  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.
