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

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.

Quantum Deep Learning: Sampling Neural Nets with a Quantum Annealer

We demonstrate the feasibility of framing a classically learned deep neural network as an energy based model that can be processed on a one-step quantum annealer in order to exploit fast sampling times. We propose approaches to overcome two hurdles for high resolution image classification on a quantum processing unit (QPU): the required number and binary nature of the model states. With this novel method we successfully transfer a convolutional neural network to the QPU and show the potential for classification speedup of at least one order of magnitude.

Deep Learning: An Introduction for Applied Mathematicians

Multilayered artificial neural networks are becoming a pervasive tool in a host of application fields. At the heart of this deep learning revolution are familiar concepts from applied and computational mathematics; notably, in calculus, approximation theory, optimization and linear algebra. This article provides a very brief introduction to the basic ideas that underlie deep learning from an applied mathematics perspective. Our target audience includes postgraduate and final year undergraduate students in mathematics who are keen to learn about the area. The article may also be useful for instructors in mathematics who wish to enliven their classes with references to the application of deep learning techniques. We focus on three fundamental questions: what is a deep neural network? how is a network trained? what is the stochastic gradient method? We illustrate the ideas with a short MATLAB code that sets up and trains a network. We also show the use of state-of-the art software on a large scale image classification problem. We finish with references to the current literature.