401-3650-19L  Numerical Analysis Seminar: Deep Neural Network Approximation

SemesterFrühjahrssemester 2021
DozierendeC. Schwab
Periodizitätjährlich wiederkehrende Veranstaltung
LehrveranstaltungFindet dieses Semester nicht statt.
LehrspracheEnglisch


KurzbeschreibungThis seminar will review recent _mathematical results_ on approximation power of deep neural networks (DNNs). The focus will be on mathematical proof techniques to obtain approximation rate estimates (in terms of neural network size and connectivity) on various classes of input data including, in particular, selected types of PDE solutions.
Lernziel
InhaltPresentation of the Seminar:
Deep Neural Networks (DNNs) have recently attracted substantial
interest and attention due to outperforming the best established
techniques in a number of tasks (Chess, Go, Shogi,
autonomous driving, language translation, image classification, etc.).
In many cases, these successes have been achieved by
heuristic implementations combined
with massive compute power and training data.

The seminar will address mathematical results on
the approximation/ expressive power of DNNs.
For a (bird's eye) overview, see
Link
and, more mathematical and closer to the seminar theme,
Link

Specifically,
this seminar will review recent _mathematical results_
on approximation power of deep neural networks (DNNs).
The focus will be on mathematical proof techniques to
obtain approximation rate estimates (in terms of neural network
size and connectivity) on various classes of input data
including, in particular, selected types of PDE solutions.
Mathematical results support that DNNs can
equalize or outperform the best mathematical results
known to date.

Particular cases comprise:
high-dimensional parametric maps,
analytic and holomorphic maps,
maps containing multi-scale features which arise as solution classes from PDEs,
classes of maps which are invariant under group actions.
Voraussetzungen / BesonderesEach seminar topic will allow expansion to a semester or a
master thesis in the MSc MATH or MSc Applied MATH.

The seminar format will be oral student presentations in
the first half of May 2021, combined with a written report.
Student presentations will be
based on a recent research paper selected in two meetings
at the start of the semester (end of February).

Disclaimer:
The seminar will _not_ address recent developments in DNN software,
such as training heuristics, or programming techniques
for DNN training in various specific applications.