TY - JOUR
T1 - Accurate Adaptive Deep Learning Method for Solving Elliptic Problems
AU - Ying , Jinyong
AU - Xie , Yaqi
AU - Li , Jiao
AU - Wang , Hongqiao
JO - Communications in Computational Physics
VL - 3
SP - 849
EP - 876
PY - 2025
DA - 2025/03
SN - 37
DO - http://doi.org/10.4208/cicp.OA-2024-0113
UR - https://global-sci.org/intro/article_detail/cicp/23925.html
KW - Deep learning methods, elliptic problems, adaptive sampling method.
AB - <p style="text-align: justify;">Deep learning method is of great importance in solving partial differential
equations. In this paper, inspired by the failure-informed idea proposed by Gao et al.
(SIAM Journal on Scientific Computing 45(4) (2023)) and as an improvement, a new
accurate adaptive deep learning method is proposed for solving elliptic problems, including interface problems and convection-dominated problems. Based on the failure
probability framework, the piece-wise uniform distribution is used to approximate
the optimal proposal distribution and a kernel-based method is proposed for efficient
sampling. Together with the improved Levenberg-Marquardt optimization method,
the proposed adaptive deep learning method shows great potential in improving solution accuracy. Numerical tests on the elliptic problems without interface conditions,
on one elliptic interface problem, and on the convection-dominated problems demonstrate the effectiveness of the proposed method, as it reduces the relative errors by a
factor varying from $10^2$ to $10^4$ for different cases.</p>