An approach based on neural networks for solving time fractional diffusion-wave equation

Abstract

This article delves into the application of a neural network (NN) to solve the time fractional diffusion-wave equation (TFDWE), wherein the fractional term is expressed in the Caputo sense of order alpha with 1 < alpha < 2. A numerical scheme of order O(t(3-alpha)) is employed to approximate the Caputo derivative. A structured multilayer NN model, encompassing an input layer, a hidden layer, and an output layer, is meticulously crafted. The proposed deep neural network (DNN) framework is also developed for the two-dimensional case on an irregular domain, and the Adam optimization algorithm is employed to enhance performance. The trial solution of the NN is described as the sum of two terms: the first term satisfies the prescribed initial and boundary conditions, while the second term corresponds to the NN's output and involves unknown weights. The back-propagation algorithm iteratively adjusts the weights of the multilayer NN to minimize the loss function through a gradient descent scheme. Additionally, the convergence analysis of the gradient descent algorithm for the TFDWE is discussed. To validate the efficacy of the proposed approach, four numerical experiments are presented, with results tested and compared using both Sigmoid and ReLU activation functions.