Rigorously Characterizing Dynamics with Machine Learning

TL;DR

This paper explores how machine learning can be used to rigorously characterize dynamical systems from time series data, leveraging algebraic topology and order theory to approximate invariant set dynamics.

Contribution

It introduces a novel approach that combines machine learning with topological and order-theoretic methods to approximate dynamics, overcoming classical non-computability issues.

Findings

Dynamics on invariant sets are non-computable.

Approximate characterizations can be identified via machine learning.

Topological and order-theoretic methods enable practical analysis.

Abstract

The identification of dynamics from time series data is a problem of general interest. It is well established that dynamics on the level of invariant sets, the primary objects of interest in the classical theory of dynamical systems, is not computable. We recall a coarser characterization of dynamics based on order theory and algebraic topology and prove that this characterization can be identified using approximations.