Koopman Representations for Non-Vanishing Time Intervals: An Optimization Approach and Sampling Effects

TL;DR

This paper introduces an optimization-based method for learning Koopman eigenfunctions from data sampled at arbitrary intervals, highlighting sampling effects like aliasing and phase cancellation.

Contribution

It formulates Koopman eigenfunction learning as an optimization problem that accounts for sampling effects, revealing aliasing limits and benefits of irregular sampling.

Findings

Irregular sampling can break aliasing and aid in spectrum recovery.

The proposed method outperforms generator extended DMD at large regular intervals.

Phase alignment near true frequencies creates optimization challenges.

Abstract

Koopman operator theory is a key tool in data assimilation of complex dynamical systems, with the potential to be applied to multimodal data. We formulate the problem of learning Koopman eigenfunctions from observations at arbitrary, possibly non-vanishing, time intervals as an optimization problem. Analysis of the formulation reveals aliasing induced by oscillatory dynamics and the sampling pattern, making an inherent identifiability limit explicit. The analysis also uncovers phase alignment near the true Koopman frequency, which creates a steep loss valley and demands careful optimization. We further show that irregular sampling can break aliasing and lead to phase cancellation. Numerical results demonstrate the efficacy of the proposed method under large regular time intervals compared to generator extended dynamic mode decomposition, and support the idea that irregular sampling can help recover the true Koopman spectrum.