Functional Flow Matching

TL;DR

Functional Flow Matching (FFM) introduces a novel function-space generative model that interpolates between Gaussian and data distributions, learning vector fields without likelihoods, and demonstrates superior performance on real-world benchmarks.

Contribution

The paper extends flow matching models to infinite-dimensional function spaces, providing a theoretical framework and empirical validation for the approach.

Findings

FFM outperforms recent function-space generative models on benchmarks.

The method operates without likelihoods or simulations, suitable for infinite-dimensional spaces.

Empirical results validate the effectiveness of the proposed approach.

Abstract

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.