Diffusion SDE Stochastic Calculus Proof

Score-Based Continuous Stochastic Differential Equations (Score SDE, pioneered by Yang Song) unifies discrete DDPMs and score-matching generative models into a continuous-time stochastic calculus framework; the forward noising process is parameterized as an Itô SDE: $\mathrm{d}x = f(x, t)\mathrm{d}t + g(t)\mathrm{d}w$ ($f$ drift coefficient, $g$ diffusion coefficient, $w$ standard Brownian motion); via Anderson's Reverse-Time Theorem, the exact continuous reverse-time generative process is proven to be another closed-form SDE: $\mathrm{d}x = [ f(x, t) - g(t)^2 \nabla_x \ln p_t(x) ] \mathrm{d}t + g(t) \mathrm{d}\bar{w}$; neural generation simplifies to learning the score function $\nabla_x \ln p_t(x)$ via score matching ($s_\theta(x, t)$); furthermore, the deterministic Probability Flow ODE allows standard Runge-Kutta numerical solvers to generate samples in 10-20 steps.

Generative AI research scientist theoretical interviews, fast numerical diffusion solver engineering, and Continuous Normalizing Flows.

Discrete 1000-step Markovian sampling is slow and mathematically fragmented; Score SDE delivers a unified continuous stochastic foundation and fast deterministic ODE trajectories.

Derive the appearance of the score function $\nabla_x \ln p_t(x)$ in reverse-time SDEs via Fokker-Planck and Anderson's theorem?

Derive the Probability Flow ODE equation and explain why it shares the exact same marginal probability trajectories as the stochastic SDE?

Why does Denoising Score Matching (DSM) bypass intractable marginal score calculations by training on known conditional Gaussian kernels?

Derive Classifier-Free Guidance (CFG) as a linear extrapolation between conditional and unconditional score estimates?

How does DPM-Solver leverage exponential integrators over the semi-linear Probability Flow ODE to achieve photorealistic sampling in under 10 steps?