Koopman Representations for Non-Vanishing Time Intervals: An Optimization Approach and Sampling Effects
For researchers in data assimilation and dynamical systems, this work clarifies identifiability limits and sampling effects in Koopman learning, but the contribution is incremental.
The paper addresses learning Koopman eigenfunctions from observations at arbitrary time intervals, revealing aliasing and phase alignment issues. It proposes an optimization method that outperforms generator extended dynamic mode decomposition under large regular time intervals and shows irregular sampling can recover the true Koopman spectrum.
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.