2025/04/28 by Till Hauser, Hauser, Till, Julian Hölz +1
Engineering · Mathematics · Physics and Astronomy · #37A05 #37M10 #37M25 #65P99 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Machine Fault Diagnosis Techniques #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Tensor decomposition and applications
paper · pdf · doi:10.48550/arxiv.2504.20269
openalex publication_date 2025/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by Dynamic Mode Decomposition algorithms, we provide lower bounds on the dimension of a finite-dimensional subspace F ⊆ L2(X) required for predicting the behavior of dynamical systems over long time horizons. We distinguish between two cases: (i) If F is determined by a finite partition of X we derive a lower bound that depends on the dynamical measure-theoretic entropy of the partition. (ii) We consider general finite-dimensional subspaces F and establish a lower bound for the dimension of F that is contingent on the spectral structure of the Koopman operator of the system, via the approximation entropy of F as studied by Voiculescu. Furthermore, we motivate the use of delay observables to improve the predictive qualities of Dynamic Mode Decomposition algorithms.