rectified flow

Rectified flow is an ODE-based generative modeling framework that learns a velocity field $v(x,t)$ transporting samples from a source distribution ${\pi }_0$ to a target distribution ${\pi }_1$ along straight paths. Given a coupling $(X_0,X_1)$, the velocity is trained by minimizing $\mathbb{E}[\Vert (X_1-X_0)-v(X_t,t){\Vert }^2]$ where $X_t=t{X}_1+(1-t)X_0$ is the linear interpolation. The optimal velocity is $v^{\ast }(z,t)=\mathbb{E}[X_1-X_0\mid X_t=z]=(\mathbb{E}[Y\mid X_t=z]-z)/(1-t)$, a pure conditional target-pulling velocity.

The rectified flow ODE $d{Z}_t=v(Z_t,t)dt$ produces a deterministic coupling $(Z_0,Z_1)$ with provably non-increasing convex transport costs — simultaneously for all convex cost functions, via Jensen’s inequality. The non-crossing property of ODE trajectories forces the flow to “rewire” intersecting interpolation paths, creating a deterministic transport from a stochastic coupling.

Rectified flow connects deeply to random bridges: the conditional expectation drift $(\mathbb{E}[Y\mid {\xi }_t]-{\xi }_t)/(T-t)$ of a Lévy random bridge has the identical form to the optimal rectified flow velocity (Shelley & Mengütürk 2025). In fact, rectified flow is the zero-volatility limit of the Gaussian random bridge, and $\mathbb{E}[{\xi }_t^{x,y}]=R_t^{x,y}$ for any Lévy driving process.

Probability flow ODEs and DDIM are special cases of a generalized nonlinear rectified flow framework with non-linear interpolation choices, but these produce curved paths that cannot be straightened by reflow.

Key Details

• Training loss = $\mathbb{E}[\Vert (X_1-X_0)-v(X_t,t){\Vert }^2]$
• Optimal $v^{\ast }(z,t)=(\mathbb{E}[Y\mid R_t=z]-z)/(T-t)$
• Non-crossing property creates deterministic coupling
• 1-RF achieves FID 2.58 on CIFAR-10
• PF-ODEs and DDIM are special cases
• Zero-volatility limit of Gaussian random bridge

Textbook References

Optimal Transport Old and New (Villani, 2009)

• Example 7.2 (pp. 127-128): The fundamental straightness argument — for strictly convex $L(v)$ in ${\mathbb{R}}^n$, Jensen forces constant-velocity straight lines. The OT displacement interpolation paths ${\gamma }_t=(1-t)x+t\nabla \phi (x)$ are the fixed point that reflow converges toward
• Theorem 7.30(iv) (p. 151): Non-crossing of OT paths at intermediate times. This is the continuous analogue of the non-crossing property of ODE flows that rectified flow exploits
• Corollary 7.22 (pp. 139-140): Displacement interpolation $=$ geodesic in $(P_p,W_p)$, confirming that the OT coupling gives $S(Z)=0$ (perfect straightness)

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