A Physics-Informed Neural Network Based on the Separation of Variables for Solving the Distributed-Order Time-Fractional Advection–Diffusion Equation

In this work, we propose a new physics-informed neural network framework based on the method of separation of variables (SVPINN) to solve the distributed-order time-fractional advection–diffusion equation. We develop a new method for calculating the distributed-order derivative, which enables the fractional integral to be modeled by a network and directly solved by combining automatic differentiation technology. In this way, the approximation of the distributed-order derivative is integrated into the parameter training system of the network, and the data-driven adaptive learning mechanism is used to replace the numerical discretization scheme. In the SVPINN framework, we decompose the kernel function of the Caputo integral into three independent functions using the method of separation of variables, and apply a neural network as a surrogate model for the modified integral and the function related to the time variable. The new physical constraint generated by the modified integral serves as an extra supervised learning task for the network. We systematically evaluated the feasibility of the SVPINN on several numerical experiments and demonstrated its performance.