Universität zu Köln

Grimm, Viktor ORCID: 0000-0001-5300-3705, Heinlein, Alexander ORCID: 0000-0003-1578-8104 and Klawonn, Axel ORCID: 0000-0003-4765-7387 (2022). A short note on solving partial differential equations using convolutional neural networks. Technical Report.

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Abstract

The approach of using physics-based machine learning to solve PDEs has recently become very popular. A recent approach to solve PDEs based on CNNs uses finite difference stencils to include the residual of the partial differential equation into the loss function. In this work, the relation between the network training and the solution of a respective finite difference linear system of equations using classical numerical solvers is discussed. It turns out that many beneficial properties of the linear equation system are neglected in the network training. Finally, numerical results which underline the benefits of classical numerical solvers are presented.

Item Type:	Preprints, Working Papers or Reports (Technical Report)
Creators:	Creators Email ORCID ORCID Put Code Grimm, Viktor viktor.grimm@uni-koeln.de orcid.org/0000-0001-5300-3705 UNSPECIFIED Heinlein, Alexander a.heinlein@tudelft.nl orcid.org/0000-0003-1578-8104 UNSPECIFIED Klawonn, Axel axel.klawonn@uni-koeln.de orcid.org/0000-0003-4765-7387 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-642271
Series Name at the University of Cologne:	Technical report series. Center for Data and Simulation Science
Volume:	2022-07
Number of Pages:	9
Date:	29 November 2022
Language:	English
Faculty:	Central Institutions / Interdisciplinary Research Centers
Divisions:	Center for Data and Simulation Science
Subjects:	Natural sciences and mathematics Mathematics Technology (Applied sciences)
Uncontrolled Keywords:	Keywords Language iterative solvers English convolutional neural networks English physics-informed machine learning English
Refereed:	No
URI:	http://kups.ub.uni-koeln.de/id/eprint/64227