Mathematical Introduction to Deep Learning: Methods, Implementations,
  and Theory
Arnulf Jentzen Benno Kuckuck Philippe von Wurstemberger
This book aims to provide an introduction to the topic of deep learning
algorithms. We review essential components of deep learning algorithms in full
mathematical detail including different artificial neural network (ANN)
architectures (such as fully-connected feedforward ANNs, convolutional ANNs,
recurrent ANNs, residual ANNs, and ANNs with batch normalization) and different
optimization algorithms (such as the basic stochastic gradient descent (SGD)
method, accelerated methods, and adaptive methods). We also cover several
theoretical aspects of deep learning algorithms such as approximation
capacities of ANNs (including a calculus for ANNs), optimization theory
(including Kurdyka-{\L}ojasiewicz inequalities), and generalization errors. In
the last part of the book some deep learning approximation methods for PDEs are
reviewed including physics-informed neural networks (PINNs) and deep Galerkin
methods. We hope that this book will be useful for students and scientists who
do not yet have any background in deep learning at all and would like to gain a
solid foundation as well as for practitioners who would like to obtain a firmer
mathematical understanding of the objects and methods considered in deep
learning.