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dc.contributor.authorGeist, Moritz
dc.contributor.authorPetersen, Philipp
dc.contributor.authorRaslan, Mones
dc.contributor.authorSchneider, Reinhold
dc.contributor.authorKutyniok, Gitta Astrid Hildegard
dc.date.accessioned2022-03-07T06:23:45Z
dc.date.available2022-03-07T06:23:45Z
dc.date.issued2021-06-05
dc.description.abstractWe perform a comprehensive numerical study of the effect of approximation-theoretical results for neural networks on practical learning problems in the context of numerical analysis. As the underlying model, we study the machine-learning-based solution of parametric partial differential equations. Here, approximation theory for fully-connected neural networks predicts that the performance of the model should depend only very mildly on the dimension of the parameter space and is determined by the intrinsic dimension of the solution manifold of the parametric partial differential equation. We use various methods to establish comparability between test-cases by minimizing the effect of the choice of test-cases on the optimization and sampling aspects of the learning problem. We find strong support for the hypothesis that approximation-theoretical effects heavily influence the practical behavior of learning problems in numerical analysis. Turning to practically more successful and modern architectures, at the end of this study we derive improved error bounds by focusing on convolutional neural networks.en_US
dc.identifier.citationGeist, Petersen, Raslan, Schneider, Kutyniok. Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks. Journal of Scientific Computing. 2021;88(1)en_US
dc.identifier.cristinIDFRIDAID 2004942
dc.identifier.doi10.1007/s10915-021-01532-w
dc.identifier.issn0885-7474
dc.identifier.issn1573-7691
dc.identifier.urihttps://hdl.handle.net/10037/24272
dc.language.isoengen_US
dc.publisherSpringeren_US
dc.relation.journalJournal of Scientific Computing
dc.rights.accessRightsopenAccessen_US
dc.rights.holderCopyright 2021 The Author(s)en_US
dc.titleNumerical Solution of the Parametric Diffusion Equation by Deep Neural Networksen_US
dc.type.versionpublishedVersionen_US
dc.typeJournal articleen_US
dc.typeTidsskriftartikkelen_US
dc.typePeer revieweden_US


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