Generative artificial intelligence (AI) refers to algorithms that create
synthetic but realistic output. Diffusion models currently offer state of the
art performance in generative AI 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 (a) to present illustrative computational examples, (b) to
give a careful derivation of the underlying mathematical formulas involved, and
(c) to 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 students 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.

生成人工智能是指生成合成但逼真的输出的算法。扩散模型在图像生成人工智能中目前达到最高水平，并在包括文本到图像生成器和大型语言模型在内的更通用工具中起着关键作用。