Modern generative models are behind today's systems for generating text, images, and video, and are increasingly used in scientific applications. Mathematically, they can be understood as solutions to a problem of transport of measure: transform a simple reference distribution into a complex target known only through samples. Many of the most successful approaches perform this transport dynamically, by learning the velocity of an ODE or the drift of an SDE whose solution carries one distribution onto the other. I will describe how flow matching with stochastic interpolants makes this construction explicit, treats deterministic flows and diffusions within a single framework, and reduces learning the dynamics to simple quadratic regression problems. I will explain how the design of the interpolation shapes the transport, and what this implies for accuracy and computational cost. I will conclude by discussing how the flow map of the ODE can be learned directly, enabling sampling in one or a few steps, and point to some open mathematical questions.

 

We consider a system of $N$ particles, subject to a mean-field type pairwise interaction kernel $K$, each driven by an independent fractional Brownian motion (idiosyncratic noises). Previous works established that, for a large class of non-Lipschitz, possibly singular kernels, the associated McKean-Vlasov equation is well-posed, and the empirical measure converges to its law as $N\to\infty$, with rate of order $N^{-1/2}$ in suitable negative Sobolev norms. In this talk I will present results concerning the Gaussian fluctuations underlying this mean field convergence, validating the optimality of this rate; they are valid for both first order interactions and for kinetic systems. In the Brownian case, the Gaussian limit field can be identified as the solution to a linear SPDE. The proofs are based on the use of Girsanov transform and the method of U-statistics first introduced by Sznitman.

Based on ongoing joint work with Avi Mayorcas (Bath) and Johanna Weinberger (MPI Leipzig).