The backward diffusion process \ref{Backward Process} is not the only reverse process for the forward process \ref{Forward Process}. We can derive a deterministic ordinary differential equation (ODE) as an alternative, removing the stochastic term d\mathbf{W} in the backward process.

In this section, we will show how to train a neural network that models the score function \mathbf{s}(\mathbf{x}, t).

In this section, we present the key processes of Denoising Diffusion Probabilistic Models (DDPMs):

When I first learned about diffusion models, I was introduced to them as a type of variational autoencoder (VAE) applied to a series of quantities \mathbf{x}_0, \dots, \mathbf{x}_T. Deriving the forward and reverse processes required lengthy derivations spanning multiple pages, dense with priors, posteriors, Bayesian theorems, and mathematical intricacies. Later, I encountered the stochastic differential equation (SDE) perspective, which frames diffusion models through Fokker-Planck and Kolmogorov backward equations—concepts no simpler to grasp than the VAE approach.