  Dynamic Mode Decomposition (DMD) and its variants, such as extended DMD
(EDMD), are broadly used to fit simple linear models to dynamical systems known
from observable data. As DMD methods work well in several situations but
perform poorly in others, a clarification of the assumptions under which DMD is
applicable is desirable. Upon closer inspection, existing interpretations of
DMD methods based on the Koopman operator are not quite satisfactory: they
justify DMD under assumptions that hold only with probability zero for generic
observables. Here, we give a justification for DMD as a local, leading-order
reduced model for the dominant system dynamics under conditions that hold with
probability one for generic observables and non-degenerate observational data.
We achieve this for autonomous and for periodically forced systems of finite or
infinite dimensions by constructing linearizing transformations for their
dominant dynamics within attracting slow spectral submanifolds (SSMs). Our
arguments also lead to a new algorithm, data-driven linearization (DDL), which
is a higher-order, systematic linearization of the observable dynamics within
slow SSMs. We show by examples how DDL outperforms DMD and EDMD on numerical
and experimental data.
