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rectified flow

Straighten the noise-to-data paths a flow model learns until sampling takes a handful of ODE steps — or just one. The flow-matching formulation behind Stable Diffusion 3.

denoiser + flow matching

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Flow matching trains a velocity field that carries noise to data along an ODE, and its appeal over diffusion is that straighter transport paths need fewer integration steps. Rectified flow takes that logic to its conclusion: make straightness the explicit goal. A perfectly straight trajectory can be integrated exactly in a single Euler step — no solver error, no hundred-step sampling loop. The method’s contribution is a procedure that provably makes the learned paths straighter each time you apply it.

How it works

Start with linear pairings. Draw a noise sample x0 and a data sample x1 independently, connect them with the straight line x_t = (1−t)·x0 + t·x1, and regress a velocity network onto the direction x1 − x0 — the same simulation-free objective as flow matching with linear paths.

Why the learned flow still curves. The individual training lines cross each other (many random noise–data pairs pass through the same point), and an ODE’s trajectories cannot cross. At crossing points the model averages the conflicting directions, so the marginal flow it learns bends even though every training path was straight. Following it accurately still takes many small steps.

Reflow — the rectification step. Sample the trained model: integrate from a noise sample x0 to its output x1'. This produces a new coupling (x0 → x1') that the flow itself generated — and these pairs, by construction, no longer cross. Retrain the velocity field on straight lines between these pairs. The result is a new flow with the same noise and data distributions at the endpoints but visibly straighter transport; each repetition (1-rectified flow, 2-rectified flow, …) straightens further. A final distillation pass collapses the nearly-straight flow into a one-step generator.

Scaling it. Stable Diffusion 3 pairs the rectified-flow objective with a diffusion transformer backbone and biases the training-time sampling of t toward the middle of the trajectory (where the velocity is hardest to learn) — the recipe that took rectified flow from paper to production text-to-image.

Relation to flow matching

The two papers appeared independently within weeks of each other, and the base training objective is the same conditional regression. What rectified flow adds is the coupling perspective: the insight that which (x0, x1) pairs you draw lines between determines how straight the marginal flow can be, plus the reflow algorithm for improving those couplings using the model itself. In practice “rectified flow” usually names the linear-path objective (as in SD3), with reflow reserved for few-step and one-step variants.

Where you’ll see it

Stable Diffusion 3 made rectified flow the highest-profile generative objective in image synthesis, and Flux continues the lineage. Reflow-style straightening and its descendants power the one- and few-step samplers that make near-realtime image generation possible.

Sources

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