VooPINN: Physics-informed neural network with variable-order operator for forward and inverse problems of time fractional partial differential equations

In this paper, we develop a novel physics-informed neural network with variable-order operator to solve variable-order time fractional partial differential equations (PDEs), termed VooPINN. In the VooPINN framework, we design an operator network to reconstruct integral kernel of the variable-order fractional derivative, converting it from a power function form to a linear function form. Moreover, the operator network is made to be associated only with the spatial variable and the temporal variable using a new integral operation. Leveraging the separable property of the linear kernel, we transform complex integral into a form amenable to direct application of integral rule, thereby eliminating the integral operation. We conduct several numerical experiments to evaluate the feasibility and effectiveness of the VooPINN. The results illustrate that our proposed method exhibits high-precision prediction capability in both froward and inverse problems, and can still maintain excellent performance especially in high-dimensional scenarios, such as 10D and 50D.