Constructing low-dimensional ordinary differential equations from chaotic time series of high/infinite-dimensional systems using radial function-based regression

TL;DR

This paper extends the radial function-based regression method to construct low-dimensional differential equations from high-dimensional chaotic time series, including noisy data, using various complex systems.

Contribution

It demonstrates the effectiveness of RfR in modeling chaotic systems from diverse high-dimensional data sources, including noise, with applications to different types of equations.

Findings

Successfully constructed differential equations for complex systems

Effective in noisy data scenarios

Identified trajectories on chaotic saddles

Abstract

In our previous study (N. Tsutsumi, K. Nakai and Y. Saiki (2022)) we proposed a method of constructing a system of differential equations of chaotic behavior only from observable deterministic time series, which we will call radial function-based regression (RfR) method. The RfR method employs a regression using Gaussian radial basis functions together with polynomial terms to facilitate the robust modeling of chaotic behavior. In this paper, we apply the RfR method to several types of relatively high-dimensional deterministic time series generated by a partial differential equation, a delay differential equation, a turbulence model, and intermittent dynamics. The case when the observation includes noise is also tested. We have effectively constructed a system of differential equations for each of these examples, which is assessed from the point of view of time series forecast, reconstruction of invariant sets, and invariant densities. We find that in some of the models, an appropriate trajectory is realized on the chaotic saddle and is identified by the Stagger-and-Step method.