# Riemannian MeanFlow

arXiv: https://arxiv.org/abs/2602.07744 · HTML: https://mlsanity.com/paper/2602.07744/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 65 references that are in the map.
- First posted on arXiv 8 Feb 2026.

## Builds on (most cited first, 30 of 38)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [Improved Techniques for Training Consistency Models](https://mlsanity.com/paper/2310.14189/) (2023)
- [Simplifying, Stabilizing and Scaling Continuous-Time Consistency Models](https://mlsanity.com/paper/2410.11081/) (2024)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [Diffusion probabilistic modeling of protein backbones in 3D for the motif-scaffolding problem](https://mlsanity.com/paper/2206.04119/) (2022)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [SE(3) diffusion model with application to protein backbone generation](https://mlsanity.com/paper/2302.02277/) (2023)
- [SE(3)-Stochastic Flow Matching for Protein Backbone Generation](https://mlsanity.com/paper/2310.02391/) (2023)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Consistency Models Made Easy](https://mlsanity.com/paper/2406.14548/) (2024)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Dirichlet Flow Matching with Applications to DNA Sequence Design](https://mlsanity.com/paper/2402.05841/) (2024)
- [Flow map matching with stochastic interpolants: A mathematical framework for consistency models](https://mlsanity.com/paper/2406.07507/) (2024)
- [How to build a consistency model: Learning flow maps via self-distillation](https://mlsanity.com/paper/2505.18825/) (2025)
- [Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts](https://mlsanity.com/paper/2503.02819/) (2025)
- [Align Your Flow: Scaling Continuous-Time Flow Map Distillation](https://mlsanity.com/paper/2506.14603/) (2025)
- [Neural Manifold Ordinary Differential Equations](https://mlsanity.com/paper/2006.10254/) (2020)
- [Fisher Flow Matching for Generative Modeling over Discrete Data](https://mlsanity.com/paper/2405.14664/) (2024)
- [Proteina: Scaling Flow-based Protein Structure Generative Models](https://mlsanity.com/paper/2503.00710/) (2025)
- [Generating Novel, Designable, and Diverse Protein Structures by Equivariantly Diffusing Oriented Residue Clouds](https://mlsanity.com/paper/2301.12485/) (2023)
- [Categorical Flow Matching on Statistical Manifolds](https://mlsanity.com/paper/2405.16441/) (2024)
- [Improved motif-scaffolding with SE(3) flow matching](https://mlsanity.com/paper/2401.04082/) (2024)
- [La-Proteina: Atomistic Protein Generation via Partially Latent Flow Matching](https://mlsanity.com/paper/2507.09466/) (2025)
- [SplitMeanFlow: Interval Splitting Consistency in Few-Step Generative Modeling](https://mlsanity.com/paper/2507.16884/) (2025)
- [Terminal Velocity Matching](https://mlsanity.com/paper/2511.19797/) (2025)

## Built on by (most cited first, 8 of 8)

- [Kernel-Gradient Drifting Models](https://mlsanity.com/paper/2605.10727/) (2026)
- [OMatG-flash: An All-Atom Flow Map with Reinforce Adjoint Matching for Scalable Materials Discovery](https://mlsanity.com/paper/2609.26402/) (2026)
- [SE(3)-MeanFlow: Few-Step Protein Backbone Generation on Lie Groups](https://mlsanity.com/paper/2607.27431/) (2026)
- [FestDPO: Few-step Generator Alignment with Direct Preference Optimization](https://mlsanity.com/paper/2609.34673/) (2026)
- [MeanFlowAdvantage: Stable Reward Fine-Tuning for Few-Step Average-Velocity Generators](https://mlsanity.com/paper/2609.37670/) (2026)
- [RW-Flow: One-Step Generation on Compact Manifolds via Wasserstein Gradient Flows](https://mlsanity.com/paper/2609.39271/) (2026)
- [MaDeL: Manifold-Decomposed Feature Losses for Generative Modeling](https://mlsanity.com/paper/2610.04419/) (2026)
- [One-Step Generation via Riemannian Wasserstein Gradient Flows](https://mlsanity.com/paper/2610.04454/) (2026)

## Abstract (arXiv)

Diffusion and flow models have become the dominant paradigm for generative modeling on Riemannian manifolds, with successful applications in protein backbone generation and DNA sequence design. However, these methods require tens to hundreds of neural network evaluations at inference time, which can become a computational bottleneck in large-scale scientific sampling workflows. We introduce Riemannian MeanFlow~(RMF), a framework for learning flow maps directly on manifolds, enabling high-quality generations with as few as one forward pass. We derive three equivalent characterizations of the manifold average velocity (Eulerian, Lagrangian, and semigroup identities), and analyze parameterizations and stabilization techniques to improve training on high-dimensional manifolds. In promoter DNA design and protein backbone generation settings, RMF achieves comparable sample quality to prior methods while requiring up to 10× fewer function evaluations. Finally, we show that few-step flow maps enable efficient reward-guided design through reward look-ahead, where terminal states can be predicted from intermediate steps at minimal additional cost.