# Generalised Flow Maps for Few-Step Generative Modelling on Riemannian Manifolds arXiv: https://arxiv.org/abs/2510.21608 ยท HTML: https://mlsanity.com/paper/2510.21608/ ## Key facts - Cited by 12 papers in the map (data through 14 Oct 2026). - Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. - Builds on 47 of its 59 references that are in the map. - First posted on arXiv 24 Oct 2025. ## Placed in - Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/) ## Builds on (most cited first, 30 of 47) - [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020) - [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) - [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019) - [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022) - [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022) - [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015) - [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016) - [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023) - [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022) - [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023) - [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023) - [On Distillation of Guided Diffusion Models](https://mlsanity.com/paper/2210.03142/) (2022) - [Movie Gen: A Cast of Media Foundation Models](https://mlsanity.com/paper/2410.13720/) (2024) - [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025) - [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022) - [Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusion](https://mlsanity.com/paper/2310.02279/) (2023) - [Improved Techniques for Training Consistency Models](https://mlsanity.com/paper/2310.14189/) (2023) - [Analyzing and Improving the Training Dynamics of Diffusion Models](https://mlsanity.com/paper/2312.02696/) (2023) - [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024) - [Knowledge Distillation in Iterative Generative Models for Improved Sampling Speed](https://mlsanity.com/paper/2101.02388/) (2021) - [Simplifying, Stabilizing and Scaling Continuous-Time Consistency Models](https://mlsanity.com/paper/2410.11081/) (2024) - [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022) - [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) - [GENIE: Higher-Order Denoising Diffusion Solvers](https://mlsanity.com/paper/2210.05475/) (2022) - [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022) - [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020) - [Parallel Sampling of Diffusion Models](https://mlsanity.com/paper/2305.16317/) (2023) - [FlowMM: Generating Materials with Riemannian Flow Matching](https://mlsanity.com/paper/2406.04713/) (2024) ## Built on by (most cited first, 12 of 12) - [Categorical Flow Maps](https://mlsanity.com/paper/2602.12233/) (2026) - [Posterior Refinement: Fast Language Generation via Any-Order Flow Maps](https://mlsanity.com/paper/2606.24773/) (2026) - [Riemannian MeanFlow](https://mlsanity.com/paper/2602.07744/) (2026) - [Kernel-Gradient Drifting Models](https://mlsanity.com/paper/2605.10727/) (2026) - [Riemannian MeanFlow for One-Step Generation on Manifolds](https://mlsanity.com/paper/2603.10718/) (2026) - [Generative Modeling in Protein Design: Neural Representations, Conditional Generation, and Evaluation Standards](https://mlsanity.com/paper/2603.26378/) (2026) - [Recursive Flow Matching](https://mlsanity.com/paper/2605.26535/) (2026) - [OMatG-flash: An All-Atom Flow Map with Reinforce Adjoint Matching for Scalable Materials Discovery](https://mlsanity.com/paper/2609.26402/) (2026) - [When Riemann flows with Wasserstein: Generative Modeling of Probability Distributions on Manifolds](https://mlsanity.com/paper/2609.25659/) (2026) - [Faster Visuomotor Policy Learning on Action Manifolds via Riemannian MeanFlow](https://mlsanity.com/paper/2609.30127/) (2026) - [RW-Flow: One-Step Generation on Compact Manifolds via Wasserstein Gradient Flows](https://mlsanity.com/paper/2609.39271/) (2026) - [One-Step Generation via Riemannian Wasserstein Gradient Flows](https://mlsanity.com/paper/2610.04454/) (2026) ## Abstract (arXiv) Geometric data and purpose-built generative models on them have become ubiquitous in high-impact deep learning application domains, ranging from protein backbone generation and computational chemistry to geospatial data. Current geometric generative models remain computationally expensive at inference -- requiring many steps of complex numerical simulation -- as they are derived from dynamical measure transport frameworks such as diffusion and flow-matching on Riemannian manifolds. In this paper, we propose Generalised Flow Maps (GFM), a new class of few-step generative models that generalises the Flow Map framework in Euclidean spaces to arbitrary Riemannian manifolds. We instantiate GFMs with three self-distillation-based training methods: Generalised Lagrangian Flow Maps, Generalised Eulerian Flow Maps, and Generalised Progressive Flow Maps. We theoretically show that GFMs, under specific design decisions, unify and elevate existing Euclidean few-step generative models, such as consistency models, shortcut models, and meanflows, to the Riemannian setting. We benchmark GFMs against other geometric generative models on a suite of geometric datasets, including geospatial data, RNA torsion angles, and hyperbolic manifolds, and achieve state-of-the-art sample quality for single- and few-step evaluations, and superior or competitive log-likelihoods using the implicit probability flow.