The Koopman operator lifts a nonlinear dynamical system into an infinite-dimensional linear one by acting on observables rather than states. Data-driven methods approximate this infinite-dimensional operator with a finite-dimensional matrix — but the approximation's quality depends on whether the chosen feature space is approximately invariant under the dynamics.
The paper develops diagnostics for this invariance. Given snapshot data, how do you know if your finite-dimensional approximation is capturing the Koopman operator or just fitting noise? The answer is principal angle decomposition (PAD): measure the angles between the image of the feature space under the dynamics and the feature space itself. Large angles mean the dynamics push observables out of your approximation space; small angles mean the space is nearly invariant.
PAD improves on standard SVD truncation by directly measuring the geometric relationship between the feature space and its image, rather than just measuring the magnitude of the discarded components. The distinction matters for chaotic systems where small SVD tails can hide large invariance violations.
The multi-step error bounds are the practical payoff: given the invariance diagnostic and the spectral decomposition, how far can the forecast drift after k steps? The bounds accumulate with each step but are computable from data, giving certified forecasting — not just a prediction but a guarantee on how wrong it might be.
Applied to Pluto-Charon dynamics and turbulent cavity flow. The diagnostic tells you where the approximation is trustworthy before the forecast fails, not after.