Diffusion Models for Generative Artificial Intelligence: An Introduction for Applied Mathematicians

SIAM Review, Volume 67, Issue 3, Page 607-623, August 2025.
Abstract.Generative artificial intelligence (GAI) refers to algorithms that create synthetic but realistic output. Diffusion models currently offer state-of-the-art performance in GAI for images. They also form a key component in more general tools, including text-to-image generators and large language models. Diffusion models work by adding noise to the available training data and then learning how to reverse the process. The reverse operation may then be applied to new random data in order to produce new outputs. We provide a brief introduction to diffusion models for applied mathematicians and statisticians. Our key aims are to (a) present illustrative computational examples, (b) give a careful derivation of the underlying mathematical formulas involved, and (c) draw a connection with partial differential equation (PDE) diffusion models. We provide code for the computational experiments. We hope that this topic will be of interest to advanced undergraduate and postgraduate students. Portions of the material may also provide useful motivational examples for those who teach courses in stochastic processes, inference, machine learning, PDEs, or scientific computing.