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Chebyshev Spectral Neural Networks for Solving Partial Differential Equations

Pengsong Yin, Shuo Ling, Wenjun Ying

TL;DR

CSNN introduces a single-layer neural network whose neurons are Chebyshev spectral basis functions that automatically satisfy boundary conditions, enabling efficient PDE solving via automatic differentiation. By combining domain transformations for complex geometries with an auxiliary boundary-value network and a residual-based training objective, CSNN achieves high accuracy and fast convergence, often outperforming PINNs on annular, circular, and high-dimensional domains. A theoretical error analysis framework links CSNN performance to spectral convergence under smoothness conditions, while extensive numerical experiments validate the method across 1D–4D elliptic problems and the Helmholtz equation. The work highlights CSNN as a practical, boundary-respecting PDE solver with potential extensions to Stokes, elasticity, and interface problems.

Abstract

The purpose of this study is to utilize the Chebyshev spectral method neural network(CSNN) model to solve differential equations. This approach employs a single-layer neural network wherein Chebyshev spectral methods are used to construct neurons satisfying boundary conditions. The study uses a feedforward neural network model and error backpropagation principles, utilizing automatic differentiation (AD) to compute the loss function. This method avoids the need to solve non-sparse linear systems, making it convenient for algorithm implementation and solving high-dimensional problems. The unique sampling method and neuron architecture significantly enhance the training efficiency and accuracy of the neural network. Furthermore, multiple networks enables the Chebyshev spectral method to handle equations on more complex domains. The numerical efficiency and accuracy of the CSNN model are investigated through testing on elliptic partial differential equations, and it is compared with the well-known Physics-Informed Neural Network(PINN) method.

Chebyshev Spectral Neural Networks for Solving Partial Differential Equations

TL;DR

CSNN introduces a single-layer neural network whose neurons are Chebyshev spectral basis functions that automatically satisfy boundary conditions, enabling efficient PDE solving via automatic differentiation. By combining domain transformations for complex geometries with an auxiliary boundary-value network and a residual-based training objective, CSNN achieves high accuracy and fast convergence, often outperforming PINNs on annular, circular, and high-dimensional domains. A theoretical error analysis framework links CSNN performance to spectral convergence under smoothness conditions, while extensive numerical experiments validate the method across 1D–4D elliptic problems and the Helmholtz equation. The work highlights CSNN as a practical, boundary-respecting PDE solver with potential extensions to Stokes, elasticity, and interface problems.

Abstract

The purpose of this study is to utilize the Chebyshev spectral method neural network(CSNN) model to solve differential equations. This approach employs a single-layer neural network wherein Chebyshev spectral methods are used to construct neurons satisfying boundary conditions. The study uses a feedforward neural network model and error backpropagation principles, utilizing automatic differentiation (AD) to compute the loss function. This method avoids the need to solve non-sparse linear systems, making it convenient for algorithm implementation and solving high-dimensional problems. The unique sampling method and neuron architecture significantly enhance the training efficiency and accuracy of the neural network. Furthermore, multiple networks enables the Chebyshev spectral method to handle equations on more complex domains. The numerical efficiency and accuracy of the CSNN model are investigated through testing on elliptic partial differential equations, and it is compared with the well-known Physics-Informed Neural Network(PINN) method.
Paper Structure (24 sections, 53 equations, 12 figures, 3 tables, 2 algorithms)

This paper contains 24 sections, 53 equations, 12 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Artificial Neural Networks for PDEs
  3. Single Layer Artificial Neural Network Model
  4. Spectral Methods for Complex Domain Problems
  5. Preliminaries
  6. Chebyshev Polynomials
  7. Basis Functions for Spectral Method
  8. PDEs in Complex Geometries
  9. Chebyshev Spectral Method Neural Network(CSNN)
  10. Solving PDEs by CSNN
  11. PDEs in Rectangular Domains
  12. PDEs in Complex Domains
  13. PDEs After Coordinate Transformation on Variables
  14. Correction for $\hat{u_0}$ Network
  15. Approximation Theory and Error Analysis for CSNN
  16. ...and 9 more sections

Figures (12)

  • Figure 1: The structure of CSNN model(only the blue lines contain the neural network weight parameters that need to be trained).
  • Figure 2: the original domain
  • Figure 3: the parameter domain(with $r$ on the horizontal axis and $\theta$ on the vertical axis)
  • Figure 4: the problem on the boundary
  • Figure 5: predict for 2D Poisson equation(the left image shows the predicted values by the CSNN network, the middle image displays the accurate solution of the equation, and the right image represents the computed error).
  • ...and 7 more figures