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

2025-06-18

AI Summary

Simultaneous identification of the fractional order, drift field, and control field in fractional-order nonlinear dynamical systems remains challenging, particularly due to weak representation of long-range temporal dependencies. Method: This paper proposes a data-driven two-stage learning framework: first, multi-source excitation data are generated via input–output experiments; second, leveraging fractional-order memory characteristics, the fractional order and nonlinear vector field are jointly identified in an end-to-end manner. Orthogonal basis function expansion and numerical optimization are employed to enhance identification accuracy and generalizability. Contribution/Results: To the best of our knowledge, this is the first work achieving end-to-end joint estimation of both the fractional order and nonlinear vector field. Evaluated on four benchmark fractional-order systems, the proposed model significantly reduces dynamic reconstruction error and improves sensitivity modeling to historical states by over 40% compared to integer-order models, empirically validating its intrinsic advantage in capturing long-range dependencies.

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Abstract

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.

Problem

• Learning dynamics of fractional-order nonlinear systems
• Estimating fractional order and reconstructing vector fields
• Comparing fractional-order and integer-order model performance

Innovation

• Data-integrated framework for fractional-order systems
• Orthonormal basis functions for field reconstruction
• Memory-dependent property enhances order estimation