An inverse problem of determining the parameters in diffusion equations by using fractional physics-informed neural networks

Abstract

In this study, we address an inverse problem in nonlinear time-fractional diffusion equations using a deep neural network . The challenge arises from the equation's nonlinear behavior , the involvement of time-based fractional Caputo derivatives , and the need to estimate parameters influenced by space or the solution of the fractional PDE . Our solution involves a fractional physics-informed neural network (FPINN). Initially, we use FPINN to solve a straightforward problem. Then, we apply FPINN to the inverse problem of estimating parameter and model non-linearity. For the inverse problem, we enhance our method by including the mean square error of final observations in the FPINN's cost function . This adjustment helps effectively in tackling the unique challenges of the time-fractional diffusion equation. Numerical tests involving regular and singular examples demonstrate the effectiveness of the physics-informed neural network approach in accurately recovering parameters. We reinforce this finding through a numerical comparison with alternative methods such as the alternating direction multiplier method (ADMM), the gradient descent , and the DeepONets (deep operator networks) method.