Chebyshev spectral approximation-based physics-informed neural network for solving higher-order nonlinear differential equations

Physics-informed neural networks (PINNs) typically involve higher-order partial derivatives with respect to their inputs, which are too costly to compute and store by using automatic differentiation (AD) even for relatively small neural networks. This paper presents an improved kind of PINN, named CD-PINN, where a Chebyshev spectral method is introduced to replace the AD method to accelerate the computation of higher-order derivatives in PDE residual, while the AD method is still used to compute the derivatives of the PDE residual with respect to weights and biases. This method directly approximates the input-output relationship of the neural network in small subdomains by using the Chebyshev series expansion. Benefiting from the superb properties of the derivatives of the Chebyshev polynomials, the higher-order derivatives of neural networks can be computed very quickly. A series of nonlinear differential equations are solved by the CD-PINN and AD-PINN simultaneously. The results show that the CD-PINN can achieve comparable accuracy to the AD-PINN and simultaneously reduce computation time significantly. To achieve the same accuracy, the time needed by the CD-PINN is much less than that of the AD-PINN. The larger the size of the neural network, the more layers it has, and the higher the derivative order in the PDEs, the greater the advantage of the CD-PINN. This research demonstrates an alternative way to improve the training speed of PINNs and a promising approach for blending traditional numerical methods with PINNs.