Year: 2025
Type: article
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 and is the object of focus of many learning techniques. However, there are many secondary aspects of dynamical systems—invariant sets, the Koopman operator, and Markov approximations—that 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 placed on methods of learning the primary feature—the dynamics law itself. The main question considered is the connection between learning this law and reconstructing the Koopman operator and the invariant set. The answers are tied to the ergodic and topological properties of the dynamics, and they reveal how these properties determine the limits of forecasting techniques.
Source: SIAM Review
DOI: 10.1137/24m1696974
Authors: Tyrus Berry, Suddhasattwa Das
Institutions: George Mason University, Texas Tech University
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4407197797