---
_id: '29240'
abstract:
- lang: eng
text: "The principle of least action is one of the most fundamental physical principle.
It says that among all possible motions connecting two points in a phase space,
the system will exhibit those motions which extremise an action functional. Many
qualitative features of dynamical systems, such as the presence of conservation
laws and energy balance equations, are related to the existence of an action functional.
Incorporating variational structure into learning algorithms for dynamical systems
is, therefore, crucial in order to make sure that the learned model shares important
features with the exact physical system. In this paper we show how to incorporate
variational principles into trajectory predictions of learned dynamical systems.
The novelty of this work is that (1) our technique relies only on discrete position
data of observed trajectories. Velocities or conjugate momenta do not need to
be observed or approximated and no prior knowledge about the form of the variational
principle is assumed. Instead, they are recovered using backward error analysis.
(2) Moreover, our technique compensates discretisation errors when trajectories
are computed from the learned system. This is important when moderate to large
step-sizes are used and high accuracy is required. For this,\r\nwe introduce and
rigorously analyse the concept of inverse modified Lagrangians by developing an
inverse version of variational backward error analysis. (3) Finally, we introduce
a method to perform system identification from position observations only, based
on variational backward error analysis."
article_type: original
author:
- first_name: Sina
full_name: Ober-Blöbaum, Sina
id: '16494'
last_name: Ober-Blöbaum
- first_name: Christian
full_name: Offen, Christian
id: '85279'
last_name: Offen
orcid: 0000-0002-5940-8057
citation:
ama: Ober-Blöbaum S, Offen C. Variational Learning of Euler–Lagrange Dynamics from
Data. Journal of Computational and Applied Mathematics. 2023;421:114780.
doi:10.1016/j.cam.2022.114780
apa: Ober-Blöbaum, S., & Offen, C. (2023). Variational Learning of Euler–Lagrange
Dynamics from Data. Journal of Computational and Applied Mathematics, 421,
114780. https://doi.org/10.1016/j.cam.2022.114780
bibtex: '@article{Ober-Blöbaum_Offen_2023, title={Variational Learning of Euler–Lagrange
Dynamics from Data}, volume={421}, DOI={10.1016/j.cam.2022.114780},
journal={Journal of Computational and Applied Mathematics}, publisher={Elsevier},
author={Ober-Blöbaum, Sina and Offen, Christian}, year={2023}, pages={114780}
}'
chicago: 'Ober-Blöbaum, Sina, and Christian Offen. “Variational Learning of Euler–Lagrange
Dynamics from Data.” Journal of Computational and Applied Mathematics 421
(2023): 114780. https://doi.org/10.1016/j.cam.2022.114780.'
ieee: 'S. Ober-Blöbaum and C. Offen, “Variational Learning of Euler–Lagrange Dynamics
from Data,” Journal of Computational and Applied Mathematics, vol. 421,
p. 114780, 2023, doi: 10.1016/j.cam.2022.114780.'
mla: Ober-Blöbaum, Sina, and Christian Offen. “Variational Learning of Euler–Lagrange
Dynamics from Data.” Journal of Computational and Applied Mathematics,
vol. 421, Elsevier, 2023, p. 114780, doi:10.1016/j.cam.2022.114780.
short: S. Ober-Blöbaum, C. Offen, Journal of Computational and Applied Mathematics
421 (2023) 114780.
date_created: 2022-01-11T13:24:00Z
date_updated: 2023-08-10T08:42:39Z
ddc:
- '510'
department:
- _id: '636'
doi: 10.1016/j.cam.2022.114780
external_id:
arxiv:
- '2112.12619'
file:
- access_level: open_access
content_type: application/pdf
creator: coffen
date_created: 2022-06-28T15:25:50Z
date_updated: 2022-06-28T15:25:50Z
description: |-
The principle of least action is one of the most fundamental physical principle. It says that among all possible motions
connecting two points in a phase space, the system will exhibit those motions which extremise an action functional.
Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equa-
tions, are related to the existence of an action functional. Incorporating variational structure into learning algorithms
for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features
with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predic-
tions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position
data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no
prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward
error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the
learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this,
we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of
variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position
observations only, based on variational backward error analysis.
file_id: '32274'
file_name: ShadowLagrangian_revision1_journal_style_arxiv.pdf
file_size: 3640770
relation: main_file
title: Variational Learning of Euler–Lagrange Dynamics from Data
file_date_updated: 2022-06-28T15:25:50Z
has_accepted_license: '1'
intvolume: ' 421'
keyword:
- Lagrangian learning
- variational backward error analysis
- modified Lagrangian
- variational integrators
- physics informed learning
language:
- iso: eng
oa: '1'
page: '114780'
publication: Journal of Computational and Applied Mathematics
publication_identifier:
issn:
- 0377-0427
publication_status: epub_ahead
publisher: Elsevier
quality_controlled: '1'
related_material:
link:
- relation: software
url: https://github.com/Christian-Offen/LagrangianShadowIntegration
status: public
title: Variational Learning of Euler–Lagrange Dynamics from Data
type: journal_article
user_id: '85279'
volume: 421
year: '2023'
...
---
_id: '29236'
abstract:
- lang: eng
text: The numerical solution of an ordinary differential equation can be interpreted
as the exact solution of a nearby modified equation. Investigating the behaviour
of numerical solutions by analysing the modified equation is known as backward
error analysis. If the original and modified equation share structural properties,
then the exact and approximate solution share geometric features such as the existence
of conserved quantities. Conjugate symplectic methods preserve a modified symplectic
form and a modified Hamiltonian when applied to a Hamiltonian system. We show
how a blended version of variational and symplectic techniques can be used to
compute modified symplectic and Hamiltonian structures. In contrast to other approaches,
our backward error analysis method does not rely on an ansatz but computes the
structures systematically, provided that a variational formulation of the method
is known. The technique is illustrated on the example of symmetric linear multistep
methods with matrix coefficients.
article_type: original
author:
- first_name: Robert
full_name: McLachlan, Robert
last_name: McLachlan
- first_name: Christian
full_name: Offen, Christian
id: '85279'
last_name: Offen
orcid: 0000-0002-5940-8057
citation:
ama: McLachlan R, Offen C. Backward error analysis for conjugate symplectic methods.
Journal of Geometric Mechanics. 2023;15(1):98-115. doi:10.3934/jgm.2023005
apa: McLachlan, R., & Offen, C. (2023). Backward error analysis for conjugate
symplectic methods. Journal of Geometric Mechanics, 15(1), 98–115.
https://doi.org/10.3934/jgm.2023005
bibtex: '@article{McLachlan_Offen_2023, title={Backward error analysis for conjugate
symplectic methods}, volume={15}, DOI={10.3934/jgm.2023005},
number={1}, journal={Journal of Geometric Mechanics}, publisher={AIMS Press},
author={McLachlan, Robert and Offen, Christian}, year={2023}, pages={98–115} }'
chicago: 'McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate
Symplectic Methods.” Journal of Geometric Mechanics 15, no. 1 (2023): 98–115.
https://doi.org/10.3934/jgm.2023005.'
ieee: 'R. McLachlan and C. Offen, “Backward error analysis for conjugate symplectic
methods,” Journal of Geometric Mechanics, vol. 15, no. 1, pp. 98–115, 2023,
doi: 10.3934/jgm.2023005.'
mla: McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate
Symplectic Methods.” Journal of Geometric Mechanics, vol. 15, no. 1, AIMS
Press, 2023, pp. 98–115, doi:10.3934/jgm.2023005.
short: R. McLachlan, C. Offen, Journal of Geometric Mechanics 15 (2023) 98–115.
date_created: 2022-01-11T12:48:39Z
date_updated: 2023-08-10T08:40:30Z
ddc:
- '510'
department:
- _id: '636'
doi: 10.3934/jgm.2023005
external_id:
arxiv:
- '2201.03911'
file:
- access_level: open_access
content_type: application/pdf
creator: coffen
date_created: 2022-08-12T16:48:59Z
date_updated: 2022-08-12T16:48:59Z
description: The numerical solution of an ordinary differential equation can be
interpreted as the exact solution of a nearby modified equation. Investigating
the behaviour of numerical solutions by analysing the modified equation is known
as backward error analysis. If the original and modified equation share structural
properties, then the exact and approximate solution share geometric features such
as the existence of conserved quantities. Conjugate symplectic methods preserve
a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian
system. We show how a blended version of variational and symplectic techniques
can be used to compute modified symplectic and Hamiltonian structures. In contrast
to other approaches, our backward error analysis method does not rely on an ansatz
but computes the structures systematically, provided that a variational formulation
of the method is known. The technique is illustrated on the example of symmetric
linear multistep methods with matrix coefficients.
file_id: '32801'
file_name: BEA_MultiStep_Matrix.pdf
file_size: 827030
relation: main_file
title: Backward error analysis for conjugate symplectic methods
file_date_updated: 2022-08-12T16:48:59Z
has_accepted_license: '1'
intvolume: ' 15'
issue: '1'
keyword:
- variational integrators
- backward error analysis
- Euler--Lagrange equations
- multistep methods
- conjugate symplectic methods
language:
- iso: eng
oa: '1'
page: 98-115
publication: Journal of Geometric Mechanics
publication_status: published
publisher: AIMS Press
quality_controlled: '1'
related_material:
link:
- relation: software
url: https://github.com/Christian-Offen/BEAConjugateSymplectic
status: public
title: Backward error analysis for conjugate symplectic methods
type: journal_article
user_id: '85279'
volume: 15
year: '2023'
...