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A Natural Deep Ritz Method for Essential Boundary Value Problems

Haijun Yu, Shuo Zhang

TL;DR

This paper proposes a novel, intrinsic approach to impose essential boundary conditions through a framework inspired by intrinsic structures and demonstrates the effectiveness of this approach using the deep Ritz method applied to Poisson problems, with the potential for extension to more general equations and other deep learning techniques.

Abstract

Deep neural network approaches show promise in solving partial differential equations. However, unlike traditional numerical methods, they face challenges in enforcing essential boundary conditions. The widely adopted penalty-type methods, for example, offer a straightforward implementation but introduces additional complexity due to the need for hyper-parameter tuning; moreover, the use of a large penalty parameter can lead to artificial extra stiffness, complicating the optimization process. In this paper, we propose a novel, intrinsic approach to impose essential boundary conditions through a framework inspired by intrinsic structures. We demonstrate the effectiveness of this approach using the deep Ritz method applied to Poisson problems, with the potential for extension to more general equations and other deep learning techniques. Numerical results are provided to substantiate the efficiency and robustness of the proposed method.

A Natural Deep Ritz Method for Essential Boundary Value Problems

TL;DR

This paper proposes a novel, intrinsic approach to impose essential boundary conditions through a framework inspired by intrinsic structures and demonstrates the effectiveness of this approach using the deep Ritz method applied to Poisson problems, with the potential for extension to more general equations and other deep learning techniques.

Abstract

Deep neural network approaches show promise in solving partial differential equations. However, unlike traditional numerical methods, they face challenges in enforcing essential boundary conditions. The widely adopted penalty-type methods, for example, offer a straightforward implementation but introduces additional complexity due to the need for hyper-parameter tuning; moreover, the use of a large penalty parameter can lead to artificial extra stiffness, complicating the optimization process. In this paper, we propose a novel, intrinsic approach to impose essential boundary conditions through a framework inspired by intrinsic structures. We demonstrate the effectiveness of this approach using the deep Ritz method applied to Poisson problems, with the potential for extension to more general equations and other deep learning techniques. Numerical results are provided to substantiate the efficiency and robustness of the proposed method.

Paper Structure

This paper contains 10 sections, 3 theorems, 12 equations, 16 figures.

Table of Contents

  1. Introduction
  2. A natural formulation of the essential boundary value problems
  3. Poisson equations of Dirichlet type
  4. Elliptic problem with varying coefficient in divergence form
  5. A simple approach for the interface problem
  6. Natural deep Ritz methods
  7. Natural deep Ritz method for Poisson equations with Dirichlet boundary conditions
  8. Natural deep Ritz method for elliptic interface problems
  9. Numerical experiments
  10. Concluding remarks

Key Result

Theorem 2.1

Let $u$ be the solution of eq:modelpoisson, and $u^*$ be obtained by the four steps below: Then $u^*=u$.

Figures (16)

  • Figure 1: Illustration of the domain and the interface
  • Figure 2: Training results : The learning rate (top-left), training loss (top-right) $L2$ Testing error on $\Omega$ (bottom-left) and boundary $\Gamma$ (bottom-right) for Example 1 using New method
  • Figure 3: The exact solution and learned solution for Example 1 using New method
  • Figure 4: Training results : The learning rate (top-left), training loss (top-right) $L2$ Testing error on $\Omega$ (bottom-left) and boundary $\Gamma$ (bottom-right) for Example 1 using Deep Ritz method
  • Figure 5: The exact solution and learned solution for for Example 1 using Deep Ritz method
  • ...and 11 more figures

Theorems & Definitions (9)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Theorem 2.5
  • proof
  • Remark 2.6