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CATEGORIES:Data Science and Computational Statistics Seminar
SUMMARY:A priori generalization error analysis of neural n
etwork methods for solving high dimensional ellipt
ic PDEs - Yulong Lu (University of Massachusetts)
DTSTART:20210629T140000Z
DTEND:20210629T150000Z
UID:TALK4536AT
URL:/talk/index/4536
DESCRIPTION:Neural network-based machine learning methods\, in
cluding the most notably deep learning have achiev
ed extraordinary successes in numerous fields. Des
pite the rapid development of learning algorithms
based on neural networks\, their mathematical anal
ysis is far from understood. In particular\, it ha
s been a big mystery that neural network-based mac
hine learning methods work extremely well for solv
ing high dimensional problems.\n\nIn this talk\, w
e will demonstrate the power of neural network met
hods for solving high dimensional elliptic PDEs. S
pecifically\, we will discuss an a priori generali
zation error analysis of the Deep Ritz Method for
solving two classes of high dimensional Schrödinge
r problems: the stationary Schrödinger equation an
d the ground state of Schrödinger operator. Assum
ing the exact solution or the ground state lies in
a low-complexity function space called spectral B
arron space\, we show that the convergence rate of
the generalization error is independent of dimens
ion. We also develop a new regularity theory for t
he PDEs of consideration on the spectral Barron sp
ace. This can be viewed as an analog of the classi
cal Sobolev regularity theory for PDEs.
LOCATION:https://bham-ac-uk.zoom.us/j/3900445385?pwd=Mk1LSm
RiUk5STEJMWVl0bE82NTM1QT09
CONTACT:Hong Duong
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