---
_id: '53793'
abstract:
- lang: eng
  text: We utilize extreme learning machines for the prediction of partial differential
    equations (PDEs). Our method splits the state space into multiple windows that
    are predicted individually using a single model. Despite requiring only few data
    points (in some cases, our method can learn from a single full-state snapshot),
    it still achieves high accuracy and can predict the flow of PDEs over long time
    horizons. Moreover, we show how additional symmetries can be exploited to increase
    sample efficiency and to enforce equivariance.
author:
- first_name: Hans
  full_name: Harder, Hans
  id: '98879'
  last_name: Harder
- first_name: Sebastian
  full_name: Peitz, Sebastian
  id: '47427'
  last_name: Peitz
  orcid: 0000-0002-3389-793X
citation:
  ama: Harder H, Peitz S. Predicting PDEs Fast and Efficiently with Equivariant Extreme
    Learning Machines.
  apa: Harder, H., &#38; Peitz, S. (n.d.). <i>Predicting PDEs Fast and Efficiently
    with Equivariant Extreme Learning Machines</i>.
  bibtex: '@article{Harder_Peitz, title={Predicting PDEs Fast and Efficiently with
    Equivariant Extreme Learning Machines}, author={Harder, Hans and Peitz, Sebastian}
    }'
  chicago: Harder, Hans, and Sebastian Peitz. “Predicting PDEs Fast and Efficiently
    with Equivariant Extreme Learning Machines,” n.d.
  ieee: H. Harder and S. Peitz, “Predicting PDEs Fast and Efficiently with Equivariant
    Extreme Learning Machines.” .
  mla: Harder, Hans, and Sebastian Peitz. <i>Predicting PDEs Fast and Efficiently
    with Equivariant Extreme Learning Machines</i>.
  short: H. Harder, S. Peitz, (n.d.).
date_created: 2024-04-30T08:43:14Z
date_updated: 2024-04-30T08:45:24Z
keyword:
- extreme learning machines
- partial differential equations
- data-driven prediction
- high-dimensional systems
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/2404.18530
oa: '1'
publication_status: unpublished
status: public
title: Predicting PDEs Fast and Efficiently with Equivariant Extreme Learning Machines
type: preprint
user_id: '98879'
year: '2024'
...
---
_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.
    <i>Journal of Geometric Mechanics</i>. 2023;15(1):98-115. doi:<a href="https://doi.org/10.3934/jgm.2023005">10.3934/jgm.2023005</a>
  apa: McLachlan, R., &#38; Offen, C. (2023). Backward error analysis for conjugate
    symplectic methods. <i>Journal of Geometric Mechanics</i>, <i>15</i>(1), 98–115.
    <a href="https://doi.org/10.3934/jgm.2023005">https://doi.org/10.3934/jgm.2023005</a>
  bibtex: '@article{McLachlan_Offen_2023, title={Backward error analysis for conjugate
    symplectic methods}, volume={15}, DOI={<a href="https://doi.org/10.3934/jgm.2023005">10.3934/jgm.2023005</a>},
    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.” <i>Journal of Geometric Mechanics</i> 15, no. 1 (2023): 98–115.
    <a href="https://doi.org/10.3934/jgm.2023005">https://doi.org/10.3934/jgm.2023005</a>.'
  ieee: 'R. McLachlan and C. Offen, “Backward error analysis for conjugate symplectic
    methods,” <i>Journal of Geometric Mechanics</i>, vol. 15, no. 1, pp. 98–115, 2023,
    doi: <a href="https://doi.org/10.3934/jgm.2023005">10.3934/jgm.2023005</a>.'
  mla: McLachlan, Robert, and Christian Offen. “Backward Error Analysis for Conjugate
    Symplectic Methods.” <i>Journal of Geometric Mechanics</i>, vol. 15, no. 1, AIMS
    Press, 2023, pp. 98–115, doi:<a href="https://doi.org/10.3934/jgm.2023005">10.3934/jgm.2023005</a>.
  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'
...
---
_id: '9568'
abstract:
- lang: eng
  text: A simple pre-stress estimate method of Langevin transducers is studied. The
    measurement setup consists of a capacitor, an impedance converter and a voltmeter.
    Based on the piezoelectric equation and the basic circuit theory, the mathematical
    expression between the pre-stress and the voltage across the capacitor is derived.
    The pre-stress level can then be calculated out of the measurement of the capacitor
    voltage. In order to evaluate the precision of this method, a force washer is
    used to measure the pre-stress of the Langevin transducer. The result shows the
    pre-stress level obtained from this method is 30-40\% higher than the pre-stress
    level measured by the force washer. This method is simple and can be used to estimate
    the pre-stress of various Langevin transducers. The precision of this method can
    be raised if d33 is identified under different pre-stress levels.
author:
- first_name: Fu
  full_name: Bo, Fu
  last_name: Bo
- first_name: Li
  full_name: Ting, Li
  last_name: Ting
- first_name: Tobias
  full_name: Hemsel, Tobias
  id: '210'
  last_name: Hemsel
citation:
  ama: 'Bo F, Ting L, Hemsel T. A simple pre-stress estimating method of langevin
    transducers. In: <i>Piezoelectricity, Acoustic Waves, and Device Applications,
    2008. SPAWDA 2008. Symposium On</i>. ; 2008:324-327. doi:<a href="https://doi.org/10.1109/SPAWDA.2008.4775801">10.1109/SPAWDA.2008.4775801</a>'
  apa: Bo, F., Ting, L., &#38; Hemsel, T. (2008). A simple pre-stress estimating method
    of langevin transducers. In <i>Piezoelectricity, Acoustic Waves, and Device Applications,
    2008. SPAWDA 2008. Symposium on</i> (pp. 324–327). <a href="https://doi.org/10.1109/SPAWDA.2008.4775801">https://doi.org/10.1109/SPAWDA.2008.4775801</a>
  bibtex: '@inproceedings{Bo_Ting_Hemsel_2008, title={A simple pre-stress estimating
    method of langevin transducers}, DOI={<a href="https://doi.org/10.1109/SPAWDA.2008.4775801">10.1109/SPAWDA.2008.4775801</a>},
    booktitle={Piezoelectricity, Acoustic Waves, and Device Applications, 2008. SPAWDA
    2008. Symposium on}, author={Bo, Fu and Ting, Li and Hemsel, Tobias}, year={2008},
    pages={324–327} }'
  chicago: Bo, Fu, Li Ting, and Tobias Hemsel. “A Simple Pre-Stress Estimating Method
    of Langevin Transducers.” In <i>Piezoelectricity, Acoustic Waves, and Device Applications,
    2008. SPAWDA 2008. Symposium On</i>, 324–27, 2008. <a href="https://doi.org/10.1109/SPAWDA.2008.4775801">https://doi.org/10.1109/SPAWDA.2008.4775801</a>.
  ieee: F. Bo, L. Ting, and T. Hemsel, “A simple pre-stress estimating method of langevin
    transducers,” in <i>Piezoelectricity, Acoustic Waves, and Device Applications,
    2008. SPAWDA 2008. Symposium on</i>, 2008, pp. 324–327.
  mla: Bo, Fu, et al. “A Simple Pre-Stress Estimating Method of Langevin Transducers.”
    <i>Piezoelectricity, Acoustic Waves, and Device Applications, 2008. SPAWDA 2008.
    Symposium On</i>, 2008, pp. 324–27, doi:<a href="https://doi.org/10.1109/SPAWDA.2008.4775801">10.1109/SPAWDA.2008.4775801</a>.
  short: 'F. Bo, L. Ting, T. Hemsel, in: Piezoelectricity, Acoustic Waves, and Device
    Applications, 2008. SPAWDA 2008. Symposium On, 2008, pp. 324–327.'
date_created: 2019-04-29T11:16:13Z
date_updated: 2022-01-06T07:04:16Z
department:
- _id: '151'
doi: 10.1109/SPAWDA.2008.4775801
keyword:
- capacitors
- impedance convertors
- piezoelectric transducers
- stress analysis
- Langevin transducers
- basic circuit theory
- capacitor
- impedance converter
- piezoelectric equation
- pre-stress estimating method
- voltmeter
- Capacitors
- Educational institutions
- Equations
- Force measurement
- Impedance measurement
- Manufacturing
- Mechatronics
- Piezoelectric transducers
- Voltage
- Voltmeters
- Langevin transducer
- capacitor
- piezoelectric element
- pre-stress
language:
- iso: eng
page: 324-327
publication: Piezoelectricity, Acoustic Waves, and Device Applications, 2008. SPAWDA
  2008. Symposium on
status: public
title: A simple pre-stress estimating method of langevin transducers
type: conference
user_id: '55222'
year: '2008'
...