A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models

Abstract

Diffusion models have emerged as a dominant framework for generative modeling, but their mathematical foundations are often presented separately through diffusion probabilistic models, score-based modeling, stochastic differential equations, and numerical sampling methods. We write this tutorial to provide a unified and self-contained account of these viewpoints from the perspective of differential equations. Starting from a conditional Gaussian noising process, we derive ordinary differential equation (ODE) and stochastic differential equation (SDE) representations, pass to the corresponding marginal forward dynamics, and then obtain the reverse-time SDE and probability-flow ODE that make generation possible. We show that the central unknown quantity in reverse sampling is the marginal score, explain how score matching becomes the standard denoising objective under a noise-prediction parameterization, and discuss practical reverse-time sampling and guidance. We further place DDPM, DDIM, flow matching, and score-based SDEs in a common framework, and conclude with diffusion language models in continuous embedding space together with a brief discussion of discrete masked-token diffusion. The tutorial is intended as a bridge between the analytical foundations of diffusion processes and the modern generative algorithms built upon them.