Entropy based lower dimension bounds for finite-time prediction of Dynamic Mode Decomposition algorithms

Motivated by Dynamic Mode Decomposition algorithms, we provide lower bounds on the dimension of a finite-dimensional subspace $F\subseteq {\mathrm{L}}^2(\mathrm{X})$ 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.