Optimal Transport (OT) equips the space of probability measures with the 2-Wasserstein distance, allowing us to interpret many PDEs as gradient flows of energy functionals. I will first recall (briefly) the Monge-Kantorovich formulation of OT and the definition of the 2-Wasserstein distance. Then, I will introduce Wasserstein gradient flows, viewing probability measures as densities of point clouds.

In the second part, I will present recent work using this framework to drive an unknown initial measure towards a Gaussian reference measure, while learning the associated vector field (the score) along the way. This connects Wasserstein gradient flows with score-based methods, a central objective in many modern generative modelling approaches.