Robust learning of Koopman spectral properties from trajectory data

Determine the necessary and sufficient conditions under which spectral properties of Koopman operators can be robustly learned from trajectory (snapshot) data of dynamical systems, and delineate the regimes in which such learning is impossible.

Background

The paper develops a framework linking the Solvability Complexity Index hierarchy with ergodic theory to classify when spectral properties of Koopman operators are learnable from data. It provides upper and lower bounds, including impossibility results, but frames the overall challenge as a fundamental open question motivating the work.

The authors present algorithms with verification for certain settings and prove universal barriers in others, suggesting that a complete, sharp classification across system classes and data regimes remains a central goal.