Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks

Author(s)
Moritz Geist, Philipp Christian Petersen, Mones Raslan, Reinhold Schneider, Gitta Kutyniok
Abstract

We 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.

Organisation(s)
Department of Mathematics, Research Network Data Science
External organisation(s)
Technische Universität Berlin, Ludwig-Maximilians-Universität München, University of Tromsø - The Arctic University of Norway
Journal
Journal of Scientific Computing
Volume
88
ISSN
0885-7474
DOI
https://doi.org/10.1007/s10915-021-01532-w
Publication date
2021
Peer reviewed
Yes
Austrian Fields of Science 2012
101014 Numerical mathematics
Keywords
ASJC Scopus subject areas
Software, Engineering(all), Computational Mathematics, Theoretical Computer Science, Applied Mathematics, Numerical Analysis, Computational Theory and Mathematics
Portal url
https://ucris.univie.ac.at/portal/en/publications/numerical-solution-of-the-parametric-diffusion-equation-by-deep-neural-networks(400c900a-3a84-4e7c-92a4-e4b8ac8d2984).html