Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Domain Decomposition Subspace Neural Network Method for Solving Linear and Nonlinear Partial Differential Equations

Zhenxing Fu, Hongliang Liu, Zhiqiang Sheng, Baixue Xing

TL;DR

The paper presents the Domain Decomposition Subspace Neural Network (DD-SNN) method, which integrates domain decomposition with subspace neural networks to solve linear and nonlinear PDEs. Local neural solvers construct basis functions in subspaces, while $C^k$ continuity at interfaces ensures a smooth global solution; nonlinear problems are tackled via Picard and Newton iterations. Across diverse linear and nonlinear tests (Helmholtz, Poisson, parabolic, boundary-layer, Burgers), DD-SNN achieves markedly higher accuracy (often reaching $ orm{e}_{L^2}$ on the order of $10^{-10}$ to $10^{-12}$) with substantially lower training costs than PINNs, LocELM, SNN, ELM, DGM, and DRM. The approach enables parallelizable local training, reduces optimization complexity, and demonstrates strong performance on challenging multiscale and boundary-layer problems, indicating significant practical impact for scalable PDE solvers.

Abstract

This paper proposes a domain decomposition subspace neural network method for efficiently solving linear and nonlinear partial differential equations. By combining the principles of domain decomposition and subspace neural networks, the method constructs basis functions using neural networks to approximate PDE solutions. It imposes $C^k$ continuity conditions at the interface of subdomains, ensuring smoothness across the global solution. Nonlinear PDEs are solved using Picard and Newton iterations, analogous to classical methods. Numerical experiments demonstrate that our method achieves exceptionally high accuracy, with errors reaching up to $10^{-13}$, while significantly reducing computational costs compared to existing approaches, including PINNs, DGM, DRM. The results highlight the method's superior accuracy and training efficiency.

Domain Decomposition Subspace Neural Network Method for Solving Linear and Nonlinear Partial Differential Equations

TL;DR

The paper presents the Domain Decomposition Subspace Neural Network (DD-SNN) method, which integrates domain decomposition with subspace neural networks to solve linear and nonlinear PDEs. Local neural solvers construct basis functions in subspaces, while continuity at interfaces ensures a smooth global solution; nonlinear problems are tackled via Picard and Newton iterations. Across diverse linear and nonlinear tests (Helmholtz, Poisson, parabolic, boundary-layer, Burgers), DD-SNN achieves markedly higher accuracy (often reaching on the order of to ) with substantially lower training costs than PINNs, LocELM, SNN, ELM, DGM, and DRM. The approach enables parallelizable local training, reduces optimization complexity, and demonstrates strong performance on challenging multiscale and boundary-layer problems, indicating significant practical impact for scalable PDE solvers.

Abstract

This paper proposes a domain decomposition subspace neural network method for efficiently solving linear and nonlinear partial differential equations. By combining the principles of domain decomposition and subspace neural networks, the method constructs basis functions using neural networks to approximate PDE solutions. It imposes continuity conditions at the interface of subdomains, ensuring smoothness across the global solution. Nonlinear PDEs are solved using Picard and Newton iterations, analogous to classical methods. Numerical experiments demonstrate that our method achieves exceptionally high accuracy, with errors reaching up to , while significantly reducing computational costs compared to existing approaches, including PINNs, DGM, DRM. The results highlight the method's superior accuracy and training efficiency.

Paper Structure

This paper contains 16 sections, 46 equations, 13 figures, 27 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Domain Decomposition Subspace Neural Network Method
  3. Linear Equations
  4. Nonlinear Equations
  5. Picard Iteration
  6. Newton Iteration
  7. Numerical Experiments
  8. Linear Example
  9. One-Dimensional Linear Helmholtz Equation
  10. Two-Dimensional Linear Poisson Equation
  11. One-Dimensional Parabolic Equation
  12. The Singular Perturbed Boundary Layer Problem
  13. Nonlinear Example
  14. One-Dimensional Nonlinear Helmholtz Equation
  15. One-Dimensional Viscous Burgers Equation
  16. ...and 1 more sections

Figures (13)

  • Figure 1: Neural network framework.
  • Figure 2: Impact of subspace dimension on $L^2$ error
  • Figure 3: Comparison of numerical and exact solutions for the one-dimensional Helmholtz equation
  • Figure 4: Impact of subspace dimension on $L^2$ error
  • Figure 5: Comparison of numerical and exact solutions for the two-dimensional Poisson equation
  • ...and 8 more figures