Functional Mean Flow in Hilbert Space

• URL: http://arxiv.org/abs/2511.12898v1
• Date: Mon, 17 Nov 2025 02:38:28 GMT
• Title: Functional Mean Flow in Hilbert Space
• Authors: Zhiqi Li, Yuchen Sun, Greg Turk, Bo Zhu,
• Abstract summary: We present Functional Mean Flow (FMF) as a one-step generative model defined in infinite-dimensional Hilbert space.<n>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.
• Score: 21.770924847119478
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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