A Data-Integrated Framework for Learning Fractional-Order Nonlinear Dynamical Systems

This paper presents a data-integrated framework for learning the dynamics of fractional-order nonlinear systems in both discrete-time and continuous-time settings. The proposed framework consists of two main steps. In the first step, input-output experiments are designed to generate the necessary datasets for learning the system dynamics, including the fractional order, the drift vector field, and the control vector field. In the second step, these datasets, along with the memory-dependent property of fractional-order systems, are used to estimate the system's fractional order. The drift and control vector fields are then reconstructed using orthonormal basis functions. To validate the proposed approach, the algorithm is applied to four benchmark fractional-order systems. The results confirm the effectiveness of the proposed framework in learning the system dynamics accurately. Finally, the same datasets are used to learn equivalent integer-order models. The numerical comparisons demonstrate that fractional-order models better capture long-range dependencies, highlighting the limitations of integer-order representations.