Flow Matching for Applied Mathematicians

Flow Matching is a recent framework for learning continuous transformations between probability measures. The method constructs a time-dependent velocity field whose flow transports a source distribution to a target distribution, and whose training reduces to a simple regression problem on paired samples. This simulation-free objective makes Flow Matching an attractive alternative to continuous normalizing flows and diffusion models. This tutorial provides a self-contained and mathematically rigorous introduction to Flow Matching, aimed at applied mathematicians. Starting from the continuity equation, we establish the theoretical foundations linking velocity fields, probability paths, and flows, and explain how Flow Matching arises from a particular construction based on couplings of probability measures. We carefully state the assumptions under which the induced ordinary differential equation defines a unique flow and yields a valid pushforward between distributions, and we illustrate the limitations of the theory through explicit counterexamples. We derive closed-form velocity fields in several important settings, including one-dimensional distributions, Gaussian and Gaussian mixture models, and semi-discrete targets, and we clarify the connections with score matching, diffusion models, and optimal transport. Throughout the paper, theoretical results are complemented by reproducible numerical experiments designed to build intuition and illustrate practical behavior. Our goal is to provide readers with both a solid mathematical understanding of Flow Matching and concrete tools for its application.