Chebyshev Spectral Neural Networks for Solving Partial Differential Equations
This work addresses solving high-dimensional PDEs for computational science and engineering, representing an incremental improvement over existing neural network methods.
The study tackled solving partial differential equations by developing a Chebyshev Spectral Neural Network (CSNN) model, which improved training efficiency and accuracy compared to the Physics-Informed Neural Network (PINN) method, as demonstrated through numerical tests on elliptic equations.
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.