: Chenguang Duan, Yuling Jiao, Jerry Zhijian Yang, Pingwen Zhang
East Asian Journal on Applied Mathematics, Vol. 14 (2024), Iss. 3 : pp. 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%.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/eajam.2023-271.290324
East Asian Journal on Applied Mathematics, Vol. 14 (2024), Iss. 3 : pp. 460–489
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 30
Keywords: Inverse source problem deep neural network stability estimate convergence rate.