Continuous Function

Legacy Concept Lab

Flow Matching & Rectified Flows

Simpler than diffusion: no noise schedule to tune

Concept 83 of 100Generative ModelsPhase 4
Open New NotebookAll Foundations
#83Flow MatchGenerative Models
key equation\frac{dx_t}{dt} = v_\theta(x_t, t)
Phase 4: Generative modeling familiesConcept 83 of 100
Migrated:view the updated version in /domains →This /foundations page is legacy during migration.

Why It Matters for Modern Models

  • Simpler than diffusion: no noise schedule to tune
  • Rectified flows enable 1-4 step generation
  • State-of-the-art: Stable Diffusion 3 uses flow matching

What Tutorials Skip

What is still poorly explained in textbooks and papers:

  • Diffusion = learn to denoise; flow = learn velocity directly
  • Straight paths are easiest to approximate with few steps
  • Simulation-free: no sampling during training

Interactive Visualization

Core Math (Optional Deep Dive)

If you want intuition first, start with the key equation and the visualization. Come back here for the full walkthrough.

Key Equation
dxtdt=vθ(xt,t)\frac{dx_t}{dt} = v_\theta(x_t, t)

Learn vector field transporting noise to data:

dxtdt=vθ(xt,t)\frac{dx_t}{dt} = v_\theta(x_t, t)

Conditional flow matching:

L=Et,x0,x1[∥vθ(xt,t)−ut∥2]\mathcal{L} = \mathbb{E}_{t, x_0, x_1}[\|v_\theta(x_t, t) - u_t\|^2]

Rectified flow: Straight paths xt=(1−t)x0+tx1x_t = (1-t)x_0 + tx_1

ut=x1−x0 (constant velocity)u_t = x_1 - x_0 \text{ (constant velocity)}

Canonical Papers

Flow Matching for Generative Modeling

Lipman et al.2023ICLR
Read paper →

Connections

Prerequisites

∂Diffusion🌊Flows

Next Moves

Explore this concept from different angles — like a mathematician would.