Research map / Probabilistic and scientific modeling / Variational and generative sampling
Reflected Flow Matching
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Extends flow matching to constrained domains by adding a boundary reflection term to CNF ODEs, training velocity fields with analytic conditional velocities on domains such as hypercubes, simplices, and cups, and evaluating on toy data and CIFAR-10/ImageNet64.
Problem
Continuous normalizing flows trained with flow matching can push trajectories outside constrained domains (e.g., image pixel ranges), producing unnatural samples such as oversaturated images; existing reflected diffusion models need complicated, biased conditional score approximations.
Approach
Add a boundary reflection term to the ODE so trajectories stay in the domain, and build conditional flows that stay inside the domain so their analytic conditional velocity fields can be regressed simulation-free (reflected flow matching), with a sampler that reflects steps at the boundary and a flow guidance scheme.
Key results
- On CIFAR-10 unconditional generation, RFM reaches FID 4.76 at 139 NFE, versus 6.35 for FM (142 NFE) and 2.72 for RDM at 2000 NFE.
- On ImageNet 64x64 with guidance weight 15, RFM achieves FID 26.67 versus 31.83 for FM, with zero digit-level constraint violations for RFM versus 51-65% for FM.
- On 2D constrained toy domains, RFM has zero constraint violations while FM yields 3-14 per thousand; RFM KL is on par or better than FM.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use reflected flow matching instead of flow matching | Low-dimensional constrained-domain generation · Hypercube, Simplex, Half Annulus, and Cup | KL divergence: Hypercube d=2: 0.0044(0.0010) vs 0.0021(0.0011); Simplex d=2: 0.0062(0.0011) vs 0.0027(0.0011); Half Annulus: 0.0083(0.0 | Helped |
| Use reflected flow matching instead of flow matching | Low-dimensional constrained-domain generation · Hypercube d=10 | KL divergence: 0.0224(0.0010) vs 0.0243(0.0011) | Hurt |
| Use reflected flow matching instead of flow matching | Low-dimensional constrained-domain generation · Simplex d=10 | KL divergence: 0.0522(0.0022) vs 0.0452(0.0023) | Helped |
| Add reflection to the flow | Flow matching on low-dimensional constrained domains · Hypercube, Simplex, Half Annulus, and Cup | constraint violation ratio (‰): FM: 3.1, 26.9, 9.0, 86.2, 10.4, 13.9; RFM: 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 | Helped |
| Use reflected flow matching instead of flow matching | Unconditional image generation · CIFAR-10 32×32 | FID: FM: 6.35; RFM: 4.76; RDM: 2.72 | Hurt |
| Use reflected flow matching instead of flow matching | Class-conditioned image generation · ImageNet 64×64, guidance w=15 | FID: 31.83 vs 26.67 | Helped |
| Add reflection to the flow | Class-conditioned generation · ImageNet 64×64, guidance weights 7.5, 15, and 25 | constraint violation ratio: FM: 51.28%, 61.12%, 64.51%; RFM: 0.0%, 0.0%, 0.0% | Helped |
| Use RK4 with NFE=100 instead of Dopri5 (ablation) | RFM · CIFAR-10 32×32 | FID: 4.76 vs 5.37 | Hurt |
| Use RK4 with NFE=200 instead of Dopri5 (ablation) | RFM · CIFAR-10 32×32 | FID: 4.76 vs 4.85 | Hurt |
| Use Heun3 with NFE=150 instead of Dopri5 (ablation) | RFM · CIFAR-10 32×32 | FID: 4.76 vs 4.92 | Hurt |
| Use Heun3 with NFE=300 instead of Dopri5 (ablation) | RFM · CIFAR-10 32×32 | FID: 4.76 vs 4.81 | Hurt |
| Use a uniform prior instead of a standard Gaussian prior (ablation) | FM · Low-dimensional constrained-domain generation | constraint violation ratio (‰): Hypercube d=2: 3.1 vs 0.2; Hypercube d=10: 26.9 vs 8.6; Simplex d=2: 9.0 vs 5.9; Simplex d=10: 86.2 vs 33.7 | Helped |
Surprising findings
- Constraint violation ratio of FM increases with more ODE discretization steps, since the solver spends more time near the boundary.
- RDM performance degrades in 10-dimensional toy cases, partly due to the need for more discretization steps.
Limitations
- Requires domains with regular shape where conditional flows and boundary intersections are analytically computable (e.g., convex polytopes, simple nonconvex shapes).
- Some non-optimal-transport conditional flows on nonconvex domains need more NFE (e.g., 206-275 vs 110-119 for FM).
- Image experiments limited to CIFAR-10 and ImageNet 64x64 hypercube domains.
Experimental setup
Datasets: CIFAR-10 ImageNet 64x64
Benchmarks: CIFAR-10 ImageNet 64x64 2D hypercube 2D simplex half annulus cup
Metrics: FID KL divergence constraint violation ratio NFE
Baselines: FM RDM NCSN++ DDPM++ Subspace NCSN++ DDIM
Builds on: flow matching reflected diffusion models classifier-free guidance EDM
Closest papers by meaning
- An Error Analysis of Flow Matching for Deep Generative Modeling (ICML 2025 · similarity 0.91)
- Variational Rectified Flow Matching (ICML 2025 · similarity 0.90)
- A Theory on Flow Matching with Neural Networks (NeurIPS 2026 · similarity 0.90)
- Coreset-Induced Conditional Velocity Flow Matching (NeurIPS 2026 · similarity 0.89)
- CAR-Flow: Condition-Aware Reparameterization Aligns Source and Target for Better Flow Matching (NeurIPS 2025 · similarity 0.89)
- Stream-level Flow Matching with Gaussian Processes (ICML 2025 · similarity 0.89)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.