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