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Error Analysis of the Deep Mixed Residual Method for High-order Elliptic Equations

Mengjia Bai, Jingrun Chen, Rui Du, Zhiwei Sun

Abstract

This paper presents an a priori error analysis of the Deep Mixed Residual method (MIM) for solving high-order elliptic equations with non-homogeneous boundary conditions, including Dirichlet, Neumann, and Robin conditions. We examine MIM with two types of loss functions, referred to as first-order and second-order least squares systems. By providing boundedness and coercivity analysis, we leverage Céa's Lemma to decompose the total error into the approximation, generalization, and optimization errors. Utilizing the Barron space theory and Rademacher complexity, an a priori error is derived regarding the training samples and network size that are exempt from the curse of dimensionality. Our results reveal that MIM significantly reduces the regularity requirements for activation functions compared to the deep Ritz method, implying the effectiveness of MIM in solving high-order equations.

Error Analysis of the Deep Mixed Residual Method for High-order Elliptic Equations

Abstract

This paper presents an a priori error analysis of the Deep Mixed Residual method (MIM) for solving high-order elliptic equations with non-homogeneous boundary conditions, including Dirichlet, Neumann, and Robin conditions. We examine MIM with two types of loss functions, referred to as first-order and second-order least squares systems. By providing boundedness and coercivity analysis, we leverage Céa's Lemma to decompose the total error into the approximation, generalization, and optimization errors. Utilizing the Barron space theory and Rademacher complexity, an a priori error is derived regarding the training samples and network size that are exempt from the curse of dimensionality. Our results reveal that MIM significantly reduces the regularity requirements for activation functions compared to the deep Ritz method, implying the effectiveness of MIM in solving high-order equations.

Paper Structure

This paper contains 33 sections, 25 theorems, 216 equations.

Table of Contents

  1. Introduction
  2. Related works
  3. Our Contributions
  4. Main results
  5. Model problem
  6. Neural networks
  7. Two-layer neural networks and Barron space
  8. Two-layer neural networks induced by first-order system
  9. Two-layer neural networks induced by second-order system
  10. Main results
  11. Proof of main results
  12. Coercivity and boundedness
  13. Case I: First-order system
  14. Case II: Second-order system
  15. Error decomposition
  16. ...and 18 more sections

Key Result

Theorem 2.1

Suppose $u_\alpha^*\in \mathcal{B}^{2n+2}(\Omega)$ with $\alpha=\mathrm{D}, \mathrm{N},\mathrm{R}$ is a solution of problem 2n-order Laplace equation satisfying boundary conditions Dirichlet boundary condition, Neumann boundary condition or Robin boundary condition. Let Then for the cases of Neumann and Robin boundary, i.e. $\alpha=\mathrm{N},\mathrm{R}$, we have In the case of the Dirichlet bou

Theorems & Definitions (38)

  • Definition 2.1
  • Theorem 2.1: First-order least squares system
  • Theorem 2.2: Second-order least square system
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3: Céa's Lemma
  • proof
  • Remark 3.1
  • Lemma 3.4: First-order system
  • Lemma 3.5: Second-order system
  • ...and 28 more