---
_id: '46469'
abstract:
- lang: eng
text: 'We show how to learn discrete field theories from observational data of fields
on a space-time lattice. For this, we train a neural network model of a discrete
Lagrangian density such that the discrete Euler--Lagrange equations are consistent
with the given training data. We, thus, obtain a structure-preserving machine
learning architecture. Lagrangian densities are not uniquely defined by the solutions
of a field theory. We introduce a technique to derive regularisers for the training
process which optimise numerical regularity of the discrete field theory. Minimisation
of the regularisers guarantees that close to the training data the discrete field
theory behaves robust and efficient when used in numerical simulations. Further,
we show how to identify structurally simple solutions of the underlying continuous
field theory such as travelling waves. This is possible even when travelling waves
are not present in the training data. This is compared to data-driven model order
reduction based approaches, which struggle to identify suitable latent spaces
containing structurally simple solutions when these are not present in the training
data. Ideas are demonstrated on examples based on the wave equation and the Schrödinger
equation. '
article_number: '013104'
article_type: original
author:
- first_name: Christian
full_name: Offen, Christian
id: '85279'
last_name: Offen
orcid: 0000-0002-5940-8057
- first_name: Sina
full_name: Ober-Blöbaum, Sina
id: '16494'
last_name: Ober-Blöbaum
citation:
ama: Offen C, Ober-Blöbaum S. Learning of discrete models of variational PDEs from
data. Chaos. 2024;34(1). doi:10.1063/5.0172287
apa: Offen, C., & Ober-Blöbaum, S. (2024). Learning of discrete models of variational
PDEs from data. Chaos, 34(1), Article 013104. https://doi.org/10.1063/5.0172287
bibtex: '@article{Offen_Ober-Blöbaum_2024, title={Learning of discrete models of
variational PDEs from data}, volume={34}, DOI={10.1063/5.0172287},
number={1013104}, journal={Chaos}, publisher={AIP Publishing}, author={Offen,
Christian and Ober-Blöbaum, Sina}, year={2024} }'
chicago: Offen, Christian, and Sina Ober-Blöbaum. “Learning of Discrete Models of
Variational PDEs from Data.” Chaos 34, no. 1 (2024). https://doi.org/10.1063/5.0172287.
ieee: 'C. Offen and S. Ober-Blöbaum, “Learning of discrete models of variational
PDEs from data,” Chaos, vol. 34, no. 1, Art. no. 013104, 2024, doi: 10.1063/5.0172287.'
mla: Offen, Christian, and Sina Ober-Blöbaum. “Learning of Discrete Models of Variational
PDEs from Data.” Chaos, vol. 34, no. 1, 013104, AIP Publishing, 2024, doi:10.1063/5.0172287.
short: C. Offen, S. Ober-Blöbaum, Chaos 34 (2024).
date_created: 2023-08-10T08:24:48Z
date_updated: 2024-08-12T13:45:43Z
ddc:
- '510'
department:
- _id: '636'
doi: 10.1063/5.0172287
external_id:
arxiv:
- '2308.05082 '
file:
- access_level: open_access
content_type: application/pdf
creator: coffen
date_created: 2024-01-09T10:48:38Z
date_updated: 2024-01-09T10:48:38Z
file_id: '50376'
file_name: Accepted manuscript with AIP banner CHA23-AR-01370.pdf
file_size: 13222105
relation: main_file
title: Accepted Manuscript Chaos
- access_level: open_access
content_type: application/pdf
creator: coffen
date_created: 2024-01-09T11:19:49Z
date_updated: 2024-01-09T11:19:49Z
description: |-
We show how to learn discrete field theories from observational data of fields on a space-time lattice. For this, we train
a neural network model of a discrete Lagrangian density such that the discrete Euler–Lagrange equations are consistent
with the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrangian
densities are not uniquely defined by the solutions of a field theory. We introduce a technique to derive regularisers for
the training process which optimise numerical regularity of the discrete field theory. Minimisation of the regularisers
guarantees that close to the training data the discrete field theory behaves robust and efficient when used in numerical
simulations. Further, we show how to identify structurally simple solutions of the underlying continuous field theory
such as travelling waves. This is possible even when travelling waves are not present in the training data. This is
compared to data-driven model order reduction based approaches, which struggle to identify suitable latent spaces
containing structurally simple solutions when these are not present in the training data. Ideas are demonstrated on
examples based on the wave equation and the Schrödinger equation.
file_id: '50390'
file_name: LDensityPDE_AIP.pdf
file_size: 12960884
relation: main_file
title: Learning of discrete models of variational PDEs from data
file_date_updated: 2024-01-09T11:19:49Z
has_accepted_license: '1'
intvolume: ' 34'
issue: '1'
language:
- iso: eng
oa: '1'
project:
- _id: '52'
name: 'PC2: Computing Resources Provided by the Paderborn Center for Parallel Computing'
publication: Chaos
publication_identifier:
issn:
- 1054-1500
publication_status: published
publisher: AIP Publishing
quality_controlled: '1'
related_material:
link:
- description: GitHub
relation: software
url: https://github.com/Christian-Offen/DLNN_pde
status: public
title: Learning of discrete models of variational PDEs from data
type: journal_article
user_id: '85279'
volume: 34
year: '2024'
...
---
_id: '37654'
abstract:
- lang: eng
text: "Recently, Hamiltonian neural networks (HNN) have been introduced to incorporate
prior physical knowledge when\r\nlearning the dynamical equations of Hamiltonian
systems. Hereby, the symplectic system structure is preserved despite\r\nthe data-driven
modeling approach. However, preserving symmetries requires additional attention.
In this research, we\r\nenhance the HNN with a Lie algebra framework to detect
and embed symmetries in the neural network. This approach\r\nallows to simultaneously
learn the symmetry group action and the total energy of the system. As illustrating
examples,\r\na pendulum on a cart and a two-body problem from astrodynamics are
considered."
article_number: '063115'
article_type: original
author:
- first_name: Eva
full_name: Dierkes, Eva
last_name: Dierkes
- first_name: Christian
full_name: Offen, Christian
id: '85279'
last_name: Offen
orcid: 0000-0002-5940-8057
- first_name: Sina
full_name: Ober-Blöbaum, Sina
id: '16494'
last_name: Ober-Blöbaum
- first_name: Kathrin
full_name: Flaßkamp, Kathrin
last_name: Flaßkamp
citation:
ama: Dierkes E, Offen C, Ober-Blöbaum S, Flaßkamp K. Hamiltonian Neural Networks
with Automatic Symmetry Detection. Chaos. 2023;33(6). doi:10.1063/5.0142969
apa: Dierkes, E., Offen, C., Ober-Blöbaum, S., & Flaßkamp, K. (2023). Hamiltonian
Neural Networks with Automatic Symmetry Detection. Chaos, 33(6),
Article 063115. https://doi.org/10.1063/5.0142969
bibtex: '@article{Dierkes_Offen_Ober-Blöbaum_Flaßkamp_2023, title={Hamiltonian Neural
Networks with Automatic Symmetry Detection}, volume={33}, DOI={10.1063/5.0142969},
number={6063115}, journal={Chaos}, publisher={AIP Publishing}, author={Dierkes,
Eva and Offen, Christian and Ober-Blöbaum, Sina and Flaßkamp, Kathrin}, year={2023}
}'
chicago: Dierkes, Eva, Christian Offen, Sina Ober-Blöbaum, and Kathrin Flaßkamp.
“Hamiltonian Neural Networks with Automatic Symmetry Detection.” Chaos
33, no. 6 (2023). https://doi.org/10.1063/5.0142969.
ieee: 'E. Dierkes, C. Offen, S. Ober-Blöbaum, and K. Flaßkamp, “Hamiltonian Neural
Networks with Automatic Symmetry Detection,” Chaos, vol. 33, no. 6, Art.
no. 063115, 2023, doi: 10.1063/5.0142969.'
mla: Dierkes, Eva, et al. “Hamiltonian Neural Networks with Automatic Symmetry Detection.”
Chaos, vol. 33, no. 6, 063115, AIP Publishing, 2023, doi:10.1063/5.0142969.
short: E. Dierkes, C. Offen, S. Ober-Blöbaum, K. Flaßkamp, Chaos 33 (2023).
date_created: 2023-01-20T09:10:06Z
date_updated: 2023-08-10T08:37:01Z
ddc:
- '510'
department:
- _id: '636'
doi: 10.1063/5.0142969
external_id:
arxiv:
- '2301.07928'
file:
- access_level: open_access
content_type: application/pdf
creator: coffen
date_created: 2023-04-26T16:20:56Z
date_updated: 2023-04-26T16:20:56Z
description: |-
Incorporating physical system knowledge into data-driven
system identification has been shown to be beneficial. The
approach presented in this article combines learning of an
energy-conserving model from data with detecting a Lie
group representation of the unknown system symmetry.
The proposed approach can improve the learned model
and reveal underlying symmetry simultaneously.
file_id: '44205'
file_name: JournalPaper_main.pdf
file_size: 5200111
relation: main_file
title: Hamiltonian Neural Networks with Automatic Symmetry Detection
file_date_updated: 2023-04-26T16:20:56Z
has_accepted_license: '1'
intvolume: ' 33'
issue: '6'
language:
- iso: eng
oa: '1'
publication: Chaos
publication_identifier:
issn:
- 1054-1500
publication_status: published
publisher: AIP Publishing
related_material:
link:
- description: GitHub
relation: software
url: https://github.com/eva-dierkes/HNN_withSymmetries
status: public
title: Hamiltonian Neural Networks with Automatic Symmetry Detection
type: journal_article
user_id: '85279'
volume: 33
year: '2023'
...
---
_id: '16552'
author:
- first_name: Michael
full_name: Dellnitz, Michael
last_name: Dellnitz
- first_name: Andreas
full_name: Hohmann, Andreas
last_name: Hohmann
- first_name: Oliver
full_name: Junge, Oliver
last_name: Junge
- first_name: Martin
full_name: Rumpf, Martin
last_name: Rumpf
citation:
ama: 'Dellnitz M, Hohmann A, Junge O, Rumpf M. Exploring invariant sets and invariant
measures. Chaos: An Interdisciplinary Journal of Nonlinear Science. 1997:221-228.
doi:10.1063/1.166223'
apa: 'Dellnitz, M., Hohmann, A., Junge, O., & Rumpf, M. (1997). Exploring invariant
sets and invariant measures. Chaos: An Interdisciplinary Journal of Nonlinear
Science, 221–228. https://doi.org/10.1063/1.166223'
bibtex: '@article{Dellnitz_Hohmann_Junge_Rumpf_1997, title={Exploring invariant
sets and invariant measures}, DOI={10.1063/1.166223},
journal={Chaos: An Interdisciplinary Journal of Nonlinear Science}, author={Dellnitz,
Michael and Hohmann, Andreas and Junge, Oliver and Rumpf, Martin}, year={1997},
pages={221–228} }'
chicago: 'Dellnitz, Michael, Andreas Hohmann, Oliver Junge, and Martin Rumpf. “Exploring
Invariant Sets and Invariant Measures.” Chaos: An Interdisciplinary Journal
of Nonlinear Science, 1997, 221–28. https://doi.org/10.1063/1.166223.'
ieee: 'M. Dellnitz, A. Hohmann, O. Junge, and M. Rumpf, “Exploring invariant sets
and invariant measures,” Chaos: An Interdisciplinary Journal of Nonlinear Science,
pp. 221–228, 1997.'
mla: 'Dellnitz, Michael, et al. “Exploring Invariant Sets and Invariant Measures.”
Chaos: An Interdisciplinary Journal of Nonlinear Science, 1997, pp. 221–28,
doi:10.1063/1.166223.'
short: 'M. Dellnitz, A. Hohmann, O. Junge, M. Rumpf, Chaos: An Interdisciplinary
Journal of Nonlinear Science (1997) 221–228.'
date_created: 2020-04-15T09:06:28Z
date_updated: 2022-01-06T06:52:52Z
department:
- _id: '101'
doi: 10.1063/1.166223
language:
- iso: eng
page: 221-228
publication: 'Chaos: An Interdisciplinary Journal of Nonlinear Science'
publication_identifier:
issn:
- 1054-1500
- 1089-7682
publication_status: published
status: public
title: Exploring invariant sets and invariant measures
type: journal_article
user_id: '15701'
year: '1997'
...