Sampling and density estimation via transport maps is a growing topic of inquiry both in the machine learning and scientific computing communities, with applications in areas such as generative modeling, lattice field theory, and molecular dynamics. The challenge in this domain is to choose what structure to impose on the transport to best reach a complex target distribution from a simple one used as base, while maintaining computational efficiency. I will discuss a recently introduced continuous time flow which connects any pair of distributions in finite integration time, whose velocity is the minimizer of a simple quadratic objective. Moreover, the flow can be optimized to achieve optimal transport in the sense of Benamou-Brenier. The method is contextualized with score-based diffusion probabilistic models. Time permitting, I will discuss motivation for these maps in MCMC sampling QCD field configurations.