In this work, we propose Functional Flow Matching (FFM), a function-space generative model that generalizes the recently-introduced Flow Matching model to operate directly 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 synthetic and real-world benchmarks that our proposed FFM method outperforms several recently proposed function-space generative models.

翻译：本文提出Functional Flow Matching (FFM)，这是一种函数空间生成模型，将近期提出的Flow Matching模型推广至可直接在无限维空间中进行操作。我们的方法首先定义一条概率测度路径，该路径在固定的高斯测度与数据分布之间进行插值，随后在底层函数空间上学习生成该测度路径的向量场。该方法既不依赖似然函数也不依赖模拟计算，使其特别适用于函数空间场景。我们既为该类模型的构建提供了理论框架，也通过实验评估验证了相关技术。在合成基准与真实世界基准上的实验表明，本文提出的FFM方法优于近期提出的多种函数空间生成模型。