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Flow Matching on General Geometries
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Proposes Riemannian Flow Matching, a simulation-free method for training continuous normalizing flows on manifolds via premetric-defined conditional vector fields, using geodesic distances on simple geometries and spectral (biharmonic/diffusion) distances on general triangular meshes.
Problem
Training continuous normalizing flows / generative models on Riemannian manifolds is either simulation-heavy, biased due to score or divergence approximations, or fails to scale to high dimensions and general geometries such as meshes and manifolds with boundaries.
Approach
Defines conditional flows through a user-specified premetric whose gradient gives a closed-form conditional vector field (linearly decreasing the premetric), using exact geodesic distances on simple manifolds and finite-k spectral distances (biharmonic, diffusion) with Neumann eigenfunctions on general meshes and manifolds with boundaries.
Key results
- On earth science datasets, Riemannian Flow Matching with geodesic distance reaches test NLL of -7.93 on Volcano and -1.86 on Fire, outperforming prior methods on those datasets.
- On the 7D RNA torus, RFM achieves test NLL -5.20 versus -3.70 for the Riemannian Diffusion Model.
- On the Stanford Bunny mesh, the biharmonic distance gives test NLL 1.06 at k=10, and on Spot the Cow the diffusion distance gives 0.87 at k=10.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use Riemannian Flow Matching with geodesic distance | Riemannian CNF · Volcano | test NLL: -6.61 ± 0.96 -> -7.93 ± 1.67 | Helped |
| use Riemannian Flow Matching with geodesic distance | Riemannian CNF · Earthquake | test NLL: -0.40 ± 0.05 -> -0.28 ± 0.08 | Hurt |
| use Riemannian Flow Matching with geodesic distance | Riemannian CNF · Flood | test NLL: 0.25 ± 0.02 -> 0.42 ± 0.05 | Hurt |
| use Riemannian Flow Matching with geodesic distance | Riemannian CNF · Fire | test NLL: -1.40 ± 0.02 -> -1.86 ± 0.11 | Helped |
| use Riemannian Flow Matching with geodesic distance | Riemannian CNF · Protein datasets | test NLL: General: 1.04 ±0.012 -> 1.01 ±0.025; Glycine: 1.97 ±0.012 -> 1.90 ±0.055; Proline: 0.12 ±0.011 -> 0.15 ±0.027; Pre-Pro: | Mixed |
| replace biharmonic distance with diffusion distance (ablation) | Riemannian CFM · Stanford Bunny and Spot the Cow meshes | test NLL: Stanford Bunny k=10: 1.06 ±0.05 -> 1.16 ±0.02; k=50: 1.55 ±0.01 -> 1.48 ±0.01; k=100: 1.49 ±0.01 -> 1.53 ±0.01; Spot the | Mixed |
| increase eigenfunction count from k=10 to k=50 (ablation) | Riemannian CFM w/ Diffusion (τ=1/4) · Stanford Bunny mesh | test NLL: 1.16 ±0.02 -> 1.48 ±0.01 | Helped |
| increase eigenfunction count from k=50 to k=100 (ablation) | Riemannian CFM w/ Diffusion (τ=1/4) · Stanford Bunny mesh | test NLL: 1.48 ±0.01 -> 1.53 ±0.01 | Hurt |
| increase eigenfunction count from k=10 to k=50 (ablation) | Riemannian CFM w/ Biharmonic · Stanford Bunny mesh | test NLL: 1.06 ±0.05 -> 1.55 ±0.01 | Hurt |
| increase eigenfunction count from k=50 to k=100 (ablation) | Riemannian CFM w/ Biharmonic · Stanford Bunny mesh | test NLL: 1.55 ±0.01 -> 1.49 ±0.01 | Helped |
| increase eigenfunction count from k=10 to k=50 (ablation) | Riemannian CFM w/ Diffusion (τ=1/4) · Spot the Cow mesh | test NLL: 0.87 ±0.07 -> 0.95 ±0.16 | Hurt |
| increase eigenfunction count from k=50 to k=100 (ablation) | Riemannian CFM w/ Diffusion (τ=1/4) · Spot the Cow mesh | test NLL: 0.95 ±0.16 -> 1.08 ±0.05 | Hurt |
Surprising findings
- Heat-kernel approximations used by Riemannian score-based models can be strongly biased, with the conditional score not correct to the first significant digit near small times even with hundreds of eigenfunctions.
Limitations
- On general manifolds the conditional flow x_t must still be simulated during training, which is time consuming.
- Spectral distance eigenfunction solvers may be computationally expensive on complex manifolds.
- The diffusion distance requires careful tuning of its hyperparameter tau, while the biharmonic distance is less sensitive.
Experimental setup
Datasets: Volcano Earthquake Flood Fire General protein Glycine Proline Pre-Pro RNA Stanford Bunny Spot the Cow
Benchmarks: Volcano Earthquake Flood Fire Protein torsion datasets RNA Stanford Bunny mesh Spot the Cow mesh
Metrics: test NLL log-likelihood in bits per dimension
Baselines: Riemannian CNF Moser Flow CNF Matching Riemannian Score-Based Riemannian Diffusion Model Mixture of Power Spherical
Builds on: Flow Matching Conditional Flow Matching Riemannian diffusion models Moser Flow
Closest papers by meaning
- Riemannian Consistency Model (NeurIPS 2025 · similarity 0.90)
- Metric Flow Matching for Smooth Interpolations on the Data Manifold (NeurIPS 2024 · similarity 0.89)
- Categorical Flow Matching on Statistical Manifolds (NeurIPS 2024 · similarity 0.89)
- Generative Modeling on Manifolds Through Mixture of Riemannian Diffusion Processes (ICML 2024 · similarity 0.88)
- Learning Distributions on Manifolds with Free-Form Flows (NeurIPS 2024 · similarity 0.88)
- Beckmann Transport Models: From Autonomous Flows to One-Step Maps (NeurIPS 2026 · similarity 0.87)
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