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dc.contributor.author
Longo, Marcello
dc.contributor.author
Mishra, Siddhartha
dc.contributor.author
Rusch, T. Konstantin
dc.contributor.author
Schwab, Christoph
dc.date.accessioned
2022-01-14T17:49:34Z
dc.date.available
2021-12-23T14:33:02Z
dc.date.available
2022-01-14T17:49:34Z
dc.date.issued
2021
dc.identifier.issn
1064-8275
dc.identifier.issn
1095-7197
dc.identifier.other
10.1137/20m1369373
en_US
dc.identifier.uri
http://hdl.handle.net/20.500.11850/522204
dc.description.abstract
We present a novel algorithmic approach and an error analysis leveraging Quasi-Monte Carlo (QMC) points for training deep neural network (DNN) surrogates of holomorphic Data-to-Observable (DtO) maps in engineering design. Our analysis reveals higher-order consistent, deterministic choices of training points in the input parameter space for both deep and shallow neural networks with holomorphic activation functions such as $\tanh$. We prove that higher-order QMC training points facilitate higher-order decay (in terms of the number of training samples) of the underlying generalization error, with consistency error bounds that are free from the curse of dimensionality in terms of the number of input parameters, provided that DNN weights in hidden layers satisfy certain summability conditions. We present numerical experiments for DtO maps from elliptic and parabolic PDEs with uncertain inputs that confirm the theoretical analysis.
en_US
dc.language.iso
en
en_US
dc.publisher
SIAM
dc.subject
Higher-order quasi-Monte Carlo
en_US
dc.subject
machine learning
en_US
dc.subject
deep neural networks
en_US
dc.subject
parametric PDEs
en_US
dc.subject
high-dimensional approximation
en_US
dc.title
Higher-Order Quasi-Monte Carlo Training of Deep Neural Networks
en_US
dc.type
Journal Article
dc.date.published
2021-12-02
ethz.journal.title
SIAM Journal on Scientific Computing
ethz.journal.volume
43
en_US
ethz.journal.issue
6
en_US
ethz.journal.abbreviated
SIAM J. Sci. Comput.
ethz.pages.start
A3938
en_US
ethz.pages.end
A3966
en_US
ethz.grant
Computation and analysis of statistical solutions of fluid flow
en_US
ethz.identifier.wos
ethz.publication.place
Philadelphia, PA
ethz.publication.status
published
en_US
ethz.leitzahl
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics::03851 - Mishra, Siddhartha / Mishra, Siddhartha
en_US
ethz.leitzahl
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics::03435 - Schwab, Christoph / Schwab, Christoph
en_US
ethz.leitzahl.certified
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics::03851 - Mishra, Siddhartha / Mishra, Siddhartha
en_US
ethz.leitzahl.certified
ETH Zürich::00002 - ETH Zürich::00012 - Lehre und Forschung::00007 - Departemente::02000 - Dep. Mathematik / Dep. of Mathematics::02501 - Seminar für Angewandte Mathematik / Seminar for Applied Mathematics::03435 - Schwab, Christoph / Schwab, Christoph
en_US
ethz.grant.agreementno
770880
ethz.grant.fundername
EC
ethz.grant.funderDoi
10.13039/501100000780
ethz.grant.program
H2020
ethz.date.deposited
2021-12-23T14:33:07Z
ethz.source
FORM
ethz.eth
yes
en_US
ethz.availability
Metadata only
en_US
ethz.rosetta.installDate
2022-01-14T17:49:40Z
ethz.rosetta.lastUpdated
2024-02-02T16:00:22Z
ethz.rosetta.versionExported
true
ethz.COinS
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