Riemannian MeanFlow for One-Step Generation on Manifolds

A new method called Riemannian MeanFlow (RMF) enables one-step generation of data on curved spaces, or manifolds, extending the MeanFlow framework to settings where standard Euclidean assumptions break down [1]. The technique, detailed in a paper by Haoliang Sun and revised in June 2026, addresses a core limitation of Flow Matching on Riemannian manifolds. While Flow Matching allows simulation-free training, generating samples has typically required numerically integrating a probability-flow ordinary differential equation, a multi-step process [1]. RMF circumvents this by defining an average-velocity field through a geometric operation known as parallel transport, which accounts for the fact that velocities on a manifold reside in tangent spaces that vary by location [2]. The authors derive a Riemannian MeanFlow identity that intrinsically links these average velocities to instantaneous ones, providing a supervisory signal without needing to simulate full trajectories or perform heavy geometric computations [2]. To make the identity practical, the team uses a log-map tangent representation [2]. For stable training, the RMF objective is split into two terms, and a conflict-aware multi-task learning strategy is applied to manage gradient interference between them [2]. The framework also incorporates classifier-free guidance, allowing it to handle conditional generation tasks [1]. The approach was tested on several manifolds, including spheres, tori, the 3D rotation group SO(3), and the special Euclidean group SE(3) [1]. Results showed competitive one-step sampling performance with an improved trade-off between quality and computational efficiency, substantially reducing the cost of generating samples compared to multi-step integrators [2]. The work represents a step forward for generative modeling on structured data, where manifold constraints are common in fields like robotics and molecular modeling. The paper has been submitted in three versions, with the latest update on June 17, 2026, totaling 10,130 KB [1].