  We propose Functional Flow Matching (FFM), a function-space generative model
that generalizes the recently-introduced Flow Matching model to operate in
infinite-dimensional spaces. Our approach works by first defining a path of
probability measures that interpolates between a fixed Gaussian measure and the
data distribution, followed by learning a vector field on the underlying space
of functions that generates this path of measures. Our method does not rely on
likelihoods or simulations, making it well-suited to the function space
setting. We provide both a theoretical framework for building such models and
an empirical evaluation of our techniques. We demonstrate through experiments
on several real-world benchmarks that our proposed FFM method outperforms
several recently proposed function-space generative models.
