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Constant Acceleration Flow
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Proposes Constant Acceleration Flow (CAF), an ODE framework with a learnable constant acceleration term plus initial velocity conditioning and reflow for the initial velocity, achieving strong one-step FID on CIFAR-10 and ImageNet 64x64.
Problem
Rectified flow assumes constant-velocity straight trajectories between noise and data couplings; this is limited in expressiveness and suffers from flow crossing after reflow, which hurts few-step generation quality and coupling preservation.
Approach
Replace constant velocity with a constant acceleration ODE, x_t = (1-t^2)x0 + t^2 x1 + v(x0)(t-t^2), learning an initial velocity model and an acceleration model; condition the acceleration model on the initial velocity (IVC) to resolve ambiguity at trajectory intersections, and apply reflow to the initial velocity model.
Key results
- CAF with adversarial training reaches FID 1.39 (conditional) on CIFAR-10 and 1.69 on ImageNet 64x64 with one step.
- Base unconditional CAF achieves FID 4.81 at N=1 on CIFAR-10 versus 7.89 for 2-Rectified Flow at N=2.
- Reconstruction PSNR via inversion is 46.68 for CAF versus 33.34 for 2-RF at N=2 on CIFAR-10.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use constant-acceleration CAF instead of 2-Rectified Flow | CIFAR-10 unconditional generation · CIFAR-10 unconditional, N=1 vs 2-Rectified Flow N=2 | FID: 7.89 -> 4.81 | Helped |
| Add adversarial training to CAF | CAF · CIFAR-10 unconditional, N=1 | FID: 4.81 -> 1.48 | Helped |
| Add adversarial training to CAF | CAF · CIFAR-10 conditional, N=1 | FID: 2.68 -> 1.39 | Helped |
| Use CAF + GAN instead of CTM | ImageNet 64×64 one-step generation · ImageNet 64×64, N=1 | FID: 1.92 -> 1.69 | Helped |
| Use CAF instead of 2-Rectified Flow | Coupling preservation · CIFAR-10 coupling preservation | LPIPS: 0.092 -> 0.041 | Helped |
| Use CAF instead of 2-Rectified Flow | Flow straightness · CIFAR-10 | NFSS: 0.043 -> 0.034 | Helped |
| Use CAF instead of 2-RF | Image reconstruction by inversion · Reconstruction | PSNR: 33.34 -> 46.68 | Helped |
| Add reflow to 1-Rectified flow (ablation) | One-step CIFAR-10 generation · CIFAR-10, N=1 | FID: 378 -> 6.88 | Helped |
| Add constant acceleration to reflowed 2-Rectified Flow, without IVC (ablation) | One-step CIFAR-10 generation · CIFAR-10, N=1 | FID: 6.88 -> 3.82 | Helped |
| Add initial-velocity conditioning (ablation) | CAF with constant acceleration and reflow · CIFAR-10, N=1 | FID: 3.82 -> 2.68 | Helped |
| Set the initial velocity scale h=1 instead of h=1.5 (ablation) | CAF with IVC and reflow · CIFAR-10, N=1 | FID: 2.68 -> 3.02 | Hurt |
| Set the initial velocity scale h=0.5 instead of h=1.5 (ablation) | CAF with IVC and reflow · CIFAR-10, N=1 | FID: 2.68 -> 2.73 | Hurt |
Surprising findings
- Reflow alone does not overcome flow crossing: 2-Rectified Flow poorly approximates target on 2D data despite reflow.
- AGM does not benefit from reflow in few-step regime, unlike CAF.
Limitations
- Requires one extra function evaluation (N+1 NFE) for the initial velocity.
- Depends on generating a large deterministic coupling dataset with a pretrained EDM, increasing compute and carbon cost.
Experimental setup
Datasets: CIFAR-10 ImageNet 64x64
Benchmarks: CIFAR-10 ImageNet 64x64
Metrics: FID IS recall PSNR LPIPS
Baselines: 2-Rectified Flow 1-Rectified Flow AGM EDM CTM Consistency Models DMD DPM-solver
Builds on: rectified flow EDM ADM StyleGAN-XL
Closest papers by meaning
- Simple ReFlow: Improved Techniques for Fast Flow Models (ICLR 2025 · similarity 0.85)
- Accelerating Rectified Flow Models via Trajectory-Aware Caching (NeurIPS 2026 · similarity 0.85)
- PeRFlow: Piecewise Rectified Flow as Universal Plug-and-Play Accelerator (NeurIPS 2024 · similarity 0.85)
- ProReflow: Progressive Reflow with Decomposed Velocity (CVPR 2025 · similarity 0.85)
- Improving the Training of Rectified Flows (NeurIPS 2024 · similarity 0.85)
- Balanced Conic Rectified Flow (NeurIPS 2025 · similarity 0.84)
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