reflow

Reflow is the procedure of recursively applying rectification to progressively straighten ODE flow paths, enabling accurate simulation with very few (even one) Euler discretization steps. Starting from a 1-rectified flow trained on independent coupling $(X_0,X_1)\sim {\pi }_0\times {\pi }_1$, the flow is simulated to generate new endpoint pairs $(Z_0^1,Z_1^1)$, and a 2-rectified flow is trained on these induced pairs. Each iteration provably reduces the straightness measure $S(Z)=\int \mathbb{E}[\Vert (Z_1-Z_0)-d{Z}_t/dt{\Vert }^2]dt$ at rate $O(1/K)$.

Algorithm

1. Train $k$-rectified flow on coupling $(Z_0^{k-1},Z_1^{k-1})$
2. Simulate flow to generate new pairs $(Z_0^k,Z_1^k)$
3. Train $(k+1)$-rectified flow on new pairs
4. Optionally distill with LPIPS loss for one-step inference

Key Properties

• $S(Z)$ decreases at $O(1/K)$ rate
• In practice one reflow step suffices for near-straight paths
• Distilled 2-rectified flow achieves FID 4.85 with single Euler step on CIFAR-10
• Slightly worsens full-solver FID due to accumulated estimation error
• Connected to one IPF half-iteration in random bridge framework

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