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Generative artificial intelligence (AI) refers to algorithms that createsynthetic but realistic output. Diffusion models currently offer state of theart performance in generative AI for images. They also form a key component inmore general tools, including text-to-image generators and large languagemodels. Diffusion models work by adding noise to the available training dataand then learning how to reverse the process. The reverse operation may then beapplied to new random data in order to produce new outputs. We provide a briefintroduction to diffusion models for applied mathematicians and statisticians.Our key aims are (a) to present illustrative computational examples, (b) togive a careful derivation of the underlying mathematical formulas involved, and(c) to draw a connection with partial differential equation (PDE) diffusionmodels. We provide code for the computational experiments. We hope that thistopic will be of interest to advanced undergraduate students and postgraduatestudents. Portions of the material may also provide useful motivationalexamples for those who teach courses in stochastic processes, inference,machine learning, PDEs or scientific computing.

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