Physics-informed neural network algorithm for solving forward and inverse problems of variable-order space-fractional advection–diffusion equations

Abstract

A new physics-informed neural network (PINN) algorithm is proposed to solve variable-order space-fractional partial differential equations (PDEs). For the forward problem, PINN algorithm based on a power series expansion is established to solve the space-fractional advection–diffusion equations with variable coefficients in one and two dimensions. The loss function is constructed by taking the coefficients in the power series expansion as the output of the network. The learning rate range is also analyzed to ensure that the error is reduced with respect to the training time. For the inverse problem, a novel dual-network architecture based on neural network parallelism is designed to estimate the order of variable fractional operator. One of the networks outputs the coefficients of the power series expansion, which is a function that depends only on time variables, and the other is fitted to the order of variable fractional operator, which is a function that is related to both space and time variables. Compared with the forward problem, we make more comparisons between the approximated solution and the exact solution at some interior points when constructing the loss function to improve the fitting ability of the inverse problem. The exact solutions are easy to find and can be readily obtained with a fine mesh. Several numerical examples are illustrated with graphs. These results confirm the effectiveness of our method in solving variable-order space-fractional partial differential equations.