---
_id: '55159'
abstract:
- lang: eng
  text: "We introduce a method based on Gaussian process regression to identify discrete
    variational principles from observed solutions of a field theory. The method is
    based on the data-based identification of a discrete Lagrangian density. It is
    a geometric machine learning technique in the sense that the variational structure
    of the true field theory is reflected in the data-driven model by design. We provide
    a rigorous convergence statement of the method. The proof circumvents challenges
    posed by the ambiguity of discrete Lagrangian densities in the inverse problem
    of variational calculus.\r\nMoreover, our method can be used to quantify model
    uncertainty in the equations of motions and any linear observable of the discrete
    field theory. This is illustrated on the example of the discrete wave equation
    and Schrödinger equation.\r\nThe article constitutes an extension of our previous
    article  arXiv:2404.19626 for the data-driven identification of (discrete) Lagrangians
    for variational dynamics from an ode setting to the setting of discrete pdes."
author:
- first_name: Christian
  full_name: Offen, Christian
  id: '85279'
  last_name: Offen
  orcid: 0000-0002-5940-8057
citation:
  ama: Offen C. Machine learning of discrete field theories with guaranteed convergence
    and uncertainty quantification.
  apa: Offen, C. (n.d.). <i>Machine learning of discrete field theories with guaranteed
    convergence and uncertainty quantification</i>.
  bibtex: '@article{Offen, title={Machine learning of discrete field theories with
    guaranteed convergence and uncertainty quantification}, author={Offen, Christian}
    }'
  chicago: Offen, Christian. “Machine Learning of Discrete Field Theories with Guaranteed
    Convergence and Uncertainty Quantification,” n.d.
  ieee: C. Offen, “Machine learning of discrete field theories with guaranteed convergence
    and uncertainty quantification.” .
  mla: Offen, Christian. <i>Machine Learning of Discrete Field Theories with Guaranteed
    Convergence and Uncertainty Quantification</i>.
  short: C. Offen, (n.d.).
date_created: 2024-07-10T13:43:50Z
date_updated: 2024-08-12T13:43:32Z
ddc:
- '510'
department:
- _id: '636'
external_id:
  arxiv:
  - '2407.07642'
file:
- access_level: open_access
  content_type: application/pdf
  creator: coffen
  date_created: 2024-07-10T13:39:32Z
  date_updated: 2024-07-10T13:39:32Z
  description: |-
    We introduce a method based on Gaussian process regression to identify discrete
    variational principles from observed solutions of a field theory. The method is based on the data-based identification of a discrete Lagrangian density. It is a geometric machine learning technique in the sense that the variational structure of the true field theory is reflected in the data-driven model by design.
    We provide a rigorous convergence statement of the method.
    The proof circumvents challenges posed by the ambiguity of discrete Lagrangian densities in the inverse problem of variational calculus.
    Moreover, our method can be used to quantify model uncertainty in the equations of motions and any linear observable of the discrete field theory.
    This is illustrated on the example of the discrete wave equation and Schrödinger equation.
    The article constitutes an extension of our previous article for the data-driven identification of (discrete) Lagrangians for variational dynamics from an ode setting to the setting of discrete pdes.
  file_id: '55160'
  file_name: L_Collocation.pdf
  file_size: 4569314
  relation: main_file
  title: Machine learning of discrete field theories with guaranteed convergence and
    uncertainty quantification
file_date_updated: 2024-07-10T13:39:32Z
has_accepted_license: '1'
keyword:
- System identification
- inverse problem of variational calculus
- Gaussian process
- Lagrangian learning
- physics informed machine learning
- geometry aware learning
language:
- iso: eng
oa: '1'
page: '28'
project:
- _id: '52'
  name: 'PC2: Computing Resources Provided by the Paderborn Center for Parallel Computing'
publication_status: submitted
related_material:
  link:
  - description: GitHub
    relation: software
    url: https://github.com/Christian-Offen/Lagrangian_GP_PDE
status: public
title: Machine learning of discrete field theories with guaranteed convergence and
  uncertainty quantification
type: preprint
user_id: '85279'
year: '2024'
...
---
_id: '42163'
abstract:
- lang: eng
  text: '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.'
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 discrete Lagrangians for variational PDEs
    from data and detection of travelling waves. In: Nielsen F, Barbaresco F, eds.
    <i>Geometric Science of Information</i>. Vol 14071. Lecture Notes in Computer
    Science (LNCS). Springer, Cham.; 2023:569-579. doi:<a href="https://doi.org/10.1007/978-3-031-38271-0_57">10.1007/978-3-031-38271-0_57</a>'
  apa: Offen, C., &#38; Ober-Blöbaum, S. (2023). Learning discrete Lagrangians for
    variational PDEs from data and detection of travelling waves. In F. Nielsen &#38;
    F. Barbaresco (Eds.), <i>Geometric Science of Information</i> (Vol. 14071, pp.
    569–579). Springer, Cham. <a href="https://doi.org/10.1007/978-3-031-38271-0_57">https://doi.org/10.1007/978-3-031-38271-0_57</a>
  bibtex: '@inproceedings{Offen_Ober-Blöbaum_2023, series={Lecture Notes in Computer
    Science (LNCS)}, title={Learning discrete Lagrangians for variational PDEs from
    data and detection of travelling waves}, volume={14071}, DOI={<a href="https://doi.org/10.1007/978-3-031-38271-0_57">10.1007/978-3-031-38271-0_57</a>},
    booktitle={Geometric Science of Information}, publisher={Springer, Cham.}, author={Offen,
    Christian and Ober-Blöbaum, Sina}, editor={Nielsen, F and Barbaresco, F}, year={2023},
    pages={569–579}, collection={Lecture Notes in Computer Science (LNCS)} }'
  chicago: Offen, Christian, and Sina Ober-Blöbaum. “Learning Discrete Lagrangians
    for Variational PDEs from Data and Detection of Travelling Waves.” In <i>Geometric
    Science of Information</i>, edited by F Nielsen and F Barbaresco, 14071:569–79.
    Lecture Notes in Computer Science (LNCS). Springer, Cham., 2023. <a href="https://doi.org/10.1007/978-3-031-38271-0_57">https://doi.org/10.1007/978-3-031-38271-0_57</a>.
  ieee: 'C. Offen and S. Ober-Blöbaum, “Learning discrete Lagrangians for variational
    PDEs from data and detection of travelling waves,” in <i>Geometric Science of
    Information</i>, Saint-Malo, Palais du Grand Large, France, 2023, vol. 14071,
    pp. 569–579, doi: <a href="https://doi.org/10.1007/978-3-031-38271-0_57">10.1007/978-3-031-38271-0_57</a>.'
  mla: Offen, Christian, and Sina Ober-Blöbaum. “Learning Discrete Lagrangians for
    Variational PDEs from Data and Detection of Travelling Waves.” <i>Geometric Science
    of Information</i>, edited by F Nielsen and F Barbaresco, vol. 14071, Springer,
    Cham., 2023, pp. 569–79, doi:<a href="https://doi.org/10.1007/978-3-031-38271-0_57">10.1007/978-3-031-38271-0_57</a>.
  short: 'C. Offen, S. Ober-Blöbaum, in: F. Nielsen, F. Barbaresco (Eds.), Geometric
    Science of Information, Springer, Cham., 2023, pp. 569–579.'
conference:
  end_date: 2023-09-01
  location: Saint-Malo, Palais du Grand Large, France
  name: '  GSI''23 6th International Conference on Geometric Science of Information'
  start_date: 2023-08-30
date_created: 2023-02-16T11:32:48Z
date_updated: 2024-08-12T13:46:29Z
ddc:
- '510'
department:
- _id: '636'
doi: 10.1007/978-3-031-38271-0_57
editor:
- first_name: F
  full_name: Nielsen, F
  last_name: Nielsen
- first_name: F
  full_name: Barbaresco, F
  last_name: Barbaresco
external_id:
  arxiv:
  - '2302.08232 '
file:
- access_level: open_access
  content_type: application/pdf
  creator: coffen
  date_created: 2023-08-02T12:04:17Z
  date_updated: 2023-08-02T12:04:17Z
  description: |-
    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.
  file_id: '46273'
  file_name: LDensityLearning.pdf
  file_size: 1938962
  relation: main_file
  title: Learning discrete Lagrangians for variational PDEs from data and detection
    of travelling waves
file_date_updated: 2023-08-02T12:04:17Z
has_accepted_license: '1'
intvolume: '     14071'
keyword:
- System identification
- discrete Lagrangians
- travelling waves
language:
- iso: eng
oa: '1'
page: 569-579
project:
- _id: '52'
  name: 'PC2: Computing Resources Provided by the Paderborn Center for Parallel Computing'
publication: Geometric Science of Information
publication_identifier:
  eisbn:
  - 978-3-031-38271-0
publication_status: published
publisher: Springer, Cham.
quality_controlled: '1'
related_material:
  link:
  - description: GitHub
    relation: software
    url: https://github.com/Christian-Offen/LagrangianDensityML
series_title: Lecture Notes in Computer Science (LNCS)
status: public
title: Learning discrete Lagrangians for variational PDEs from data and detection
  of travelling waves
type: conference
user_id: '85279'
volume: 14071
year: '2023'
...
---
_id: '26539'
abstract:
- lang: eng
  text: In control design most control strategies are model-based and require accurate
    models to be applied successfully. Due to simplifications and the model-reality-gap
    physics-derived models frequently exhibit deviations from real-world-systems.
    Likewise, purely data-driven methods often do not generalise well enough and may
    violate physical laws. Recently Physics-Guided Neural Networks (PGNN) and physics-inspired
    loss functions separately have shown promising results to conquer these drawbacks.
    In this contribution we extend existing methods towards the identification of
    non-autonomous systems and propose a combined approach PGNN-L, which uses a PGNN
    and a physics-inspired loss term (-L) to successfully identify the system's dynamics,
    while maintaining the consistency with physical laws. The proposed method is demonstrated
    on two real-world nonlinear systems and outperforms existing techniques regarding
    complexity and reliability.
author:
- first_name: Ricarda-Samantha
  full_name: Götte, Ricarda-Samantha
  id: '43992'
  last_name: Götte
- first_name: Julia
  full_name: Timmermann, Julia
  id: '15402'
  last_name: Timmermann
citation:
  ama: 'Götte R-S, Timmermann J. Composed Physics- and Data-driven System Identification
    for Non-autonomous Systems in Control Engineering. In: <i>2022 3rd International
    Conference on Artificial Intelligence, Robotics and Control (AIRC)</i>. ; 2022:67-76.
    doi:<a href="https://doi.org/10.1109/AIRC56195.2022.9836982">10.1109/AIRC56195.2022.9836982</a>'
  apa: Götte, R.-S., &#38; Timmermann, J. (2022). Composed Physics- and Data-driven
    System Identification for Non-autonomous Systems in Control Engineering. <i>2022
    3rd International Conference on Artificial Intelligence, Robotics and Control
    (AIRC)</i>, 67–76. <a href="https://doi.org/10.1109/AIRC56195.2022.9836982">https://doi.org/10.1109/AIRC56195.2022.9836982</a>
  bibtex: '@inproceedings{Götte_Timmermann_2022, title={Composed Physics- and Data-driven
    System Identification for Non-autonomous Systems in Control Engineering}, DOI={<a
    href="https://doi.org/10.1109/AIRC56195.2022.9836982">10.1109/AIRC56195.2022.9836982</a>},
    booktitle={2022 3rd International Conference on Artificial Intelligence, Robotics
    and Control (AIRC)}, author={Götte, Ricarda-Samantha and Timmermann, Julia}, year={2022},
    pages={67–76} }'
  chicago: Götte, Ricarda-Samantha, and Julia Timmermann. “Composed Physics- and Data-Driven
    System Identification for Non-Autonomous Systems in Control Engineering.” In <i>2022
    3rd International Conference on Artificial Intelligence, Robotics and Control
    (AIRC)</i>, 67–76, 2022. <a href="https://doi.org/10.1109/AIRC56195.2022.9836982">https://doi.org/10.1109/AIRC56195.2022.9836982</a>.
  ieee: 'R.-S. Götte and J. Timmermann, “Composed Physics- and Data-driven System
    Identification for Non-autonomous Systems in Control Engineering,” in <i>2022
    3rd International Conference on Artificial Intelligence, Robotics and Control
    (AIRC)</i>, Cairo, Egypt, 2022, pp. 67–76, doi: <a href="https://doi.org/10.1109/AIRC56195.2022.9836982">10.1109/AIRC56195.2022.9836982</a>.'
  mla: Götte, Ricarda-Samantha, and Julia Timmermann. “Composed Physics- and Data-Driven
    System Identification for Non-Autonomous Systems in Control Engineering.” <i>2022
    3rd International Conference on Artificial Intelligence, Robotics and Control
    (AIRC)</i>, 2022, pp. 67–76, doi:<a href="https://doi.org/10.1109/AIRC56195.2022.9836982">10.1109/AIRC56195.2022.9836982</a>.
  short: 'R.-S. Götte, J. Timmermann, in: 2022 3rd International Conference on Artificial
    Intelligence, Robotics and Control (AIRC), 2022, pp. 67–76.'
conference:
  end_date: 2021-12-10
  location: Cairo, Egypt
  name: 3rd International Conference on Artificial Intelligence, Robotics and Control
  start_date: 2021-12-08
date_created: 2021-10-19T14:47:17Z
date_updated: 2024-11-13T08:43:28Z
department:
- _id: '153'
- _id: '880'
doi: 10.1109/AIRC56195.2022.9836982
keyword:
- data-driven
- physics-based
- physics-informed
- neural networks
- system identification
- hybrid modelling
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2112.08148
oa: '1'
page: 67-76
publication: 2022 3rd International Conference on Artificial Intelligence, Robotics
  and Control (AIRC)
quality_controlled: '1'
status: public
title: Composed Physics- and Data-driven System Identification for Non-autonomous
  Systems in Control Engineering
type: conference
user_id: '43992'
year: '2022'
...
---
_id: '31066'
abstract:
- lang: eng
  text: 'While trade-offs between modeling effort and model accuracy remain a major
    concern with system identification, resorting to data-driven methods often leads
    to a complete disregard for physical plausibility. To address this issue, we propose
    a physics-guided hybrid approach for modeling non-autonomous systems under control.
    Starting from a traditional physics-based model, this is extended by a recurrent
    neural network and trained using a sophisticated multi-objective strategy yielding
    physically plausible models. While purely data-driven methods fail to produce
    satisfying results, experiments conducted on real data reveal substantial accuracy
    improvements by our approach compared to a physics-based model. '
author:
- first_name: Oliver
  full_name: Schön, Oliver
  last_name: Schön
- first_name: Ricarda-Samantha
  full_name: Götte, Ricarda-Samantha
  id: '43992'
  last_name: Götte
- first_name: Julia
  full_name: Timmermann, Julia
  id: '15402'
  last_name: Timmermann
citation:
  ama: 'Schön O, Götte R-S, Timmermann J. Multi-Objective Physics-Guided Recurrent
    Neural Networks for Identifying Non-Autonomous Dynamical Systems. In: <i>14th
    IFAC Workshop on Adaptive and Learning Control Systems (ALCOS 2022)</i>. Vol 55.
    ; 2022:19-24. doi:<a href="https://doi.org/10.1016/j.ifacol.2022.07.282">https://doi.org/10.1016/j.ifacol.2022.07.282</a>'
  apa: Schön, O., Götte, R.-S., &#38; Timmermann, J. (2022). Multi-Objective Physics-Guided
    Recurrent Neural Networks for Identifying Non-Autonomous Dynamical Systems. <i>14th
    IFAC Workshop on Adaptive and Learning Control Systems (ALCOS 2022)</i>, <i>55</i>(12),
    19–24. <a href="https://doi.org/10.1016/j.ifacol.2022.07.282">https://doi.org/10.1016/j.ifacol.2022.07.282</a>
  bibtex: '@inproceedings{Schön_Götte_Timmermann_2022, title={Multi-Objective Physics-Guided
    Recurrent Neural Networks for Identifying Non-Autonomous Dynamical Systems}, volume={55},
    DOI={<a href="https://doi.org/10.1016/j.ifacol.2022.07.282">https://doi.org/10.1016/j.ifacol.2022.07.282</a>},
    number={12}, booktitle={14th IFAC Workshop on Adaptive and Learning Control Systems
    (ALCOS 2022)}, author={Schön, Oliver and Götte, Ricarda-Samantha and Timmermann,
    Julia}, year={2022}, pages={19–24} }'
  chicago: Schön, Oliver, Ricarda-Samantha Götte, and Julia Timmermann. “Multi-Objective
    Physics-Guided Recurrent Neural Networks for Identifying Non-Autonomous Dynamical
    Systems.” In <i>14th IFAC Workshop on Adaptive and Learning Control Systems (ALCOS
    2022)</i>, 55:19–24, 2022. <a href="https://doi.org/10.1016/j.ifacol.2022.07.282">https://doi.org/10.1016/j.ifacol.2022.07.282</a>.
  ieee: 'O. Schön, R.-S. Götte, and J. Timmermann, “Multi-Objective Physics-Guided
    Recurrent Neural Networks for Identifying Non-Autonomous Dynamical Systems,” in
    <i>14th IFAC Workshop on Adaptive and Learning Control Systems (ALCOS 2022)</i>,
    Casablanca, Morocco, 2022, vol. 55, no. 12, pp. 19–24, doi: <a href="https://doi.org/10.1016/j.ifacol.2022.07.282">https://doi.org/10.1016/j.ifacol.2022.07.282</a>.'
  mla: Schön, Oliver, et al. “Multi-Objective Physics-Guided Recurrent Neural Networks
    for Identifying Non-Autonomous Dynamical Systems.” <i>14th IFAC Workshop on Adaptive
    and Learning Control Systems (ALCOS 2022)</i>, vol. 55, no. 12, 2022, pp. 19–24,
    doi:<a href="https://doi.org/10.1016/j.ifacol.2022.07.282">https://doi.org/10.1016/j.ifacol.2022.07.282</a>.
  short: 'O. Schön, R.-S. Götte, J. Timmermann, in: 14th IFAC Workshop on Adaptive
    and Learning Control Systems (ALCOS 2022), 2022, pp. 19–24.'
conference:
  end_date: 2022-07-01
  location: Casablanca, Morocco
  name: 14th IFAC Workshop on Adaptive and Learning Control Systems (ALCOS 2022)
  start_date: 2022-06-29
date_created: 2022-05-05T06:22:55Z
date_updated: 2024-11-13T08:43:16Z
department:
- _id: '153'
- _id: '880'
doi: https://doi.org/10.1016/j.ifacol.2022.07.282
intvolume: '        55'
issue: '12'
keyword:
- neural networks
- physics-guided
- data-driven
- multi-objective optimization
- system identification
- machine learning
- dynamical systems
language:
- iso: eng
page: 19-24
publication: 14th IFAC Workshop on Adaptive and Learning Control Systems (ALCOS 2022)
quality_controlled: '1'
status: public
title: Multi-Objective Physics-Guided Recurrent Neural Networks for Identifying Non-Autonomous
  Dynamical Systems
type: conference
user_id: '43992'
volume: 55
year: '2022'
...