Accurate Adaptive Deep Learning Method for Solving Elliptic Problems
Year: 2025
Author: Jinyong Ying, Yaqi Xie, Jiao Li, Hongqiao Wang
Communications in Computational Physics, Vol. 37 (2025), Iss. 3 : pp. 849–876
Abstract
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 102 to 104 for different cases.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2024-0113
Communications in Computational Physics, Vol. 37 (2025), Iss. 3 : pp. 849–876
Published online: 2025-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 28
Keywords: Deep learning methods elliptic problems adaptive sampling method.
Author Details
Jinyong Ying Email
Yaqi Xie Email
Jiao Li Email
Hongqiao Wang Email