Revisiting Fourier and Chebyshev Spectral Methods with Physics-Informed Machine Learning

Spectral methods solve a system of ordinary differential equations (ODEs) by writing their solution as a finite composition of certain basis functions and optimizing the coefficients in the sum to satisfy the differential equations as much as possible. The well-known Fourier and Chebyshev spectral methods utilize the strong approximation capability of the Fourier and Chebyshev series. By placing the vectorization of Fourier and Chebyshev basis functions at the network edge, we revisit these spectral methods with modern machine learning and automatic differentiation. This leads to new physics-informed neural network (PINN) layers that can learn the solution of arbitrary ODEs as an explicit sum of known basic functions rather than as an implicit function in traditional neural network layers. Numerical results show that our approach can provide continuous and differentiable solutions for ODEs with good interpolation in a computationally effective manner using a smaller number of network parameters compared to traditional neural network layers in existing PINNs.