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

• URL: http://arxiv.org/abs/2605.22586v3
• Date: Wed, 27 May 2026 21:34:42 GMT
• Title: A Tutorial on Diffusion Theory: From Differential Equations to Diffusion Models
• Abstract summary: Diffusion models have emerged as a dominant framework for generative modeling.<n>This tutorial provides a unified and self-contained account of these viewpoints from the perspective of differential equations.<n>The tutorial is intended as a bridge between the analytical foundations of diffusion processes and the modern generative algorithms built upon them.
• Score: 15.927114890047035
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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