Learning Discrete Lagrangians for Variational PDEs from Data and Detection of Travelling Waves

Christian Offen; Sina Ober-Blöbaum

doi:10.1007/978-3-031-38271-0_57

Learning Discrete Lagrangians for Variational PDEs from Data and Detection of Travelling Waves

Christian Offen^*, Sina Ober-Blöbaum

^*Corresponding author for this work

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density L_d that is modelled as a neural network. Careful regularisation of the loss function for training L_d is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler–Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE.

Original language	English
Title of host publication	Geometric Science of Information
Subtitle of host publication	6th International Conference, GSI 2023, St. Malo, France, August 30 – September 1, 2023, Proceedings, Part I
Editors	Frank Nielsen, Frédéric Barbaresco
Publisher	Springer
Pages	569-579
Number of pages	11
Edition	1
ISBN (Electronic)	9783031382710
ISBN (Print)	9783031382703
DOIs	https://doi.org/10.1007/978-3-031-38271-0_57
Publication status	Published - 1 Aug 2023
Event	The 6th International Conference on Geometric Science of Information, GSI 2023 - St. Malo, France Duration: 30 Aug 2023 → 1 Sept 2023

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	14071
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	The 6th International Conference on Geometric Science of Information, GSI 2023
Country/Territory	France
City	St. Malo
Period	30/08/23 → 1/09/23

Bibliographical note

Publisher Copyright:
© 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.

Keywords

discrete Lagrangians
System identification
travelling waves

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/978-3-031-38271-0_57Licence: None: All rights reserved

Cite this

Offen, C., & Ober-Blöbaum, S. (2023). Learning Discrete Lagrangians for Variational PDEs from Data and Detection of Travelling Waves. In F. Nielsen, & F. Barbaresco (Eds.), Geometric Science of Information: 6th International Conference, GSI 2023, St. Malo, France, August 30 – September 1, 2023, Proceedings, Part I (1 ed., pp. 569-579). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 14071). Springer. https://doi.org/10.1007/978-3-031-38271-0_57

Offen, Christian ; Ober-Blöbaum, Sina. / Learning Discrete Lagrangians for Variational PDEs from Data and Detection of Travelling Waves. Geometric Science of Information: 6th International Conference, GSI 2023, St. Malo, France, August 30 – September 1, 2023, Proceedings, Part I. editor / Frank Nielsen ; Frédéric Barbaresco. 1. ed. Springer, 2023. pp. 569-579 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

@inproceedings{7fde9ed352da47cc82ac52a66fc41533,

title = "Learning Discrete Lagrangians for Variational PDEs from Data and Detection of Travelling Waves",

abstract = "The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density Ld that is modelled as a neural network. Careful regularisation of the loss function for training Ld is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler–Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE.",

keywords = "discrete Lagrangians, System identification, travelling waves",

author = "Christian Offen and Sina Ober-Bl{\"o}baum",

note = "Publisher Copyright: {\textcopyright} 2023, The Author(s), under exclusive license to Springer Nature Switzerland AG.; The 6th International Conference on Geometric Science of Information, GSI 2023 ; Conference date: 30-08-2023 Through 01-09-2023",

year = "2023",

month = aug,

day = "1",

doi = "10.1007/978-3-031-38271-0\_57",

language = "English",

isbn = "9783031382703",

series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

publisher = "Springer",

pages = "569--579",

editor = "Frank Nielsen and Fr{\'e}d{\'e}ric Barbaresco",

booktitle = "Geometric Science of Information",

edition = "1",

}

Offen, C & Ober-Blöbaum, S 2023, Learning Discrete Lagrangians for Variational PDEs from Data and Detection of Travelling Waves. in F Nielsen & F Barbaresco (eds), Geometric Science of Information: 6th International Conference, GSI 2023, St. Malo, France, August 30 – September 1, 2023, Proceedings, Part I. 1 edn, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 14071, Springer, pp. 569-579, The 6th International Conference on Geometric Science of Information, GSI 2023, St. Malo, France, 30/08/23. https://doi.org/10.1007/978-3-031-38271-0_57

Learning Discrete Lagrangians for Variational PDEs from Data and Detection of Travelling Waves. / Offen, Christian; Ober-Blöbaum, Sina.
Geometric Science of Information: 6th International Conference, GSI 2023, St. Malo, France, August 30 – September 1, 2023, Proceedings, Part I. ed. / Frank Nielsen; Frédéric Barbaresco. 1. ed. Springer, 2023. p. 569-579 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 14071).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - Learning Discrete Lagrangians for Variational PDEs from Data and Detection of Travelling Waves

AU - Offen, Christian

AU - Ober-Blöbaum, Sina

PY - 2023/8/1

Y1 - 2023/8/1

N2 - The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density Ld that is modelled as a neural network. Careful regularisation of the loss function for training Ld is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler–Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE.

AB - The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density Ld that is modelled as a neural network. Careful regularisation of the loss function for training Ld is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler–Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE.

KW - discrete Lagrangians

KW - System identification

KW - travelling waves

UR - http://www.scopus.com/inward/record.url?scp=85168819950&partnerID=8YFLogxK

U2 - 10.1007/978-3-031-38271-0_57

DO - 10.1007/978-3-031-38271-0_57

M3 - Conference contribution

AN - SCOPUS:85168819950

SN - 9783031382703

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 569

EP - 579

BT - Geometric Science of Information

A2 - Nielsen, Frank

A2 - Barbaresco, Frédéric

PB - Springer

T2 - The 6th International Conference on Geometric Science of Information, GSI 2023

Y2 - 30 August 2023 through 1 September 2023

ER -

Offen C, Ober-Blöbaum S. Learning Discrete Lagrangians for Variational PDEs from Data and Detection of Travelling Waves. In Nielsen F, Barbaresco F, editors, Geometric Science of Information: 6th International Conference, GSI 2023, St. Malo, France, August 30 – September 1, 2023, Proceedings, Part I. 1 ed. Springer. 2023. p. 569-579. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-031-38271-0_57