用实验数据和记忆特性，精准学习分数阶非线性系统的动态行为。
A Data-Integrated Framework for Learning Fractional-Order Nonlinear Dynamical Systems
- 通过输入输出实验生成数据，结合记忆依赖性估计分数阶阶数。
- 利用正交基函数重构漂移与控制向量场，精度高且适用多类系统。
- 分数阶模型比整数阶更擅长捕捉长期依赖，适合复杂动态建模者。
本文提出一种数据融合框架，用于在离散和连续时间设置下学习分数阶非线性系统的动力学。该框架包含两个步骤：首先设计输入输出实验，生成包含分数阶阶数、漂移向量场和控制向量场的学习数据集；其次，利用这些数据集及分数阶系统的记忆依赖特性，估计系统分数阶阶数，并通过正交基函数重构漂移与控制向量场。为验证方法有效性，算法应用于四个基准分数阶系统，结果表明其能准确学习系统动态。进一步使用相同数据集学习等效整数阶模型，数值对比显示分数阶模型更优，能更好刻画长程依赖，揭示整数阶表示的局限性。
原文摘要 · Abstract (English)
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.
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