Accurate adaptive deep learning method for solving elliptic problems

TL;DR

This paper introduces an accurate adaptive deep learning approach for elliptic PDEs, leveraging failure probability, kernel sampling, and improved optimization to significantly enhance solution accuracy across various elliptic problems.

Contribution

It presents a novel adaptive deep learning method that combines failure-informed sampling and advanced optimization for solving complex elliptic PDEs more accurately.

Findings

Reduces relative errors by 100 to 10,000 times in tests

Effective for interface and convection-dominated problems

Demonstrates significant accuracy improvements over existing methods

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 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 $10^2$ to $10^4$ for different cases.