  We describe the new field of mathematical analysis of deep learning. This
field emerged around a list of research questions that were not answered within
the classical framework of learning theory. These questions concern: the
outstanding generalization power of overparametrized neural networks, the role
of depth in deep architectures, the apparent absence of the curse of
dimensionality, the surprisingly successful optimization performance despite
the non-convexity of the problem, understanding what features are learned, why
deep architectures perform exceptionally well in physical problems, and which
fine aspects of an architecture affect the behavior of a learning task in which
way. We present an overview of modern approaches that yield partial answers to
these questions. For selected approaches, we describe the main ideas in more
detail.
