Standard system identification methods rely on system trajectories, where the model dynamics are matched with trajectory data. In this talk, I will discuss a different approach based on the physical measures of dynamical systems. This method helps when trajectory data are sampled infrequently, rendering estimations of time derivatives challenging or impossible. I will present PDE-based approximation methods for the physical measures and regularity results for optimal-transportation-based fidelity functions necessary for efficient gradient-based optimization. I will conclude with some remarks on future work and open questions.