Deep neural network model for fractional-order cancer chemotherapy dynamics.

This article presents a novel hybrid computational framework for a fractional-order cancer chemotherapy model based on Caputo reaction–diffusion equations. Its main novelty is the coupling of a high-fidelity fractional Adams–Bashforth–Moulton (ABM) predictor–corrector solver with a carefully designed hyperbolic tangent sigmoid deep neural network (HTS–DNN) adapted to nonlocal, memory-dependent tumor–immune–drug dynamics. Unlike standard feed-forward networks using ReLU-type activations, the proposed model employs four hidden layers with (50, 30, 20, 10) neurons, smooth HTS activations, and Levenberg–Marquardt optimization, providing an efficient approximation framework for hereditary fractional trajectories. High-fidelity data generated by the ABM solver are used to train the HTS–DNN with an 80%/10%/10% train/validation/test split. For several fractional orders and initial conditions, the model accurately reproduces the numerical solutions, achieving mean squared errors below [Formula: see text] and uniformly small time-dependent errors. It also provides more than five orders of magnitude reduction in prediction time compared with direct fractional simulation, while preserving key biological features such as memory-driven tumor suppression, delayed immune activation, and concentration-dependent drug effects. These results show that the proposed ABM–HTS–DNN framework is an accurate, robust, and computationally efficient surrogate for fractional cancer chemotherapy dynamics, enabling rapid exploration of therapy schedules and parameter regimes.