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

Document Type : Research article

Authors

1 Department of Mathematics, Kongu Engineering College, Erode, India

2 Clinical Research Development Unit of Rouhani Hospital, Babol University of Medical Sciences, Babol, Iran

3 Payame Noor University (PNU), Tehran, Iran

4 Department of Physics, Sari Branch, Islamic Azad University, Sari, Iran

Abstract

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{bvp4c} solver demonstrates that the ANN surrogate achieves high-accuracy predictions with error magnitudes below $\mathcal{O}(10^{-4})$ while reducing computational cost by approximately 45\%. 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.

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Analytical and Numerical Solutions for Nonlinear Equations
Volume 11, Issue 2
September 2026
Pages 220-237

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History

  • Receive Date: 21 April 2026
  • Revise Date: 09 May 2026
  • Accept Date: 02 June 2026
  • Publish Date: 04 September 2026

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