Limits of Learning Dynamical Systems

TL;DR

This paper explores the fundamental limits of learning dynamical systems by analyzing how different aspects like the transformation law, invariant sets, and operators influence forecast accuracy and the inherent challenges in approximating these facets.

Contribution

It provides a comprehensive analysis of the relationships and obstructions between various dynamical system aspects and their impact on learning and forecasting capabilities.

Findings

Forecast performance depends on approximating various dynamical facets.

Connections and obstructions exist between different dynamical aspects.

Properties like ergodicity influence the limits of learning methods.

Abstract

A dynamical system is a transformation of a phase space, and the transformation law is the primary means of defining as well as identifying the dynamical system. It is the object of focus of many learning techniques. Yet there are many secondary aspects of dynamical systems - invariant sets, the Koopman operator, and Markov approximations, which provide alternative objectives for learning techniques. Crucially, while many learning methods are focused on the transformation law, we find that forecast performance can depend on how well these other aspects of the dynamics are approximated. These different facets of a dynamical system correspond to objects in completely different spaces - namely interpolation spaces, compact Hausdorff sets, unitary operators and Markov operators respectively. Thus learning techniques targeting any of these four facets perform different kinds of approximations. We examine whether an approximation of any one of these aspects of the dynamics could lead to an approximation of another facet. Many connections and obstructions are brought to light in this analysis. Special focus is put on methods of learning of the primary feature - the dynamics law itself. The main question considered is the connection of learning this law with reconstructing the Koopman operator and the invariant set. The answers are tied to the ergodic and topological properties of the dynamics, and reveal how these properties determine the limits of forecasting techniques.