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

Zhongjian Wang, Zhiwen Zhang

TL;DR

This work presents a mesh-free deep learning framework to solve interface problems by reformulating PDEs as variational problems and solving them with deep neural networks in the deep Ritz paradigm. It handles both elliptic equations with discontinuous, high-contrast coefficients and linear elasticity with discontinuous stress tensors, including inhomogeneous boundary data via a shallow network that augments a homogeneous auxiliary problem. The approach does not require explicit interface meshing or specialized basis functions, and optimization is performed with SGD/Adam on a neural-parameter space; numerical experiments demonstrate accurate solutions and robustness, albeit with observed nonconvexity-related convergence behavior. The method offers a simple, flexible, and scalable mesh-free alternative for complex interface problems with potential impact in engineering and physics simulations.

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.

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

TL;DR

This work presents a mesh-free deep learning framework to solve interface problems by reformulating PDEs as variational problems and solving them with deep neural networks in the deep Ritz paradigm. It handles both elliptic equations with discontinuous, high-contrast coefficients and linear elasticity with discontinuous stress tensors, including inhomogeneous boundary data via a shallow network that augments a homogeneous auxiliary problem. The approach does not require explicit interface meshing or specialized basis functions, and optimization is performed with SGD/Adam on a neural-parameter space; numerical experiments demonstrate accurate solutions and robustness, albeit with observed nonconvexity-related convergence behavior. The method offers a simple, flexible, and scalable mesh-free alternative for complex interface problems with potential impact in engineering and physics simulations.

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.

Paper Structure

This paper contains 12 sections, 1 theorem, 31 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Some preliminaries
  3. The DNN and its approximation property
  4. Formulation of the deep Ritz method
  5. Inhomogeneous boundary condition
  6. Derivation of the methodology
  7. Elliptic PDEs with discontinuous and high-contrast coefficients
  8. Linear elasticity with discontinuous stress tensors
  9. Numerical Example
  10. 2D high-contrast elliptic problems
  11. 2D Linear elasticity interface problem
  12. Conclusions

Key Result

Proposition 2.1

Given a locally convex finite element grid ${\cal T}_h$, any linear finite element function with $N$ degrees of freedom, can be written as a ReLU-DNN with at most $k = \lceil\log_2 k_h\rceil+1$ hidden layers and at most $\mathcal{O}(k_hN)$ number of the neurons, where $k_h$ denotes the maximum numbe

Figures (7)

  • Figure 1: Network Layout for $\tilde{g}$.
  • Figure 3: Profile of the high-contrast coefficient $\alpha$ on $D$.
  • Figure 4: High contrast problem, $\alpha_0=1000$, $\alpha_1=1$ case: (a) profile of $g$; (b) profile of $u'$; (c) decay of the Lagrangian during the training process; (d) profile of the DNN solution $u$ at the final step; (e) profile of the reference solution; (f) decay of the $L_2$ relative error during the training process.
  • Figure 5: High contrast problem, $\alpha_0=1$, $\alpha_1=1000$ case: (a) profile of $g$; (b) profile of $u'$; (c) decay of the Lagrangian during the training process; (d) profile of the DNN solution $u$ at the final step; (e) profile of the reference solution; (f) decay of the $L_2$ relative error during the training process.
  • Figure 6: Histogram of the number of steps to obtain a convergence result.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Proposition 2.1
  • Remark 4.1