In this presentation, we discuss how dynamical systems estimated through linear operators (linear, kernel, or deep learning) can be modeled as distributions. We propose SGOT, a novel optimal-transport-based distance between these operators that simultaneously encodes the spectrum and the Grassmann subspace of the operator decomposition. This distance provides a new geometry for comparing operators and computing interpolations and barycenters between them. Finally, we present a novel dictionary-learning framework that learns nonlinear representations from many estimated operators using SGOT as data fitting term. We show that this framework encodes variations in physical parameters in simulated and fluid-mechanics systems. We also propose a novel estimator that can be trained from short observations of a system.