Despite the outstanding success of deep neural networks in
real-world applications, ranging from science to public life, most
of the related research is empirically driven and a comprehensive
mathematical foundation is still missing. At the same time, these
methods have already shown their impressive potential in
mathematical research areas such as imaging sciences, inverse
problems, or numerical analysis of partial differential equations,
sometimes by far outperforming classical mathematical approaches
for particular problem classes.
The goal of this lecture is to first provide an introduction into
this new vibrant research area. We will then survey recent advances
in two directions, namely the development of a mathematical
foundation of deep learning and the introduction of novel deep
learning-based approaches to mathematical problem settings.