A theoretical analysis of deep neural networks and parametric PDEs

Author(s)
Gitta Kutyniok, Philipp Christian Petersen, Mones Raslan, Reinhold Schneider
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.

Organisation(s)
Department of Mathematics, Research Network Data Science
External organisation(s)
Ludwig-Maximilians-Universität München, University of Tromsø - The Arctic University of Norway, Technische Universität Berlin
Journal
Constructive Approximation
Volume
55
Pages
73-125
ISSN
0176-4276
DOI
https://doi.org/10.1007/s00365-021-09551-4
Publication date
06-2021
Peer reviewed
Yes
Austrian Fields of Science 2012
101031 Approximation theory
Keywords
ASJC Scopus subject areas
Computational Mathematics, Analysis, Mathematics(all)
Portal url
https://ucris.univie.ac.at/portal/en/publications/a-theoretical-analysis-of-deep-neural-networks-and-parametric-pdes(087c2187-2cfb-4213-9b0a-0ed8d43de6fd).html