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 - <p style="text-align: justify;">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%.</p>