Artificial Neural Network and Physics-Informed Neural Network Assisted Analytical Solutions for Nonlinear Fractional Biological Models Using Local Fractional Operators

This study presents a hybrid computational framework integrating analytical methods, artificial neural networks (ANN), and physics-informed neural networks (PINNs) for solving nonlinear fractional biological models governed by local fractional operators. In contrast to conventional approaches, the proposed framework incorporates local fractional calculus to effectively represent fractal and heterogeneous biological structures. An analytical solution is first derived using the Natural transform in conjunction with the Adomian decomposition method, yielding closed-form expressions in terms of Mittag--Leffler functions. This analytical formulation is subsequently utilized as a knowledge-driven prior to train an ANN-based surrogate model, thereby improving convergence efficiency and reducing computational complexity. Furthermore, a physics-informed neural network is constructed by embedding the governing fractional differential equation into the loss function via a Gr"unwald--Letnikov approximation, enabling accurate learning of long-memory effects without requiring explicit analytical solutions. Comparative analysis with classical numerical solutions obtained using the \texttt{bvp۴c} solver demonstrates that the ANN surrogate achieves high-accuracy predictions with error magnitudes below \mathcal{O}(۱۰^{-۴}) while reducing computational cost by approximately ۴۵\%. The PINN framework further exhibits strong capability in capturing intrinsic fractional dynamics under limited data conditions. Overall, the proposed ADM--ANN--PINN hybrid architecture provides a robust, efficient, and physically consistent framework for modeling complex nonlinear fractional biological systems, thereby advancing the state-of-the-art in fractional computational intelligence.

کلیدواژه ها:

Fractional differential equations ، Adomian decomposition method ، Artificial Neural Networks ، Physics-informed neural networks ، Mittag-Leffler function ، Reaction-diffusion systems