# Manifold-Stable Flow Matching

arXiv: https://arxiv.org/abs/2609.35454 · HTML: https://mlsanity.com/paper/2609.35454/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 38 of its 53 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## 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 38)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Diffusion Policy: Visuomotor Policy Learning via Action Diffusion](https://mlsanity.com/paper/2303.04137/) (2023)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [What Matters in Learning from Offline Human Demonstrations for Robot Manipulation](https://mlsanity.com/paper/2108.03298/) (2021)
- [robosuite: A Modular Simulation Framework and Benchmark for Robot Learning](https://mlsanity.com/paper/2009.12293/) (2020)
- [Flow Matching on General Geometries](https://mlsanity.com/paper/2302.03660/) (2023)
- [Metric Flow Matching for Smooth Interpolations on the Data Manifold](https://mlsanity.com/paper/2405.14780/) (2024)
- [EquiBot: SIM(3)-Equivariant Diffusion Policy for Generalizable and Data Efficient Learning](https://mlsanity.com/paper/2407.01479/) (2024)
- [Equivariant Diffusion Policy](https://mlsanity.com/paper/2407.01812/) (2024)
- [Learning Robotic Manipulation Policies from Point Clouds with Conditional Flow Matching](https://mlsanity.com/paper/2409.07343/) (2024)
- [Riemannian Flow Matching Policy for Robot Motion Learning](https://mlsanity.com/paper/2403.10672/) (2024)
- [Diffusion-EDFs: Bi-equivariant Denoising Generative Modeling on SE(3) for Visual Robotic Manipulation](https://mlsanity.com/paper/2309.02685/) (2023)
- [ManiFlow: A General Robot Manipulation Policy via Consistency Flow Training](https://mlsanity.com/paper/2509.01819/) (2025)
- [Contraction Theory for Nonlinear Stability Analysis and Learning-based Control: A Tutorial Overview](https://mlsanity.com/paper/2110.00675/) (2021)
- [Meta Flow Matching: Integrating Vector Fields on the Wasserstein Manifold](https://mlsanity.com/paper/2408.14608/) (2024)
- [Much Ado About Noising: Dispelling the Myths of Generative Robotic Control](https://mlsanity.com/paper/2512.01809/) (2025)
- [Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints](https://mlsanity.com/paper/2506.08604/) (2025)
- [Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling](https://mlsanity.com/paper/2504.10612/) (2025)
- [ET-SEED: Efficient Trajectory-Level SE(3) Equivariant Diffusion Policy](https://mlsanity.com/paper/2411.03990/) (2024)
- [Fast and Robust Visuomotor Riemannian Flow Matching Policy](https://mlsanity.com/paper/2412.10855/) (2024)
- [Neural Contractive Dynamical Systems](https://mlsanity.com/paper/2401.09352/) (2024)
- [DDAT: Diffusion Policies Enforcing Dynamically Admissible Robot Trajectories](https://mlsanity.com/paper/2502.15043/) (2025)
- [Carr'e du champ flow matching: better quality-generalisation tradeoff in generative models](https://mlsanity.com/paper/2510.05930/) (2025)
- [SafeFlowMatcher: Safe and Fast Planning using Flow Matching with Control Barrier Functions](https://mlsanity.com/paper/2509.24243/) (2025)
- [Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees](https://mlsanity.com/paper/2402.08090/) (2024)
- [Riemannian Variational Flow Matching for Material and Protein Design](https://mlsanity.com/paper/2502.12981/) (2025)
- [Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problems](https://mlsanity.com/paper/2508.09156/) (2025)
- [Flow Matching is Adaptive to Manifold Structures](https://mlsanity.com/paper/2602.22486/) (2026)
- [Bures-Wasserstein Flow Matching for Graph Generation](https://mlsanity.com/paper/2506.14020/) (2025)
- [Pullback Flow Matching on Data Manifolds](https://mlsanity.com/paper/2410.04543/) (2024)

## Abstract (arXiv)

Flow matching (FM) learns generative dynamics through velocity regression. Geometric FM variants commonly assume a prior supported on the data manifold, requiring geometric knowledge that is often unavailable. Without such knowledge, low regression error alone does not guarantee manifold adherence. Adherence keeps generated samples within valid configurations and is empirically associated with better task performance. We introduce manifold-stable flow matching (MSFM), which can start from an arbitrary ambient prior, not necessarily supported on the manifold. Using tools from nonlinear dynamics, namely contraction theory, MSFM combines learned tangential transport with prescribed normal contraction. The construction uses analytical projectors for known manifolds and local affine proxies estimated by principal component analysis for unknown data geometry. By implementing contraction theory in both cases of known and unknown manifolds, we guarantee manifold invariance and transverse convergence to the manifold within a desired time window (e.g., one second). We derive a family of compatible probability paths and decompose the training loss into a learnable tangential term and a normal residual. An ellipse experiment attains a mean terminal off-manifold error of order 10-6. In Push-T robotic experiments, MSFM raises success from 74% to 82%. In the Robomimic Square task, success increases from 60% to 72%, while rotation-manifold deviation decreases from order 10-2 to 10-7. The MSFM terminal geometric errors are controlled by the chosen numerical tolerance. These results demonstrate stronger geometric adherence and higher observed task performance, supporting prescribed normal contraction as a complement to learned generative transport.