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.

该研究论文介绍了深度学习算法的主题，包括不同的人工神经网络架构、优化算法和理论方面，同时讨论了用于偏微分方程的深度学习逼近方法。

深度学习的数学导引：方法、实现和理论

Mathematical Introduction to Deep Learning: Methods, Implementations,  and Theory

The problem of finding groups in data (cluster analysis) has been extensively
studied by researchers from the fields of Statistics and Computer Science,
among others. However, despite its popularity it is widely recognized that the
investigation of some theoretical aspects of clustering has been relatively
sparse. One of the main reasons for this lack of theoretical results is surely
the fact that, unlike the situation with other statistical problems as
regression or classification, for some of the cluster methodologies it is quite
difficult to specify a population goal to which the data-based clustering
algorithms should try to get close. This paper aims to provide some insight
into the theoretical foundations of the usual nonparametric approach to
clustering, which understands clusters as regions of high density, by
presenting an explicit formulation for the ideal population clustering.

本文旨在通过为最理想的种群聚类提供一个明确的公式来深入探讨通常的非参数聚类方法的理论基础，该方法将聚类视为高密度区域。