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

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.

Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains

“…The exact solution is 25) and, correspondingly The reference solution is again calculated by a FDM with a very fine mesh. DNN results.…”

A Multi-Scale DNN Algorithm for Nonlinear Elliptic Equations with Multiple Scales

“…A conventional DNN model can achieve a satisfactory solution for PDE problems when the coefficient κ(x) is homogeneous (e.g., smooth or possessing few scales) [4,22,39,43]. However, it is difficult to solve PDEs (1.1) with multi-scale κ(x) due to the complex interaction of nonlinearity and multiple scales.…”

Deep Learning for Solving Partial Differential Equations

“…Далее -отыскание оптимальных значений параметров θ нейронной сети, при которых значение функционала (7) минимально будет производиться по методу стохастического градиентного спуска [8]:…”