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<math display="block">\min_{\theta} \int_0^1 \mathbb{E}_{\pi_0, \pi_1, p_t}\left [\lVert{(x_1-x_0) - v_t(x_t)}\rVert^2\right] \,\mathrm{d}t.</math>
The data pair <math>(x_0, x_1)</math> can be any coupling of <math>\pi_0</math> and <math>\pi_1</math>, typically independent (i.e., <math>(x_0,x_1) \sim \pi_0 \times \pi_1</math>) obtained by randomly combining observations from <math>\pi_0</math> and <math>\pi_1</math>. This process ensures that the trajectories closely mirror the density map of <math>x_t</math> trajectories but ''reroute'' at intersections to ensure causality. This rectifying process is also known as Flow Matching,<ref>{{cite arXiv |last1=Lipman |first1=Yaron |title=Flow Matching for Generative Modeling |date=2023-02-08 |eprint=2210.02747 |last2=Chen |first2=Ricky T. Q. |last3=Ben-Hamu |first3=Heli |last4=Nickel |first4=Maximilian |last5=Le |first5=Matt|class=cs.LG }}</ref> Stochastic Interpolation,<ref>{{cite arXiv |last1=Albergo |first1=Michael S. |title=Building Normalizing Flows with Stochastic Interpolants |date=2023-03-09 |eprint=2209.15571 |last2=Vanden-Eijnden |first2=Eric|class=cs.LG }}</ref> and
[[File:Reflow Illustration.png|thumb|390px|The reflow process<ref name=":0"/>]]
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