Neural network method for solving nonlinear fractional advection-diffusion equation with spatiotemporal variable-order

•Neural networks based on Legendre polynomials are established to solve the variable order fractional advection-diffusion equation.•The error functions containing adjustable parameters for the training sets are constructed.•Under the Lipschitz hypothesis, the range of learning rate is analyzed. In this article, neural network method (NNM) is presented to solve the spatiotemporal variable-order fractional advection-diffusion equation with a nonlinear source term. The network is established by using shifted Legendre orthogonal polynomials with adjustable coefficients. According to the properties of variable fractional derivative, the loss function of neural network is deduced theoretically. Assume that the source function satisfies the Lipschitz hypothesis, the reasonable range for learning rate is discussed in details. Then neural networks are trained repeatedly on the training set to reduce the loss functions for two numerical examples. Numerical results show that the neural network method is better than the finite difference method in solving some nonlinear variable fractional order problems. Finally, several graphs and some numerical analysis are given to confirm our conclusions.