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Learning to Differentiate
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.
Department of Mechanical Engineering and Institute for Computational Mathematical Engineering, Stanford University, Stanford, California, USA.
Linköping University, Department of Mathematics, Computational Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-7972-6183
2020 (English)Report (Other academic)
Abstract [en]

Artificial neural networks together with associated computational libraries provide a powerful framework for constructing both classification and regression algorithms. In this paper we use neural networks to design linear and non-linear discrete differential operators. We show that neural network based operators can be used to construct stable discretizations of initial boundary-value problems by ensuring that the operators satisfy a discrete analogue of integration-byparts known as summation-by-parts. Furthermore we demonstrate the benefits of building the summation-by-parts property into the network by weight restriction, rather than enforcing it through a regularizer. We conclude that, if possible, known structural elements of an operation are best implemented as innate—rather than learned—properties of the network. The strategy developed in this work also opens the door for constructing stable differential operators on general meshes.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2020. , p. 27
Series
LiTH-MAT-R, ISSN 0348-2960 ; 2020:1
Keywords [en]
Neural network, discrete differential operators, stability, initial boundary value problem, summation-by-parts, regularization, weight restriction, general mesh
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-163120ISRN: LiTH-MAT-R--2020/01--SEOAI: oai:DiVA.org:liu-163120DiVA, id: diva2:1385295
Available from: 2020-01-14 Created: 2020-01-14 Last updated: 2020-01-14Bibliographically approved

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Learning to Differentiate(808 kB)33 downloads
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Ålund, OskarNordström, Jan

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CiteExportLink to record
Permanent link

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Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • oxford
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf