A Short Note on Solving Partial Differential Equations Using Convolutional Neural Networks

Viktor Grimm; Alexander Heinlein; Axel Klawonn

doi:10.1007/978-3-031-50769-4_1

A Short Note on Solving Partial Differential Equations Using Convolutional Neural Networks

Viktor Grimm^*, Alexander Heinlein, Axel Klawonn

^*Corresponding author for this work

Numerical Analysis

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

Abstract

Solving partial differential equations (PDEs) is a common task in numerical mathematics and scientific computing. Typical discretization schemes, for example, finite element (FE), finite volume (FV), or finite difference (FD) methods, have the disadvantage that the computations have to be repeated once the boundary conditions (BCs) or the geometry change slightly; typical examples requiring the solution of many similar problems are time-dependent and inverse problems or uncertainty quantification.

Original language	English
Title of host publication	Domain Decomposition Methods in Science and Engineering XXVII
Editors	Zdenek Dostal, Tomas Kozubek, Axel Klawonn, Luca F. Pavarino, Olof B. Widlund, Ulrich Langer, Jakub Sístek
Publisher	Springer
Pages	3-14
Number of pages	12
ISBN (Print)	9783031507687
DOIs	https://doi.org/10.1007/978-3-031-50769-4_1
Publication status	Published - 2024
Event	27th International Conference on Domain Decomposition Methods in Science and Engineering, DD 2022 - Prague, Czech Republic Duration: 25 Jul 2022 → 29 Jul 2022

Publication series

Name	Lecture Notes in Computational Science and Engineering
Volume	149
ISSN (Print)	1439-7358
ISSN (Electronic)	2197-7100

Conference

Conference	27th International Conference on Domain Decomposition Methods in Science and Engineering, DD 2022
Country/Territory	Czech Republic
City	Prague
Period	25/07/22 → 29/07/22

Bibliographical note

Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care
Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

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10.1007/978-3-031-50769-4_1

Cite this

Grimm, V., Heinlein, A., & Klawonn, A. (2024). A Short Note on Solving Partial Differential Equations Using Convolutional Neural Networks. In Z. Dostal, T. Kozubek, A. Klawonn, L. F. Pavarino, O. B. Widlund, U. Langer, & J. Sístek (Eds.), Domain Decomposition Methods in Science and Engineering XXVII (pp. 3-14). (Lecture Notes in Computational Science and Engineering; Vol. 149). Springer. https://doi.org/10.1007/978-3-031-50769-4_1

Grimm, Viktor ; Heinlein, Alexander ; Klawonn, Axel. / A Short Note on Solving Partial Differential Equations Using Convolutional Neural Networks. Domain Decomposition Methods in Science and Engineering XXVII. editor / Zdenek Dostal ; Tomas Kozubek ; Axel Klawonn ; Luca F. Pavarino ; Olof B. Widlund ; Ulrich Langer ; Jakub Sístek. Springer, 2024. pp. 3-14 (Lecture Notes in Computational Science and Engineering).

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title = "A Short Note on Solving Partial Differential Equations Using Convolutional Neural Networks",

abstract = "Solving partial differential equations (PDEs) is a common task in numerical mathematics and scientific computing. Typical discretization schemes, for example, finite element (FE), finite volume (FV), or finite difference (FD) methods, have the disadvantage that the computations have to be repeated once the boundary conditions (BCs) or the geometry change slightly; typical examples requiring the solution of many similar problems are time-dependent and inverse problems or uncertainty quantification.",

author = "Viktor Grimm and Alexander Heinlein and Axel Klawonn",

note = "Green Open Access added to TU Delft Institutional Repository {\textquoteleft}You share, we take care!{\textquoteright} – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public. ; 27th International Conference on Domain Decomposition Methods in Science and Engineering, DD 2022 ; Conference date: 25-07-2022 Through 29-07-2022",

year = "2024",

doi = "10.1007/978-3-031-50769-4_1",

language = "English",

isbn = "9783031507687",

series = "Lecture Notes in Computational Science and Engineering",

publisher = "Springer",

pages = "3--14",

editor = "Zdenek Dostal and Tomas Kozubek and Axel Klawonn and Pavarino, {Luca F.} and Widlund, {Olof B.} and Ulrich Langer and Jakub S{\'i}stek",

booktitle = "Domain Decomposition Methods in Science and Engineering XXVII",

}

Grimm, V, Heinlein, A & Klawonn, A 2024, A Short Note on Solving Partial Differential Equations Using Convolutional Neural Networks. in Z Dostal, T Kozubek, A Klawonn, LF Pavarino, OB Widlund, U Langer & J Sístek (eds), Domain Decomposition Methods in Science and Engineering XXVII. Lecture Notes in Computational Science and Engineering, vol. 149, Springer, pp. 3-14, 27th International Conference on Domain Decomposition Methods in Science and Engineering, DD 2022, Prague, Czech Republic, 25/07/22. https://doi.org/10.1007/978-3-031-50769-4_1

A Short Note on Solving Partial Differential Equations Using Convolutional Neural Networks. / Grimm, Viktor; Heinlein, Alexander; Klawonn, Axel.
Domain Decomposition Methods in Science and Engineering XXVII. ed. / Zdenek Dostal; Tomas Kozubek; Axel Klawonn; Luca F. Pavarino; Olof B. Widlund; Ulrich Langer; Jakub Sístek. Springer, 2024. p. 3-14 (Lecture Notes in Computational Science and Engineering; Vol. 149).

Research output: Chapter in Book/Conference proceedings/Edited volume › Conference contribution › Scientific › peer-review

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AU - Heinlein, Alexander

AU - Klawonn, Axel

N1 - Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.

PY - 2024

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N2 - Solving partial differential equations (PDEs) is a common task in numerical mathematics and scientific computing. Typical discretization schemes, for example, finite element (FE), finite volume (FV), or finite difference (FD) methods, have the disadvantage that the computations have to be repeated once the boundary conditions (BCs) or the geometry change slightly; typical examples requiring the solution of many similar problems are time-dependent and inverse problems or uncertainty quantification.

AB - Solving partial differential equations (PDEs) is a common task in numerical mathematics and scientific computing. Typical discretization schemes, for example, finite element (FE), finite volume (FV), or finite difference (FD) methods, have the disadvantage that the computations have to be repeated once the boundary conditions (BCs) or the geometry change slightly; typical examples requiring the solution of many similar problems are time-dependent and inverse problems or uncertainty quantification.

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A2 - Dostal, Zdenek

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A2 - Klawonn, Axel

A2 - Pavarino, Luca F.

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A2 - Langer, Ulrich

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Grimm V, Heinlein A, Klawonn A. A Short Note on Solving Partial Differential Equations Using Convolutional Neural Networks. In Dostal Z, Kozubek T, Klawonn A, Pavarino LF, Widlund OB, Langer U, Sístek J, editors, Domain Decomposition Methods in Science and Engineering XXVII. Springer. 2024. p. 3-14. (Lecture Notes in Computational Science and Engineering). doi: 10.1007/978-3-031-50769-4_1

A Short Note on Solving Partial Differential Equations Using Convolutional Neural Networks

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