Study of fractional-order reaction-advection-diffusion equation using neural network method

Abstract

In the present article the spatio-temporal fractional-order nonlinear reaction-advection-diffusion equation is solved using the neural network method (NNM). Shifted Legendre orthogonal polynomials with variable coefficients are used in the network’s construction. The characteristics of a fractional-order derivative are used to determine the loss function of a neural network. The permissible learning rate range is discussed in detail, assuming that the Lipschitz hypothesis is accurate for the nonlinearity in reaction term. We have demonstrated the application of the NNM on two numerical examples by utilizing the neural networks which had been repeatedly trained on the training set. In other words, we have validated the effectiveness of the method for such problems. The effects of reaction term and also the degree of nonlinearity in reaction and advection terms on the solution profile are visualized through graphical presentations for specific test cases.