A Mesh-free Method Using Piecewise Deep Neural Network for Elliptic Interface Problems

TL;DR

This paper introduces a mesh-free deep learning approach for elliptic interface problems, employing multiple neural networks across sub-domains and an adaptive sampling strategy to enhance efficiency and accuracy in 2D and 3D cases.

Contribution

The paper presents a novel deep neural network-based mesh-free method with adaptive sampling for solving complex elliptic interface problems without meshing.

Findings

Effective in 2D and 3D problems

No meshing or complex numerical integration needed

High accuracy demonstrated in numerical experiments

Abstract

In this paper, we propose a novel mesh-free numerical method for solving the elliptic interface problems based on deep learning. We approximate the solution by the neural networks and, since the solution may change dramatically across the interface, we employ different neural networks in different sub-domains. By reformulating the interface problem as a least-squares problem, we discretize the objective function using mean squared error via sampling and solve the proposed deep least-squares method by standard training algorithms such as stochastic gradient descent. The discretized objective function utilizes only the point-wise information on the sampling points and thus no underlying mesh is required. Doing this circumvents the challenging meshing procedure as well as the numerical integration on the complex interface. To improve the computational efficiency for more challenging problems, we further design an adaptive sampling strategy based on the residual of the least-squares function and propose an adaptive algorithm. Finally, we present several numerical experiments in both 2D and 3D to show the flexibility, effectiveness, and accuracy of the proposed deep least-square method for solving interface problems.