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dc.contributor.author
Opschoor, Joost A. A.
dc.contributor.author
Petersen, Philipp C.
dc.contributor.author
Schwab, Christoph
dc.date.accessioned
2020-09-07T07:53:44Z
dc.date.available
2020-09-04T19:56:10Z
dc.date.available
2020-09-07T07:53:44Z
dc.date.issued
2020-09
dc.identifier.issn
0219-5305
dc.identifier.issn
1793-6861
dc.identifier.other
10.1142/S0219530519410136
en_US
dc.identifier.uri
http://hdl.handle.net/20.500.11850/438541
dc.description.abstract
Approximation rate bounds for emulations of real-valued functions on intervals by deep neural networks (DNNs) are established. The approximation results are given for DNNs based on ReLU activation functions. The approximation error is measured with respect to Sobolev norms. It is shown that ReLU DNNs allow for essentially the same approximation rates as nonlinear, variable-order, free-knot (or so-called "hp-adaptive") spline approximations and spectral approximations, for a wide range of Sobolev and Besov spaces. In particular, exponential convergence rates in terms of the DNN size for univariate, piecewise Gevrey functions with point singularities are established. Combined with recent results on ReLU DNN approximation of rational, oscillatory, and high-dimensional functions, this corroborates that continuous, piecewise affine ReLU DNNs afford algebraic and exponential convergence rate bounds which are comparable to "best in class" schemes for several important function classes of high and infinite smoothness. Using composition of DNNs, we also prove that radial-like functions obtained as compositions of the above with the Euclidean norm and, possibly, anisotropic affine changes of co-ordinates can be emulated at exponential rate in terms of the DNN size and depth without the curse of dimensionality. © 2020 World Scientific Publishing Company.
en_US
dc.language.iso
en
en_US
dc.publisher
World Scientific
en_US
dc.subject
Deep neural networks
en_US
dc.subject
Finite element methods
en_US
dc.subject
Exponential covergence
en_US
dc.subject
Gevrey regularity
en_US
dc.subject
Singularities
en_US
dc.title
Deep ReLU networks and high-order finite element methods
en_US
dc.type
Journal Article
dc.date.published
2020-02-21
ethz.journal.title
Analysis and Applications
ethz.journal.volume
18
en_US
ethz.journal.issue
5
en_US
ethz.journal.abbreviated
Anal. appl.
ethz.pages.start
715
en_US
ethz.pages.end
770
en_US
ethz.identifier.wos
ethz.identifier.scopus
ethz.publication.place
New Jersey, NY
en_US
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::03435 - Schwab, Christoph / Schwab, Christoph
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
ethz.date.deposited
2020-09-04T19:56:19Z
ethz.source
WOS
ethz.eth
yes
en_US
ethz.availability
Metadata only
en_US
ethz.rosetta.installDate
2020-09-07T07:53:55Z
ethz.rosetta.lastUpdated
2021-02-15T17:01:23Z
ethz.rosetta.versionExported
true
ethz.COinS
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