  We study an optimal transportation approach for recovering parameters in
dynamical systems with a single smoothly varying attractor. We assume that the
data is not sufficient for estimating time derivatives of state variables but
enough to approximate the long-time behavior of the system through an
approximation of its physical measure. Thus, we fit physical measures by taking
the Wasserstein distance from optimal transportation as a misfit function
between two probability distributions. In particular, we analyze the regularity
of the resulting loss function for general transportation costs and derive
gradient formulas. Physical measures are approximated as fixed points of
suitable PDE-based Perron--Frobenius operators. Test cases discussed in the
paper include common low-dimensional dynamical systems.
