基于RfR方法的理想数据驱动动力学模型的几何结构
Geometric structure of ideal data-driven dynamical model using RfR method
AI总结：
本文基于RfR方法研究理想数据驱动模型的几何结构，通过Lyapunov指数和向量揭示吸引子的动态特性，提出构建稳健理想模型的方法。
摘要：
Gaussian radial function-based Regression (RfR)方法是一种利用物理可理解变量的数据驱动建模方法，通过延迟坐标和高斯径向基函数构建标量时间序列。即使模型能近似描述原始系统的轨迹，数据驱动模型仍难以重建混沌动力学的负Lyapunov指数。理想模型应重建动态结构，包括主导的负Lyapunov指数。通过比较理想模型与非理想模型，利用Lyapunov指数和相应向量研究吸引子的几何结构。研究发现理想模型将原始系统的吸引子重构为时间延迟嵌入。通过应用结果，提出一种构建稳健理想模型的方法，该方法在超参数变化时保持稳定。
英文摘要：
The Gaussian radial function-based Regression (RfR) method is a data-driven modeling approach that utilizes physically understandable variables from scalar time series, constructed using delay coordinates and Gaussian radial basis functions. Even when a model successfully describes an approximate trajectory of the original system, data-driven models rarely reconstruct negative Lyapunov exponents of chaotic dynamics. An ''ideal model'' should reconstruct the dynamical structure, including the negative (physically dominant) Lyapunov exponents. Comparing the ideal model and the non-ideal model, we investigate the geometric structure of the attractor of such models using the Lyapunov exponents and the corresponding Lyapunov vectors. Our investigation suggests that the ideal model reconstructs the original system's attractor as a time-delay embedding. By applying the results, we search for a method to construct an ideal model, which persists against the change in hyperparameters.
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