A deep Legendre polynomial neural network for nonlinear time–space fractional reaction–diffusion equations

In this article, an investigation has been carried out on nonlinear time-space fractional reaction–diffusion equations governed by Caputo derivatives. A deep neural network approach based on shifted Legendre polynomial expansions has been developed to obtain the numerical solution of the considered model. The proposed architecture consists of a four-layer neural network-comprising input, hidden, and output layers—where Legendre polynomials of various degrees are employed as activation functions. The exact solution is encoded into the model through an appropriate forcing term, allowing the network to capture the underlying dynamics of the fractional system effectively. Furthermore, a detailed comparison with the classical finite difference method has been conducted, which highlights the superior accuracy and efficiency of the proposed approach. The high accuracy of the method has been demonstrated through absolute error plots and comparative numerical tables, confirming its effectiveness in solving time–space fractional partial differential equations.