Titel
A Theoretical Analysis of Deep Neural Networks and Parametric PDEs
Autor*in
Gitta Kutyniok
Mathematisches Institut, Universität München
Mones Raslan
Institut für Mathematik, Technische Universität Berlin
... show all
Abstract
We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent low dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical neural network approximation results. Concretely, we use the existence of a small reduced basis to construct, for a large variety of parametric partial differential equations, neural networks that yield approximations of the parametric solution maps in such a way that the sizes of these networks essentially only depend on the size of the reduced basis.
Stichwort
Deep neural networksParametric PDEsApproximation ratesReduced basis method
Objekt-Typ
Sprache
Englisch [eng]
Persistent identifier
phaidra.univie.ac.at/o:1536438
Erschienen in
Titel
Constructive Approximation
Band
55
ISSN
0176-4276
Erscheinungsdatum
2021
Seitenanfang
73
Seitenende
125
Publication
Springer Science and Business Media LLC
Erscheinungsdatum
2021
Zugänglichkeit
Rechte
Rechteangabe
© The Author(s) 2021
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