A mesh-free method for interface problems using the deep learning approach

In this paper, we propose a mesh-free method to solve interface problems using the deep learning approach. Two types of PDEs are considered. The first one is an elliptic PDE with a discontinuous and high-contrast coefficient. While the second one is a linear elasticity equation with discontinuous stress tensor. In both cases, we represent the solutions of the PDEs using the deep neural networks (DNNs) and formulate the PDEs into variational problems, which can be solved via the deep learning approach. To deal with inhomogeneous boundary conditions, we use a shallow neural network to approximate the boundary conditions. Instead of using an adaptive mesh refinement method or specially designed basis functions or numerical schemes to compute the PDE solutions, the proposed method has the advantages that it is easy to implement and is mesh-free. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method for interface problems.