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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Beating the Curse of Dimensionality: A Theoretical
Analysis of Deep Neural Networks and Parametric P
DEs - Gitta Kutyniok (Technische Universität Berli
n)
DTSTART;TZID=Europe/London:20190620T142000
DTEND;TZID=Europe/London:20190620T151000
UID:TALK126286AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/126286
DESCRIPTION:High-dimensional parametric partial differential e
quations (PDEs) appear in various contexts includi
ng control and optimization problems\, inverse pro
blems\, risk assessment\, and uncertainty quantifi
cation. In most such scenarios the set of all admi
ssible solutions associated with the parameter spa
ce is inherently low dimensional. This fact forms
the foundation for the so-called reduced basis met
hod.

Recently\, numerical experiments dem
onstrated the remarkable efficiency of using deep
neural networks to solve parametric problems. In t
his talk\, we will present a theoretical justifica
tion for this class of approaches. More precisely\
, we will derive upper bounds on the complexity of
ReLU neural networks approximating the solution m
aps of parametric PDEs. In fact\, without any know
ledge of its concrete shape\, we use the inherent
low-dimensionality of the solution manifold to obt
ain approximation rates which are significantly su
perior to those provided by classical approximatio
n results. We use this low-dimensionality to guara
ntee the existence of a reduced basis. Then\, for
a large variety of parametric PDEs\, we construct
neural networks that yield approximations of the p
arametric maps not suffering from a curse of dimen
sionality and essentially only depending on the si
ze of the reduced basis.

This is joint w
ork with Philipp Petersen (Oxford)\, Mones Raslan\
, and Reinhold Schneider.

LOCATION:Seminar Room 1\, Newton Institute
CONTACT:info@newton.ac.uk
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