Given two probability measures $\rho_0$ and $\rho_1$, both with densities, how to construct a map sending one onto the other?
An idea is to start at the identity and construct a map iteratively via the flow of some time-dependent vector field, hoping that it pushes $\rho_0$ onto $\rho_1$ in the infinite-time limit. For instance, this is the case when this flow is chosen to be the gradient flow of the relative entropy w.r.t. $\rho_1$. This procedure recently gained traction in the machine learning community in the context of diffusion-based models, as it allows one to sample from a relatively large class of measures quite easily (by simply making samples from a normal distribution flow).
This transport map has nice properties (e.g. Lipschitz under conditions), but it has no reason to be the optimal transport map between our two measures in the sense of the Monge problem; and except in very specific situations, it is not.
Yet, dealing with optimal transport maps is of interest because of their nice structure: by Brenier's theorem, they are gradients of convex functions. Preserving the optimality along the flow also allows to estimate optimal transport maps in a computationally efficient way, without solving an optimization problem.
After a brief summary of the literature, I will expose some early results of a work in progress. Namely, I will present two ways of enforcing optimality of the transport map along the flow, while preserving its convergence and allowing for efficient computations.