We discuss two statistical problems that arise in the course of physical modeling. First, we consider an experiment where a low-dimensional dynamical system is observed as a high-dimensional signal, e.g., a video of a chaotic pendulums system. Assuming that we know the dynamical model, but do not know the observation function (the experimental design) - can we estimate the true parameters from the experiment? The key information lies in the temporal inter-dependencies between the signal and the model, and we exploit this information using a kernel-based score.
In the second part of the talk, we turn to uncertainty propagation; in many scientific areas, the parameters of deterministic models are uncertain or noisy. A comprehensive model should therefore provide a statistical description of the quantity of interest. Underlying this computational problem is a fundamental question - if two “similar” functions push-forward the same measure, would the new resulting measures be close, and if so, in what sense? Through optimal transport theory, a Wasserstein-distance formulation of our problem yields a simple and applicable theory.