Chebyshev Spectral Neural Networks for Solving Partial Differential Equations

TL;DR

This paper introduces Chebyshev Spectral Neural Networks (CSNN), a novel approach combining spectral methods and neural networks to efficiently solve high-dimensional partial differential equations with improved accuracy and ease of implementation.

Contribution

The study presents a new CSNN model that integrates Chebyshev spectral methods into neural networks, enhancing training efficiency and accuracy for complex PDEs compared to existing methods.

Findings

CSNN achieves higher accuracy than PINN on test PDEs.

The method effectively handles high-dimensional and complex domain problems.

CSNN avoids solving non-sparse linear systems, simplifying implementation.

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.