Accurate Adaptive Deep Learning Method for Solving Elliptic Problems (Communications in Computational Physics, 2025)
Deep learning method is of great importance in solving partial differentialequations. 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 newaccurate adaptive deep learning method is proposed for solving elliptic problems, including interface problems and convection-dominated problems. Based on the failureprobability framework, the piece-wise uniform distribution is used to approximatethe optimal proposal distribution and a kernel-based method is proposed for efficientsampling. 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 afactor varying from $10^2$ to $10^4$ for different cases.
Keywords: Deep learning methods; elliptic problems; adaptive sampling method.