A Caputo fractional physics-informed neural network for Lorenz chaotic systems

Fractional-order dynamical systems provide an effective mathematical framework for describing nonlinear processes with memory and hereditary effects. Although physics-informed neural networks have demonstrated promising performance for solving differential equations, their application to fractional-order chaotic dynamical systems remains relatively limited. In this study, a Caputo fractional physics-informed neural networks (FPINNs) framework is developed to approximate the fractional-order Lorenz system by incorporating the governing fractional differential equations into a multi-objective loss function. The reference solutions are generated using the Caputo Adams-Bashforth-Moulton (ABM) predictor-corrector method to ensure an appropriate numerical treatment of the fractional derivative. The proposed framework is evaluated for several fractional orders over the considered simulation interval, and its predictive performance is assessed using quantitative error metrics, including mean squared error, mean absolute error, sum of squared errors, and the coefficient of determination. The numerical results demonstrate that the FPINN accurately reconstructs the fractional Lorenz trajectories while maintaining good agreement with the ABM reference solutions. Furthermore, the study investigates the influence of the fractional order on prediction accuracy and computational performance, providing a systematic numerical assessment of the proposed framework. The results indicate that FPINNs constitute an effective and data-efficient approach for approximating nonlinear fractional dynamical systems within the investigated computational setting.