Functional Mean Flow in Hilbert Space

TL;DR

This paper introduces Functional Mean Flow, a novel one-step generative model in Hilbert space, extending flow matching to functional data with improved stability and broad applicability.

Contribution

It provides a theoretical framework and practical implementation for functional flow matching, including an $x_1$-prediction variant for enhanced stability.

Findings

Effective functional data generation across various domains

Improved stability with the $x_1$-prediction variant

Versatile application to time series, images, PDEs, and 3D geometry

Abstract

We present Functional Mean Flow (FMF) as a one-step generative model defined in infinite-dimensional Hilbert space. FMF extends the one-step Mean Flow framework to functional domains by providing a theoretical formulation for Functional Flow Matching and a practical implementation for efficient training and sampling. We also introduce an $x_1$-prediction variant that improves stability over the original $u$-prediction form. The resulting framework is a practical one-step Flow Matching method applicable to a wide range of functional data generation tasks such as time series, images, PDEs, and 3D geometry.