Data-driven modelling of nonlinear dynamics by barycentric coordinates and memory

TL;DR

This paper introduces a data-driven approach for modeling nonlinear dynamical systems using barycentric coordinates, probabilistic projections, and memory to accurately capture complex behaviors including chaos.

Contribution

The method combines SPA projections, nonlinear transformations, and delay-embedding to create stable, low-dimensional models of complex dynamical systems from data.

Findings

Successfully models chaotic dynamics and attractors with multiple components

Produces stable models with reduced dimensionality

Capable of reproducing complex nonlinear behaviors

Abstract

We present a numerical method to model dynamical systems from data. We use the recently introduced method Scalable Probabilistic Approximation (SPA) to project points from a Euclidean space to convex polytopes and represent these projected states of a system in new, lower-dimensional coordinates denoting their position in the polytope. We then introduce a specific nonlinear transformation to construct a model of the dynamics in the polytope and to transform back into the original state space. To overcome the potential loss of information from the projection to a lower-dimensional polytope, we use memory in the sense of the delay-embedding theorem of Takens. By construction, our method produces stable models. We illustrate the capacity of the method to reproduce even chaotic dynamics and attractors with multiple connected components on various examples.