A Stable Spectral Bernstein Physics-Informed Neural Network for Solving Fractional Allee Population Models with Caputo Derivatives

Abstract

Fractional-order population models have attracted considerable attention because they effectively capture memory and hereditary effects that are absent in classical integer-order models. Among these, the fractional Allee effect model provides a realistic framework for describing population persistence and extinction by incorporating a critical population threshold. In this work, we develop a Stable Spectral Bernstein Physics-Informed Neural Network (SBNN-PINN) to solve the Caputo fractional-order Allee effect model. The proposed framework employs a trainable Bernstein polynomial expansion with a logarithmically stable basis and an analytical Bernstein differentiation matrix to evaluate the Caputo fractional derivative efficiently through matrix--vector multiplication, thereby eliminating repeated numerical discretization during training. Chebyshev--Gauss--Lobatto collocation points are used to improve numerical stability, while a hybrid Adam--L-BFGS optimization strategy accelerates convergence. Stability of the equilibrium points is investigated using the Jacobian matrix and Matignon's stability theorem, confirming that the extinction and carrying-capacity equilibria are locally stable, whereas the Allee threshold equilibrium is unstable. The proposed method is validated using the Method of Manufactured Solutions over fractional orders $0.1 \leq \alpha \leq 0.9$. Numerical experiments demonstrate excellent agreement with analytical solutions, achieving residuals of 3.04 \times 10^-29, maximum absolute errors of $3.55\times 10^{-15}$, and relative $L_2$ errors of $9.51\times 10^{-17}$ for $\alpha=0.9$. The results demonstrate that SBNN-PINN is an accurate, stable, and computationally efficient framework for solving nonlinear fractional population models.