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.

翻译：深度学习方法在求解偏微分方程中具有重要意义。本文受Gao等人（SIAM Journal on Scientific Computing 45(4)(2023)）提出的失效信息思想启发并加以改进，提出了一种新的精确自适应深度学习方法，用于求解椭圆问题，包括界面问题和对流主导问题。基于失效概率框架，采用分段均匀分布逼近最优提议分布，并提出一种基于核的方法实现高效采样。结合改进的Levenberg-Marquardt优化方法，所提出的自适应深度学习方法在提升求解精度方面展现出巨大潜力。对无界面条件的椭圆问题、椭圆界面问题以及对流主导问题的数值测试验证了该方法的有效性，在不同算例中将相对误差降低了$10^2$至$10^4$倍。