Constructing accurate data-driven models of dynamical systems in the face of data-sparsity, measurement errors, and uncertainty is of crucial importance across a wide range of scientific disciplines. In this talk, we propose a variety of techniques rooted in the concept of measure-transport, designed to be robust against such data imperfections. In the first half of the talk, we introduce a novel approach for performing system identification in which synthetic invariant measures, approximated as fixed points of a Fokker—Planck equation, are aligned with invariant measures extracted from observed trajectory data during optimization. We then use Takens' embedding theory to introduce a critical data-dependent coordinate transformation which can guarantee unique system identifiability from the invariant measure alone. In the second half of the talk, we consider the problem of forecasting the full state of a dynamical system from partial measurement data. While Takens' theorem provides the justification for a host of computational methods for data-driven attractor reconstruction, the classical theory assumes the dynamics are deterministic and that observations are noise-free. Motivated by this limitation, we leverage recent advances in optimal transportation theory to establish a measure-theoretic generalization and robust computational framework that recasts the embedding map as a pushforward between probability spaces. Throughout, we showcase the effectiveness of our proposed methods on synthetic test examples, including the Lorenz-63 system and Kuramoto—Sivashinsky equation, as well as large-scale, real-world applications, including Hall-effect thruster dynamics, a NOAA sea surface temperature dataset, and the ERA5 wind field dataset.