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Functional Mean Flow in Hilbert Space

Zhiqi Li, Yuchen Sun, Greg Turk, Bo Zhu

TL;DR

Functional Mean Flow (FMF) delivers a unified one-step generative framework for infinite-dimensional function spaces by marrying a mean-velocity transport in Hilbert space with a two-parameter flow, enabling efficient functional data generation across time series, images, PDEs, and 3D geometry. The authors introduce an $x_1$-prediction variant to improve stability over the original $u$-prediction form, derive an equivalent conditional training objective with a stop-gradient, and prove theoretical links between conditional and marginal dynamics in the Fréchet-derivative setting. Empirically, FMF achieves state-of-the-art one-step performance on several functional tasks, including real-world time series and Navier–Stokes data, function-based image generation with arbitrary resolutions, and SDF-based 3D shape reconstruction, while maintaining stability where prior methods falter. This framework advances scalable, resolution-agnostic generative modeling in functional spaces and broadens the practical impact of one-step transport methods for complex data modalities.

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.

Functional Mean Flow in Hilbert Space

TL;DR

Functional Mean Flow (FMF) delivers a unified one-step generative framework for infinite-dimensional function spaces by marrying a mean-velocity transport in Hilbert space with a two-parameter flow, enabling efficient functional data generation across time series, images, PDEs, and 3D geometry. The authors introduce an -prediction variant to improve stability over the original -prediction form, derive an equivalent conditional training objective with a stop-gradient, and prove theoretical links between conditional and marginal dynamics in the Fréchet-derivative setting. Empirically, FMF achieves state-of-the-art one-step performance on several functional tasks, including real-world time series and Navier–Stokes data, function-based image generation with arbitrary resolutions, and SDF-based 3D shape reconstruction, while maintaining stability where prior methods falter. This framework advances scalable, resolution-agnostic generative modeling in functional spaces and broadens the practical impact of one-step transport methods for complex data modalities.

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 -prediction variant that improves stability over the original -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.

Paper Structure

This paper contains 47 sections, 9 theorems, 86 equations, 13 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Functional Flow Matching
  3. Functional Mean Flow
  4. FMF with $u$-prediction
  5. FMF with $x_1$-prediction
  6. Remark.
  7. Algorithm
  8. Experiment
  9. Real-World Functional Generation
  10. Image Generation Based on Functional
  11. 3D Shape Generation
  12. Related Work
  13. Functional Generation.
  14. Few-step Diffusion/Flow Models.
  15. Conclusion
  16. ...and 32 more sections

Key Result

Theorem 3.1

Assume that the dataset measure $\nu$ satisfies $\int_{\mathcal{F}}\|f\|_\mathcal{F}^2 \mathrm{d}\nu(f)<\infty$, and the conditions of Functional Flow Matching kerrigan2024functional hold. With the conditional flow and conditional velocity chosen in eq:selection_conditional, the corresponding two-p where $D\phi_{t\to r}(g):\mathcal{F}\to \mathcal{F}$ is the Fréchet derivative of $\phi_{t\to r}$ a

Figures (13)

  • Figure 1: Illustration of Functional Mean Flow. The figure shows a 2D projection of the infinite-dimensional function space. During generation, the flow transports a Gaussian measure to the target function measure. The $u$-prediction FMF models the mean velocity $\bar{u}_{t\to r}(f_t)$ between any two points $f_t$ and $f_r$ along the flow trajectory, while the $x_1$-prediction FMF estimates the expected position $\hat{f}_{1,t\to r}(f_t)$ reached by continuing the mean velocity $\bar{u}_{t\to r}(f_t)$ for the remaining distance $1-t$. Both $u$- and $x_1$-prediction FMFs support one-step generation, formulated respectively as $f_1 = f_0 + \bar{u}_{0\to 1}(f_0)$ and $f_1 = \hat{f}_{1,0\to 1}(f_0)$.
  • Figure 2: Representing data as functions enables the same model to synthesize images at arbitrary resolutions with different noise levels. The model is trained only on randomly sampled 1/4 subsets of pixels from 256×256 CelebA-HQ images and performs one-step generation. Left to right: 64×64, 128×128, 256×256, 512×512, and 1024×1024.
  • Figure 3: From the $x_1$-prediction of Functional Flow Matching to the $x_1$-prediction of Functional Mean Flow. In the left figure, we illustrate the relationship between the $u$-prediction (predicting $u_t(f_t)$) and the $x_1$-prediction (predicting $\hat{f}_{1,t}(f_t)$) in flow matching, which satisfies $\hat{f}_{1,t}(f_t) = (1-t)u_t(f_t) + f_t$. Based on this relationship, we can analogously define the $x_1$-prediction of functional Mean Flow (predict $\hat{f}_{1,t\to r}(f_t)$), satisfying $\hat{f}_{1,t}(f_t) = (1-t)u_{t\to r}(f_t) + f_t$.
  • Figure 4: Results on AFHQ, LSUN-Church, and FFHQ. The model is trained on a random 1/4 pixel subset of 256×256 images and evaluated at 256×256 and 512×512 via one-step generation.
  • Figure 5: In Functional Mean Flow, both the input and output are modeled as continuous functions, enabling training and image generation to be defined over arbitrary pixel coordinates instead of being restricted to a discrete grid.
  • ...and 8 more figures

Theorems & Definitions (16)

  • Theorem 3.1: Initial-Time Derivative of Two-Parameter Flow
  • Theorem 3.2: Equivalence of Mean Flow Conditional and Marginal Losses
  • Theorem 3.3: Equivalence of Mean Flow Conditional and Marginal Losses for $x_1$-prediction
  • Theorem A.1
  • Theorem A.2
  • Theorem A.3
  • proof
  • Lemma B.1: Fréchet differentiability of $\phi_{t\to r}$ in Hilbert space
  • proof
  • Lemma B.2
  • ...and 6 more