Manifold-Stable Flow Matching

Amirhossein Nazerian; Ali Pezeshki; Jianguo Zhao

cs.LGSep 28, 2026

Manifold-Stable Flow Matching

Authors: Amirhossein Nazerian, Ali Pezeshki, Jianguo Zhao

Organizations: Department of Mechanical Engineering Colorado State University Fort Collins, CO 80523, USA · Department of Electrical and Computer Engineering Colorado State University Fort Collins, CO 80523, USA

Abstract

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.

Figures & tables

Figure 1: Schematic comparison. Linear conditional FM paths do not necessarily stay near a curved target manifold (top-left panel). MSFM (top-right panel) learns tangential transport and prescribes normal contraction, which guarantees convergence to the manifold. The learned MSFM sampling trajectories can curve because MSFM learns tangential motion, while it prescribes normal contraction (bottom panel). The normal component of the flow uniquely minimizes kinetic action.

Figure 2: Ellipse: training/validation losses over 1,600 epochs, sampling trajectories, and terminal off-proxy errors for 700 samples. Here, we approximate the manifold (the ellipse) by local affine proxies using PCA.

Figure 3: Push-T Task. (a,b) Total FM and parallel-only MSFM training/validation losses; red lines retain the original checkpoint selection. (c,d) Disturbed success and mean maximum reward. (e) Shows that off-manifold-proxy error is 4 orders of magnitude larger for the conventional FM. (f) A matched episode run where MSFM is successful in performing the task, while the conventional FM fails. Here, we approximate the manifold (expert actions) by local affine proxies using PCA.

Figure 4: Robomimic Square Task. (a,b) Training/velocity-validation losses; red lines mark the checkpoint selection. (c) Success on paired nominal rollouts using best integrated-policy checkpoints. (d) Cumulative success rate vs. Robomimic environment time step. (e) Pre-projection rotation deviation, nearly five orders of magnitude lower with MSFM. (f) shows a completed episode from MSFM, while a conventional FM failed. Here, the manifold for rotation poses is SO(3).

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