A Semi-Analytical Fractional Order Neural Network Framework for Two-Dimensional Time Fractional Reaction-Diffusion Problems

In this research, we provide a convergence of the Adomian decomposition based neural network method for time fractional reaction diffusion initial boundary value problems in 2D, where the fractional time derivative is considered in the Caputo sense. We propose an idea of a proper combination of Adomian decomposition method (ADM) and Physics-informed neural network (PINN), after semi-discretization of the time fractional derivative. PINN uses the ADM based partial sum inside the loss function before generating the appropriate ADM-PINN approximation. In addition, we produce the required sufficient conditions on the given data under which the usual Adomian decomposition method converges in a bounded domain. Furthermore, we analyze the error bound of the proposed method for the time fractional model and demonstrate the convergence. Several numerical examples are produced to show the effectiveness of the present ADM-PINN approach. It is observed that the proposed method is highly effective for several two-dimensional time fractional problems including the Schrodinger equation for convergent approximation of the computed solution.