Feedforward Neural Network for Solving Caputo Fractional Differential Equations with Application to Tumor Dynamics

Abstract

Fractional differential equations (FDEs) play an important role in modeling complex dynamical systems because they capture memory and hereditary effects more effectively than classical ordinary differential equations (ODEs). However, FDEs are generally more difficult to solve due to their nonlocal characteristics and higher computational complexity. Traditional numerical methods such as the predictor-corrector (PC) and Euler methods have been widely used but may face limitations in terms of computational cost and accuracy. Recently, artificial neural network (ANN) has emerged as a promising alternative for solving differential equations due to their strong approximation capability. This study proposes a neural network-based framework for solving FDEs by implementing a feedforward neural network (FNN) integrated with the Broyden-Fletcher-Goldfarb-Shanno(BFGS) optimization algorithm. The neural network parameters are initialized and iteratively updated using the BFGS algorithm by minimizing a loss function defined from the residual of the governing equation. The proposed method is evaluated on several linear and nonlinear FDE problems and further applied to a fractional tumor dynamics model. Numerical results show that the proposed method achieves higher accuracy and lower computational cost compared to the Adomian Decomposition Method (ADM) and previously published methods, while yielding a continuous and differentiable solution throughout the domain.