The most challenging problems in science often involve the learning and
accurate computation of high dimensional functions.
High-dimensionality is a typical feature for a multitude of problems
in various areas of science.
The so-called curse of dimensionality typically negates the use of
traditional numerical techniques for the solution of
high-dimensional problems. Instead, novel theoretical and
computational approaches need to be developed to make them tractable
and to capture fine resolutions and relevant features. Paradoxically,
increasing computational power may even serve to heighten this demand,
since the wealth of new computational data itself becomes a major
obstruction. Extracting essential information from complex
problem-inherent structures and developing rigorous models to quantify
the quality of information in a high-dimensional setting pose
challenging tasks from both theoretical and numerical perspective.
This has led to the emergence of several new computational methodologies,
accounting for the fact that by now well understood methods drawing on
spatial localization and mesh-refinement are in their original form no longer viable.
Common to these approaches is the nonlinearity of the solution method.
For certain problem classes, these methods have
drastically advanced the frontiers of computability.
The most visible of these new methods is deep learning. Although the use of deep neural
networks has been extremely successful in certain
application areas, their mathematical understanding is far from complete.
This workshop proposed to deepen the understanding of
the underlying mathematical concepts that drive this new evolution of
computational methods and to promote the exchange of ideas emerging in various
disciplines about how to treat multiscale and high-dimensional problems.