Functional Mean Flow in Hilbert Space

2025-11-16

AI Summary

This work addresses the challenge of generative modeling for infinite-dimensional functional data—such as time series, images, PDE solutions, and 3D geometry—by proposing Functional Mean Flow (FMF), the first one-step generative model rigorously defined on a Hilbert space. FMF extends the classical mean flow framework to the functional domain, establishing a comprehensive theory of functional flow matching. It introduces an innovative *x₁-prediction* variant that preserves mathematical rigor while substantially improving training stability and sampling efficiency. Experiments demonstrate that FMF achieves high-fidelity, single-step generation across diverse functional data modalities. The method combines theoretical soundness—including well-posedness in infinite dimensions—with strong empirical generalization. By unifying functional generative modeling under a principled, scalable paradigm, FMF provides a foundational framework for learning distributions over function spaces.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Natural Language Processing: Generation
• Computer Vision: Generative Adversarial Networks (GANs) for Vision

Application Category

• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Web Mining and Content Analysis: Web data generation and simulation
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

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.

Problem

• Extends one-step generative models to infinite-dimensional Hilbert space
• Provides theoretical formulation for Functional Flow Matching
• Enables efficient generation of functional data like time series

Innovation

• One-step generative model in infinite-dimensional Hilbert space
• Extends Mean Flow to functional domains with efficient training
• Introduces x1-prediction variant for improved stability