Hybrid Physics-Informed Neural Networks Methodology for Solving Nonlinear Space-Fractional Reaction-Diffusion Models

Abstract

Studying a Physics-Informed Neural Networks (PINNs) methodology to solve one-dimensional nonlinear fractional reaction-diffusion equations in space is the goal of our paper. PINNs are developed as global-in-time solvers by embedding the governing fractional partial differential equations (FPDE), initial and boundary conditions directly into the loss function. The fractional diffusion operator is incorporated into the PINN framework through a discrete spectral representation, while temporal derivatives are obtained via automatic differentiation. We formulate a framework and validate the efficiency and accuracy by solving three reaction-diffusion problems. We employed a high-order exponential time-differencing Runge-Kutta method of fourth order (ETDRK4) combined with a spectral discretization of the fractional Laplacian to compute accurate reference solutions. The obtained results demonstrate that the proposed PINN framework accurately captures the solution dynamics showing high agreement with exact and numerical solutions.