Flow Straight and Fast in Hilbert Space: Functional Rectified Flow

Jianxin Zhang, Clayton Scott

Abstract:

Many generative models originally developed in finite-dimensional Euclidean
space have functional generalizations in infinite-dimensional settings. However,
the extension of rectified flow to infinite-dimensional spaces remains unexplored.
In this work, we establish a rigorous functional formulation of rectified flow in an
infinite-dimensional Hilbert space. Our approach builds upon the superposition
principle for continuity equations in an infinite-dimensional space. We further show
that this framework extends naturally to functional flow matching and functional
probability flow ODEs, interpreting them as nonlinear generalizations of rectified
flow. Notably, our extension to functional flow matching removes the restrictive
measure-theoretic assumptions in the existing theory of Kerrigan et al. [37].
Furthermore, we demonstrate experimentally that our method achieves superior
performance compared to existing functional generative models.