Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds

TL;DR

This paper introduces a validation framework for Koopman operator approximations, providing diagnostics and error bounds to ensure trustworthy spectral analysis and forecasting in nonlinear dynamical systems.

Contribution

It develops a unified methodology for quantifying invariance and projection errors in Koopman methods, including principal angle diagnostics and multi-step error bounds, with practical guarantees.

Findings

Principal angle decomposition improves invariance diagnostics.

Error bounds enable certified spectral analysis and forecasting.

Validated methods demonstrated on complex real-world datasets.

Abstract

Koopman operator theory provides a global linear representation of nonlinear dynamics and underpins many data-driven methods. In practice, however, finite-dimensional feature spaces induced by a user-chosen dictionary are rarely invariant, so closure failures and projection errors lead to spurious eigenvalues, misleading Koopman modes, and overconfident forecasts. This paper addresses a central validation problem in data-driven Koopman methods: how to quantify invariance and projection errors for an arbitrary feature space using only snapshot data, and how to use these diagnostics to produce actionable guarantees and guide dictionary refinement? A unified a posteriori methodology is developed for certifying when a Koopman approximation is trustworthy and improving it when it is not. Koopman invariance is quantified using principal angles between a subspace and its Koopman image, yielding principal observables and a principal angle decomposition (PAD), a dynamics-informed alternative to SVD truncation with significantly improved performance. Multi-step error bounds are derived for Koopman and Perron--Frobenius mode decompositions, including RKHS-based pointwise guarantees, and are complemented by Gaussian process expected error surrogates. The resulting toolbox enables validated spectral analysis, certified forecasting, and principled dictionary and kernel learning, demonstrated on chaotic and high-dimensional benchmarks and real-world datasets, including cavity flow and the Pluto--Charon system.