Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow

Problem

Framing

Continuous-time generative transport had strong quality but expensive inference, while one-step models lacked a clean ODE view. The paper closes this gap by learning velocity fields that follow straight interpolation directions, then recursively straightening them with reflow. On CIFAR-10, the distilled 2-rectified flow reaches FID 4.85 with one Euler step.

Currently Used Methods

Foundational

• @goodfellowGAN2014 — adversarial training for implicit high-fidelity generation.
  • Limitation in context: unstable optimization and no reversible transport dynamics.
• @kingmaVAE2013 — likelihood-based latent-variable generation with variational inference.
  • Limitation in context: approximate inference constrains sample sharpness and transport flexibility.
• @songScoreSDE2020 — score-based SDE generation with continuous-time denoising.
  • Limitation in context: inference needs many solver steps and schedule choices.
• @DenoisingDiffusionImplicitModels2020 — deterministic DDIM sampling from diffusion-trained models.
  • Limitation in context: induced paths remain curved and weak in one-step regimes.
• @lipmanFlowMatching2022 — direct regression of continuous-time vector fields for generative transport.
  • Limitation in context: this paper adds reflow to reduce transport cost and straighten trajectories.

Proposed Method

Architecture

The method learns a time-dependent velocity field $v_\theta(\mathbf{z}, t)$ for the ODE $d\mathbf{Z}_t = v_\theta(\mathbf{Z}_t,t)dt$. For image generation, it uses the DDPM++ U-Net backbone from @songScoreSDE2020; the paper changes the training target, not the backbone.

Loss / Objective

The core objective regresses the velocity on straight-line interpolation directions between paired endpoints.

The exact population optimum is

Sampling Rule / Algorithm

After fitting $v_\theta$, sampling solves the learned ODE forward from $\pi_0$; reflow retrains on the induced endpoint coupling.

Training Procedure

• $t \sim \mathrm{Uniform}([0,1])$.
• CIFAR-10: DDPM++ U-Net.
• Optimizer: Adam.
• EMA decay: $0.999999$.
• Inference: Euler with step size $1/N$ or adaptive RK45.
• Distillation target: one-step map $\mathbf{z}_1 = \mathbf{z}_0 + v(\mathbf{z}_0,0)$.

Evaluation

Datasets

• CIFAR-10, unconditional generation.
• LSUN Bedroom, LSUN Church, CelebA-HQ, AFHQ Cat, $256 \times 256$ generation.
• AFHQ, MetFace, CelebA-HQ for unpaired image translation.
• OfficeHome, DomainNet for domain adaptation.

Metrics

• FID.
• Inception Score.
• Recall.
• NFE for inference cost.
• Classification accuracy for domain adaptation.

Headline results

• CIFAR-10, RK45, 1-rectified flow: FID 2.58, IS 9.60, Recall 0.57, NFE 127.
• CIFAR-10, RK45, 2-rectified flow: FID 3.36, IS 9.24, Recall 0.54, NFE 110.
• CIFAR-10, one-step distilled 2-rectified flow: FID 4.85, Recall 0.50.
• CIFAR-10, one-step distilled 3-rectified flow: FID 5.21, Recall 0.51.
• OfficeHome domain adaptation: 69.2 $\pm$ 0.5 accuracy.

Results table

Table 2: Domain adaptation accuracy on transferred testing data

Ablations

• Reflow depth: one reflow sharply improves small-step generation; too many reflows hurt large-step accuracy.
• Solver steps $N$: rectified flow degrades gracefully at low $N$; VP and sub-VP ODEs collapse faster.
• Distillation: one-step performance improves strongly after distilling reflowed models.
• Straightness: reflow lowers the straightness metric and makes endpoint extrapolations nearly time-invariant.

Method Strengths and Weaknesses

Strengths

• One regression loss replaces adversarial games and diffusion score matching.
• Reflow targets inference efficiency directly by straightening trajectories.
• One-step distilled 2-rectified flow reaches FID 4.85 on CIFAR-10.
• Same framework handles generation, translation, and domain adaptation.

Weaknesses

• Best FID still trails full SDE solvers on CIFAR-10.
• Repeated reflow accumulates estimation error and can hurt large-step quality.
• Training relies on sampled couplings from previous flows, adding extra stages.
• Theory assumes exact regression and well-behaved ODE solutions.

Suggestions from the authors

• Study nonlinear interpolations with non-Euclidean geometry constraints.
• Improve one-step inference with final-stage distillation schemes.
• Combine rectified flow with augmentation or GAN-style refinements.
• Extend theory beyond exact regression and idealized rectifiable flows.

Links

Prior Papers

• @lipmanFlowMatching2022 — both learn transport vector fields directly; rectified flow adds reflow-based trajectory straightening.
• @songScoreSDE2020 — rectified flow reinterprets probability-flow ODEs against simpler straight-path interpolation.
• @DenoisingDiffusionImplicitModels2020 — the paper contrasts rectified straight paths with DDIM-style curved deterministic sampling.

Further Papers

• @albergoStochasticInterpolants2023 — extends the interpolation view of transport and generative dynamics beyond the paper's linear rectification setup.
• @songConsistencyModels2023 — pursues the same fast-sampling goal with direct few-step or one-step generation.
• @karrasEDM2022 — studies diffusion design choices that also target improved solver efficiency and sample quality.