Rectified Flow

Rectified flow learns a transport that follows straight-line paths between a noise distribution and the data distribution, enabling fast, few-step sampling.

The straight-line interpolation is often:

$$x_t=(1-t)x_0+t{x}_1$$

with target velocity:

$$u_t=x_1-x_0$$

The model learns a velocity field that approximates this transport.

The training objective can be written as:

$${\mathcal{L}}_{\mathrm{RF}}={\mathbb{E}}_{t,x_0,x_1}\left[{\Vert v_{\theta }(x_t,t)-(x_1-x_0)\Vert }_2^2\right]$$

where $x_0$ is sampled from the base distribution, $x_1$ from the data distribution, and $t\in [0,1]$.

Sampling solves the ODE:

$$\frac{d{x}_t}{dt}=v_{\theta }(x_t,t),x_0\sim p_0$$

from the base distribution toward the data distribution.

Reflow

Reflow or distillation can train a new model on trajectories generated by an earlier model. The goal is to make the learned path straighter so fewer numerical integration steps are needed.

Flow-Matching View

Rectified flow is a specific transport construction inside the broader flow-matching family. The training pair defines an interpolation path:

$$x_t=\alpha (t)x_0+\sigma (t)x_1$$

and a target velocity:

$$u_t=\frac{d{x}_t}{dt}$$

For the straight path $\alpha (t)=1-t$ and $\sigma (t)=t$:

$$u_t=x_1-x_0$$

The model claim should state the endpoint distributions, pairing rule, interpolation, and ODE solver budget.

Evidence Contract

Field	Question
Base distribution	What is $p_0$?
Data distribution	What empirical data defines $p_1$?
Pairing	Are endpoints paired independently, optimally, conditionally, or by teacher trajectories?
Path	Is interpolation linear, variance-preserving, or domain-specific?
Solver	Euler, Heun, adaptive ODE, or fixed-step schedule?
Step budget	How many function evaluations are used?
Metric	sample quality, validity, likelihood proxy, downstream utility, or speed

Without this contract, “rectified flow” does not specify enough to compare papers.

Geometry and Structured Data

Straight lines in Euclidean space can violate object constraints:

Object	Constraint
molecule graph	atoms and bonds are discrete
conformer	bond lengths, torsions, chirality
protein coordinates	residue indexing, chain geometry, equivariance
protein-ligand pose	receptor frame, pocket constraints, clashes
sequence	tokens are discrete and often length-dependent

For coordinates, the velocity should obey the same symmetry as the coordinate target:

$$v_{\theta }(Rx+t,t)=R{v}_{\theta }(x,t)$$

for rotation $R$ and translation $t$ when the coordinate frame is arbitrary.

Why It Matters

• Straighter trajectories reduce the number of sampling steps.
• Closely related to flow matching and ODE-based generation.
• A practical route to fast diffusion-style samplers.

Failure Modes

• Independent endpoint pairing may define paths that cross or average incompatible modes.
• Few-step sampling can reduce diversity even when samples look sharper.
• A Euclidean straight line may be inappropriate for constrained or geometric data.
• Reflow can improve speed while preserving teacher errors.

Checks

• Are the learned paths actually close to straight?
• How much does reflow/distillation improve few-step quality?
• Does few-step sampling preserve diversity?
• Does the interpolation respect the object constraints?
• Is the ODE solver budget reported with sample quality?
• Are endpoint pairing and reflow data generation described?
• For molecules or proteins, are invalid intermediate or final samples counted?

Related

• Flow matching
• Probability flow ODE
• Score-based model
• Consistency model
• Equivariance
• Coordinate modeling contract