2024/04/29 by Jingyong Ying, Ying, Jingyong, Yaqi Xie +5
Computer Science · Engineering · #Advanced Numerical Analysis Techniques #Advanced Numerical Methods in Computational Mathematics #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Image and Signal Denoising Methods #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2404.18838
openalex publication_date 2024/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 the interface problems and the convection-dominated problems. Based on the failure probability framework, the piece-wise uniform distribution is used to approximate the optimal proposal distribution and an 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 the 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.