Why use flow-matching v as the score?

[hjkl33]

Hi TSDSR team,
I noticed that the SD3-based pipeline feeds the flow-matching predictor v directly into the score-based loss. But in my understanding, v isn’t usually interchangeable with the standard diffusion noise prediction. Could you briefly explain the rationale behind this choice and whether any extra scaling or conversion is applied? I might be missing something obvious, so any pointers to the relevant lines in the code or paper would be greatly appreciated. Thanks a lot for your great work and for clarifying!

[Microtreei]

Following Yang Song et al.,Jie Liu et al., an equivalent SDE that matches the original model’s marginal probability density function at all timesteps is specified as:

$${d}\boldsymbol{x}_t=\left(\boldsymbol{v}_t(\boldsymbol{x}_t)-\frac{\sigma_t^2}{2}\nabla\log p_t(\boldsymbol{x}_t)\right){d}t+\sigma_t{d}\boldsymbol{w}$$

For rectified flow, the formula can be expressed as：

$${d}\boldsymbol{x}_t=\left[\boldsymbol{v}_t(\boldsymbol{x}_t)+\frac{\sigma_t^2}{2t}\left(\boldsymbol{x}_t+(1\boldsymbol{-}t)\boldsymbol{v}_t(\boldsymbol{x}_t)\right)\right]{d}t+\sigma_t{d}\boldsymbol{w}$$

There is a connection between score and v-predict.

According to ProlificDreamer,DMD, the VSD loss was Initially modeled as：

$$\nabla_\theta D_{KL}=\underset{z\sim\mathcal{N}(0;\mathbf{I})} {{{\mathbb{E}}}}\left[-\left(\nabla \log p_{{real}}(x)-\nabla \log p_{{fake}}(x)\right)\frac{dG}{d\theta}\right]$$

The loss is expressed as the difference between the two score, resulting in some identical terms being either eliminated or extracted as common terms. And this common factor is only affected by $t$.