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Variational Rectified Flow Matching
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Proposes Variational Rectified Flow Matching, which adds a latent variable with a VAE-style posterior so the flow velocity field can model multi-modal directions instead of averaging them, evaluated on synthetic data, MNIST, CIFAR-10 and ImageNet.
Problem
Classic rectified flow matching regresses velocity with an MSE loss on randomly coupled source-target pairs, so the learned velocity field averages multi-modal ground-truth directions at the same location, producing curved trajectories that are harder to integrate.
Approach
Introduce a latent variable z with a learned posterior q_phi(z|x0,x1,xt,t) and condition the velocity network on z, trained with a VAE-style ELBO so each latent selects a different flow direction at the same point, allowing trajectories to intersect while keeping a Gaussian prior for inference.
Key results
- On ImageNet 256x256, V-SiT-XL reaches FID 10.6 vs 13.1 for SiT-XL at 800K steps without guidance, and 3.22 vs 3.43 with cfg=1.5.
- On CIFAR-10, the best V-RFM variant reaches FID 3.478 with adaptive Dopri5 solver vs 3.655 for OT-FM, and 104.634 vs 166.655 for OT-FM at 2 NFE.
- On MNIST, V-RFM achieves better FID than I-CFM at most evaluation steps with a 2-dimensional latent; at 2 NFE, Consistency-FM is best.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace classic rectified flow loss with variational rectified flow loss and a conditional latent | SiT-XL · ImageNet 256×256, 400K steps | FID-50K: 17.2 -> 14.6 | Helped |
| Replace classic rectified flow loss with variational rectified flow loss and a conditional latent | SiT-XL · ImageNet 256×256, 800K steps | FID-50K: 13.1 -> 10.6 | Helped |
| Replace classic rectified flow loss with variational rectified flow loss and a conditional latent | SiT-XL cfg=1.5 · ImageNet 256×256, cfg=1.5, 400K steps | FID-50K: 5.40 -> 4.91 | Helped |
| Replace classic rectified flow loss with variational rectified flow loss and a conditional latent | SiT-XL cfg=1.5 · ImageNet 256×256, cfg=1.5, 800K steps | FID-50K: 3.43 -> 3.22 | Helped |
| Use variational rectified flow matching instead of I-CFM | CIFAR-10 generation · CIFAR-10, 10 NFE | FID: 13.788 -> 13.226 | Helped |
| Increase KL loss weight from 2e-3 to 5e-3 (ablation) | V-RFM with adaptive norm and x1 posterior input · CIFAR-10, higher function evaluations | FID: 4.430 -> 4.349 (100 NFE); 3.642 -> 3.582 (1000 NFE) | Helped |
| Increase KL loss weight from 2e-3 to 5e-3 (ablation) | V-RFM with adaptive norm and x1 posterior input · CIFAR-10, lower function evaluations | FID: 135.275 -> 159.940 (2 NFE); 28.912 -> 35.293 (5 NFE) | Hurt |
| Add time t to posterior input (ablation) | V-RFM with adaptive norm and KL weight 5e-3 · CIFAR-10, adaptive NFE | FID: 3.561 -> 3.478 | Helped |
| Use bottleneck sum fusion instead of adaptive normalization (ablation) | V-RFM with x1 + t posterior input and KL weight 2e-3 · CIFAR-10, 2 NFE | FID: 135.275 -> 104.634 | Helped |
| Use bottleneck sum fusion instead of adaptive normalization (ablation) | V-RFM with x1 + t posterior input and KL weight 2e-3 · CIFAR-10, adaptive NFE | FID: 3.614 -> 3.520 | Helped |
| Reduce posterior model size to 17.5% (ablation) | V-RFM-L · CIFAR-10, adaptive NFE | FID: 3.545 -> 3.607 | Hurt |
| Reduce posterior model size to 6.7% (ablation) | V-RFM-L · CIFAR-10, adaptive NFE | FID: 3.545 -> 3.639 | Hurt |
Surprising findings
- Consistency flow matching performs well at low NFE but its best performance does not surpass classic rectified flow or V-RFM, and its performance degrades as NFE increases.
- Shrinking the posterior model to 6.7% of its size changes results only modestly, suggesting robustness to posterior capacity.
Limitations
- Prior distribution over z is fixed to a standard Gaussian; learned priors left to future work.
- Theoretical analysis of marginal preservation is limited to an extension of a known theorem; no quantified compute overhead is reported.
- Distillation and combination with consistency models are not explored.
Experimental setup
Datasets: synthetic 1D data synthetic 2D data MNIST CIFAR-10 ImageNet
Benchmarks: CIFAR-10 MNIST ImageNet 256x256 synthetic 1D synthetic 2D
Metrics: FID True Log-Likelihood Parzen Window Log-Likelihood Inception Score Wasserstein Distance
Baselines: OT-FM I-CFM Consistency-FM Consistency-FM-XL rectified flow DiT-XL SiT-XL
Builds on: rectified flow matching variational auto-encoders SiT I-CFM OT-FM
Closest papers by meaning
- Reflected Flow Matching (ICML 2024 · similarity 0.90)
- A Theory on Flow Matching with Neural Networks (NeurIPS 2026 · similarity 0.89)
- Structured Coupling for Flow Matching (NeurIPS 2026 · similarity 0.88)
- Value Gradient Guidance for Flow Matching Alignment (NeurIPS 2025 · similarity 0.88)
- On the Relation between Rectified Flows and Optimal Transport (NeurIPS 2025 · similarity 0.87)
- Towards Hierarchical Rectified Flow (ICLR 2025 · similarity 0.87)
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