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Subspace method based on neural networks for solving the partial differential equation

Zhaodong Xu, Zhiqiang Sheng

TL;DR

The paper tackles the challenge of achieving high-accuracy PDE solutions with neural networks at reasonable cost, a area where many NN-based methods struggle with either precision or efficiency. It proposes a subspace method (SNN) that builds a finite-dimensional subspace from neural-network base functions and then determines a solution within that subspace, via two algorithms: SNN-D (discrete collocation) and SNN-I (integral $L^2$ residual). Key contributions include a general training framework that avoids hyperparameter penalties, a clear separation between base-function training and coefficient fitting, and demonstrated superiority over PINN and DGM across Helmholtz, Poisson, Advection, Parabolic, and Anisotropic diffusion problems, with errors reaching as low as $10^{-10}$ and training typically requiring only a few hundred to a few thousand epochs. The approach is robust to problem features like high anisotropy and interfaces, and it establishes a practical, high-precision alternative for PDE solving in scientific computing contexts.

Abstract

We present a subspace method based on neural networks (SNN) for solving the partial differential equation with high accuracy. The basic idea of our method is to use some functions based on neural networks as base functions to span a subspace, then find an approximate solution in this subspace. We design two special algorithms in the strong form of partial differential equation. One algorithm enforces the equation and initial boundary conditions to hold on some collocation points, and another algorithm enforces $L^2$-norm of the residual of the equation and initial boundary conditions to be $0$. Our method can achieve high accuracy with low cost of training. Moreover, our method is free of parameters that need to be artificially adjusted. Numerical examples show that the cost of training these base functions of subspace is low, and only one hundred to two thousand epochs are needed for most tests. The error of our method can even fall below the level of $10^{-10}$ for some tests. The performance of our method significantly surpasses the performance of PINN and DGM in terms of the accuracy and computational cost.

Subspace method based on neural networks for solving the partial differential equation

TL;DR

The paper tackles the challenge of achieving high-accuracy PDE solutions with neural networks at reasonable cost, a area where many NN-based methods struggle with either precision or efficiency. It proposes a subspace method (SNN) that builds a finite-dimensional subspace from neural-network base functions and then determines a solution within that subspace, via two algorithms: SNN-D (discrete collocation) and SNN-I (integral residual). Key contributions include a general training framework that avoids hyperparameter penalties, a clear separation between base-function training and coefficient fitting, and demonstrated superiority over PINN and DGM across Helmholtz, Poisson, Advection, Parabolic, and Anisotropic diffusion problems, with errors reaching as low as and training typically requiring only a few hundred to a few thousand epochs. The approach is robust to problem features like high anisotropy and interfaces, and it establishes a practical, high-precision alternative for PDE solving in scientific computing contexts.

Abstract

We present a subspace method based on neural networks (SNN) for solving the partial differential equation with high accuracy. The basic idea of our method is to use some functions based on neural networks as base functions to span a subspace, then find an approximate solution in this subspace. We design two special algorithms in the strong form of partial differential equation. One algorithm enforces the equation and initial boundary conditions to hold on some collocation points, and another algorithm enforces -norm of the residual of the equation and initial boundary conditions to be . Our method can achieve high accuracy with low cost of training. Moreover, our method is free of parameters that need to be artificially adjusted. Numerical examples show that the cost of training these base functions of subspace is low, and only one hundred to two thousand epochs are needed for most tests. The error of our method can even fall below the level of for some tests. The performance of our method significantly surpasses the performance of PINN and DGM in terms of the accuracy and computational cost.
Paper Structure (14 sections, 25 equations, 11 figures, 16 tables)

This paper contains 14 sections, 25 equations, 11 figures, 16 tables.

Table of Contents

  1. Introduction
  2. Subspace method based on neural networks
  3. Neural networks architecture
  4. A general frame of subspace method based on neural networks
  5. Two special algorithms
  6. The algorithm in discrete form: SNN-D
  7. The algorithm in integral form: SNN-I
  8. Numerical results
  9. Helmholtz equation
  10. Poisson equation
  11. Advection equation
  12. Parabolic equation
  13. Anisotropic diffusion equation
  14. Conclusion

Figures (11)

  • Figure 1: The neural networks architecture.
  • Figure 2: Solution obtained by SNN-D and SNN-I for Helmholtz Equation.
  • Figure 3: Point-wise errors of SNN-D and SNN-I for Helmholtz equation.
  • Figure 4: Error variation with subspace dimension at a fixed number of 1000 sampling points and error variation with the number of sampling points at a fixed subspace dimension of 300 for SNN-D.
  • Figure 5: Error variation with subspace dimension at a fixed number of 300 integration points and error variation with the number of integration points at a fixed subspace dimension of 300 for SNN-I.
  • ...and 6 more figures

Theorems & Definitions (5)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5