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

## TL;DR

This paper introduces a mesh-free deep learning method for solving interface problems involving PDEs with discontinuities, offering an easy-to-implement alternative to traditional mesh-based techniques.

## Contribution

It develops a novel mesh-free deep learning framework for interface problems, handling discontinuities without adaptive meshes or special basis functions.

## Key findings

- Demonstrates high accuracy in solving interface PDEs
- Shows efficiency and ease of implementation
- Validates approach with numerical experiments

## Abstract

In this paper, we propose a mesh-free method to solve interface problems using the deep learning approach. Two interface problems 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 formulate the PDEs into variational problems, which can be solved via the deep learning approach. To deal with the inhomogeneous boundary conditions, we use a shallow neuron 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 mesh-free. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method for interface problems.