A question on the equation (43) from the paper.

[aylive]

Great thanks for your effort in explaining this work with this repo. It helps me a lot.

Just a small question for the derivation in the paper, in Eq(43), it derives the score matching loss as below,

$\mathcal{L}_{SM}(\theta) = \mathbb{E}\left[\lambda(t)\Vert{s_t(x) - \nabla{\log{p_t(x|x_1)}}}\Vert^2\right] = \mathbb{E}\left[\lambda(t)\Vert{s_t(x) - \frac{x-\mu_t(x_1)}{\sigma_t^2(x_1)}}\Vert^2\right]$,

where $p_t(x|x_1) = \mathcal{N}(x|\mu_t(x_1), \sigma_t^2(x_1)\mathbf{I})$. But given the derivation of Gaussian, this seems should be,

$\mathcal{L}_{SM}(\theta) = \mathbb{E}\left[\lambda(t)\Vert{s_t(x) + \frac{x-\mu_t(x_1)}{\sigma_t^2(x_1)}}\Vert^2\right]$.

Right? It confuses me to connect the theories behind ddpm, score-based, sde, flow-mathcing together. :(