Geometric structure of ideal data-driven dynamical model using RfR method

TL;DR

This paper investigates the geometric structure of data-driven dynamical models using the RfR method, emphasizing the importance of reconstructing the attractor's structure and Lyapunov exponents.

Contribution

It reveals that ideal models reconstruct the original attractor as a time-delay embedding and proposes a method to construct such models resilient to hyperparameter changes.

Findings

Ideal models reconstruct the attractor as a time-delay embedding.

The study compares ideal and non-ideal models using Lyapunov exponents.

A method is proposed to build models that persist despite hyperparameter variations.

Abstract

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.