ScoreSDE

Score-SDE's contribution is that two families which looked separate are the same object viewed at different resolutions.

Noise-conditional score networks perturb data with a geometric ladder of Gaussian noise; denoising diffusion perturbs it with a Markov chain of small Gaussian steps. Take the number of levels to infinity and both become stochastic differential equations:

$$dx = f(x, t)\,dt + g(t)\,dw.$$

NCSN's limit has zero drift and an exploding variance (VE); DDPM's has a contracting drift and a variance held at one (VP). The paper adds a third, sub-VP, whose variance is bounded by the VP process at every instant and which gives its best likelihoods.

What the continuous view buys is not a better model but better machinery, and three pieces of it.

Any solver. Sampling is integrating the reverse-time SDE

$$dx = \big[f(x, t) - g(t)^2 \nabla_x \log p_t(x)\big] dt + g(t)\, d\bar{w},$$

so any SDE solver works, and the number of steps stops being a property of the trained model.

Predictor-Corrector. A solver step can be followed by Langevin steps at fixed $t$, which correct the discretisation error the solver leaves behind rather than letting it accumulate.

The probability flow ODE. Every diffusion has a deterministic process sharing its marginals exactly:

$$dx = \Big[f(x, t) - \tfrac{1}{2} g(t)^2 \nabla_x \log p_t(x)\Big] dt.$$

That turns a generative model into a continuous normalising flow — adaptive step sizes, an invertible encoding, and exact likelihoods — without retraining anything.