  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.
