family · generative-framework

flow matching

A generative training objective that learns a velocity field transporting noise to data along straight paths — fewer sampling steps than diffusion, same backbone architectures (SD3, Flux).

denoiser + flow matching

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Diffusion models generate images by reversing a noising process across hundreds of small steps. Flow matching is a closely related — and in many formulations more general — alternative: instead of learning to denoise step by step, it trains a network to regress the velocity field that, if you follow it as an ordinary differential equation, carries a noise sample directly to a data sample. Because the learned paths are straighter, accurate samples typically need far fewer integration steps at generation time.

Flow matching is the training objective behind Stable Diffusion 3 and Flux, where it is paired with a diffusion transformer as the backbone network.

How it works

The probability path. Define an interpolation between a noise sample x0 and a data sample x1. The simplest choice is a straight line: x_t = (1−t)·x0 + t·x1 for t between 0 and 1. This defines a family of distributions morphing from noise to data, and a corresponding velocity field — the instantaneous direction and speed that transports probability mass along the path.

The objective. Train a network v(x, t) to match that velocity. For the straight-line path the target is simply x1 − x0, a constant for each pair — so training is a plain regression onto known per-sample vectors. Crucially, this regression is simulation-free: you never have to integrate the ODE during training. This is the conditional flow matching trick — conditioning on a specific (x0, x1) pair makes the otherwise-intractable marginal velocity tractable to regress.

Sampling. At generation time, draw x0 from noise and integrate the learned velocity field from t = 0 to t = 1 using a standard ODE solver (Euler, Heun, etc.). Straight paths mean fewer steps are needed for accurate results.

Rectified flow. A flow-matching variant that explicitly pushes transport paths to be as straight as possible. Straight enough paths can be integrated in a single step, underpinning fast one-step samplers. SD3 and Flux use this formulation.

Relation to diffusion

Diffusion’s score-based reverse process has an equivalent deterministic probability-flow ODE. Flow matching can be understood as directly learning that ODE’s velocity field, sidestepping the noise-schedule machinery. The two paradigms share backbone architectures — U-Net or DiT — and differ in how they define the training target.

Where you’ll see it

Flow matching is the loss function behind the current frontier of image and video generation models. Stable Diffusion 3 and Flux are the highest-profile deployments. Expect it to displace vanilla diffusion training in most new generative pipelines, given its sampling efficiency.

Sources

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