Functional Mean Flow in Hilbert Space
This provides a practical one-step generative model for functional data generation, addressing a domain-specific need.
The paper tackles the problem of generative modeling for functional data by introducing Functional Mean Flow (FMF), a one-step method in infinite-dimensional Hilbert space, which achieves efficient training and sampling for tasks like time series and images.
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.