Limits and Powers of Koopman Learning

TL;DR

Limits and Powers of Koopman Learning develops a formal framework for when spectral properties of Koopman operators can be learned from trajectory data. It introduces the Solvability Complexity Index (SCI) to classify the number of successive limits required in data-driven spectral learning, providing sharp upper and lower bounds across dynamical-system classes such as $\Omega_X^{\alpha}$ and $\Omega_X^{m}$. The results show universal barriers to single-limit learning for general systems, while enabling robust learning via towers of limits in restricted settings, and identify two- and three-limit regimes for spectra and eigenfunctions, respectively. The authors present constructive upper-bound algorithms based on pseudospectra and the resolvent, plus RAGE-based eigenfunction extraction, and demonstrate that methods like EDMD can fail to converge, thereby establishing a unified theory that clarifies when and how Koopman spectral properties can be learned from data.

Abstract

Dynamical systems provide a comprehensive way to study complex and changing behaviors across various sciences. Many modern systems are too complicated to analyze directly or we do not have access to models, driving significant interest in learning methods. Koopman operators have emerged as a dominant approach because they allow the study of nonlinear dynamics using linear techniques by solving an infinite-dimensional spectral problem. However, current algorithms face challenges such as lack of convergence, hindering practical progress. This paper addresses a fundamental open question: \textit{When can we robustly learn the spectral properties of Koopman operators from trajectory data of dynamical systems, and when can we not?} Understanding these boundaries is crucial for analysis, applications, and designing algorithms. We establish a foundational approach that combines computational analysis and ergodic theory, revealing the first fundamental barriers -- universal for any algorithm -- associated with system geometry and complexity, regardless of data quality and quantity. For instance, we demonstrate well-behaved smooth dynamical systems on tori where non-trivial eigenfunctions of the Koopman operator cannot be determined by any sequence of (even randomized) algorithms, even with unlimited training data. Additionally, we identify when learning is possible and introduce optimal algorithms with verification that overcome issues in standard methods. These results pave the way for a sharp classification theory of data-driven dynamical systems based on how many limits are needed to solve a problem. These limits characterize all previous methods, presenting a unified view. Our framework systematically determines when and how Koopman spectral properties can be learned.

Figures (3)

• Figure 1: The SCI hierarchy (SCI shaded gray) for the computational problem of learning spectra of Koopman operators from trajectory data. The $\Sigma$ and $\Pi$ classes capture complementary versions of verification in the final limit. Results are given in \ref{['thm1', 'thm2', 'thm3']}, which provide both upper and lower bounds and are expanded upon in the Appendix. The case of finite statespace is $\Delta_1$ since the Koopman operator reduces to a finite matrix.
• Figure 2: The SCI hierarchy for the computational problem of learning spectral types of Koopman operators for measure-preserving invertible systems (i.e., the unitary parts of Koopman operators for $\Omega_{\mathcal{X}}^{\alpha}\cap \Omega_{\mathcal{X}}^{m}$) from trajectory data. Results are detailed in \ref{['thm4']}, which provides both upper and lower bounds and are further expanded upon in the Appendix. (The full spectral measure can be computed with error control with respect to the $1$-Wasserstein distance that captures weak convergence.) For each problem, we have presented a representative dynamical system to underscore that the classification of these computational problems fundamentally depends on the flavor of dynamics.
• Figure 3: Application of the discrete-time RAGE theorem (\ref{['thm:RAGE_discrete_time']}) to the kinetic energy of the lid-driven cavity flow. The plots show $\frac{1}{2L+1}\sum_{\ell=-L}^L\|\mathcal{P}_nA^{\ell}g\|^2$ (for normalized $g$), which in the double limit $\lim_{n\rightarrow\infty}\lim_{L\rightarrow\infty}$ converge to the fraction of $g$ that is pure point. This double limit procedure is used in \ref{['thm:RAGE_discrete_time']} to prove upper bounds.

Theorems & Definitions (51)

• Example 1.1
• Example 1.2
• Example 1.3
• Theorem 2.1
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Example 2.6
• Definition A.1: Computational problem
• Example A.2