Flow Matching essentially models the expectation over all possibilities, called the marginal velocity $v(z_t,t)$:

$v(z_t,t)={\mathbb{E}}_{p_t(v_t|z_t)}[v_t]$ Given a marginal velocity field: $\frac{d}{dt}{z}_t=v(z_t,t)$

The solution can be written as: $z_r=z_t-{\int }_r^{t}v(z_{\tau },\tau )d\tau$

• starting from $z_1=ϵ\sim p_{prior}$

In practice, we just use the forward euler integration method, computing each step as $z_{t_{i+1}}=z_{t_i}+(t_{i+1}-t_i)v(z_{t_i},t_i)$

t is worth noting that even when the conditional flows are designed to be straight (“rectified”), the marginal velocity field (Eq. (1)) typically induces a curved trajectory.

“The core idea of our approach is to introduce a new field representing average velocity, whereas the velocity modeled in Flow Matching represents the instantaneous velocity”.

$$u(z_t,r,t)\triangleq \frac{1}{t-r}{\int }_r^{t}v\left(z_{\tau },\tau \right)d\tau$$