---
_id: '21600'
abstract:
- lang: eng
text: Many problems in science and engineering require an efficient numerical approximation
of integrals or solutions to differential equations. For systems with rapidly
changing dynamics, an equidistant discretization is often inadvisable as it results
in prohibitively large errors or computational effort. To this end, adaptive schemes,
such as solvers based on Runge–Kutta pairs, have been developed which adapt the
step size based on local error estimations at each step. While the classical schemes
apply very generally and are highly efficient on regular systems, they can behave
suboptimally when an inefficient step rejection mechanism is triggered by structurally
complex systems such as chaotic systems. To overcome these issues, we propose
a method to tailor numerical schemes to the problem class at hand. This is achieved
by combining simple, classical quadrature rules or ODE solvers with data-driven
time-stepping controllers. Compared with learning solution operators to ODEs directly,
it generalizes better to unseen initial data as our approach employs classical
numerical schemes as base methods. At the same time it can make use of identified
structures of a problem class and, therefore, outperforms state-of-the-art adaptive
schemes. Several examples demonstrate superior efficiency. Source code is available
at https://github.com/lueckem/quadrature-ML.
author:
- first_name: Michael
full_name: Dellnitz, Michael
last_name: Dellnitz
- first_name: Eyke
full_name: Hüllermeier, Eyke
id: '48129'
last_name: Hüllermeier
- first_name: Marvin
full_name: Lücke, Marvin
last_name: Lücke
- 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
- first_name: Sebastian
full_name: Peitz, Sebastian
id: '47427'
last_name: Peitz
orcid: 0000-0002-3389-793X
- first_name: Karlson
full_name: Pfannschmidt, Karlson
id: '13472'
last_name: Pfannschmidt
orcid: 0000-0001-9407-7903
citation:
ama: Dellnitz M, Hüllermeier E, Lücke M, et al. Efficient time stepping for numerical
integration using reinforcement learning. SIAM Journal on Scientific Computing.
2023;45(2):A579-A595. doi:10.1137/21M1412682
apa: Dellnitz, M., Hüllermeier, E., Lücke, M., Ober-Blöbaum, S., Offen, C., Peitz,
S., & Pfannschmidt, K. (2023). Efficient time stepping for numerical integration
using reinforcement learning. SIAM Journal on Scientific Computing, 45(2),
A579–A595. https://doi.org/10.1137/21M1412682
bibtex: '@article{Dellnitz_Hüllermeier_Lücke_Ober-Blöbaum_Offen_Peitz_Pfannschmidt_2023,
title={Efficient time stepping for numerical integration using reinforcement
learning}, volume={45}, DOI={10.1137/21M1412682},
number={2}, journal={SIAM Journal on Scientific Computing}, author={Dellnitz,
Michael and Hüllermeier, Eyke and Lücke, Marvin and Ober-Blöbaum, Sina and Offen,
Christian and Peitz, Sebastian and Pfannschmidt, Karlson}, year={2023}, pages={A579–A595}
}'
chicago: 'Dellnitz, Michael, Eyke Hüllermeier, Marvin Lücke, Sina Ober-Blöbaum,
Christian Offen, Sebastian Peitz, and Karlson Pfannschmidt. “Efficient Time Stepping
for Numerical Integration Using Reinforcement Learning.” SIAM Journal on Scientific
Computing 45, no. 2 (2023): A579–95. https://doi.org/10.1137/21M1412682.'
ieee: 'M. Dellnitz et al., “Efficient time stepping for numerical integration
using reinforcement learning,” SIAM Journal on Scientific Computing, vol.
45, no. 2, pp. A579–A595, 2023, doi: 10.1137/21M1412682.'
mla: Dellnitz, Michael, et al. “Efficient Time Stepping for Numerical Integration
Using Reinforcement Learning.” SIAM Journal on Scientific Computing, vol.
45, no. 2, 2023, pp. A579–95, doi:10.1137/21M1412682.
short: M. Dellnitz, E. Hüllermeier, M. Lücke, S. Ober-Blöbaum, C. Offen, S. Peitz,
K. Pfannschmidt, SIAM Journal on Scientific Computing 45 (2023) A579–A595.
date_created: 2021-04-09T07:59:19Z
date_updated: 2023-08-25T09:24:50Z
ddc:
- '510'
department:
- _id: '101'
- _id: '636'
- _id: '355'
- _id: '655'
doi: 10.1137/21M1412682
external_id:
arxiv:
- arXiv:2104.03562
has_accepted_license: '1'
intvolume: ' 45'
issue: '2'
language:
- iso: eng
main_file_link:
- url: https://epubs.siam.org/doi/reader/10.1137/21M1412682
page: A579-A595
publication: SIAM Journal on Scientific Computing
publication_status: published
related_material:
link:
- description: GitHub
relation: software
url: https://github.com/lueckem/quadrature-ML
status: public
title: Efficient time stepping for numerical integration using reinforcement learning
type: journal_article
user_id: '47427'
volume: 45
year: '2023'
...
---
_id: '15111'
author:
- first_name: Karlson
full_name: Pfannschmidt, Karlson
id: '13472'
last_name: Pfannschmidt
orcid: 0000-0001-9407-7903
- first_name: Eyke
full_name: Hüllermeier, Eyke
id: '48129'
last_name: Hüllermeier
- first_name: S.
full_name: Held, S.
last_name: Held
- first_name: R.
full_name: Neiger, R.
last_name: Neiger
citation:
ama: 'Pfannschmidt K, Hüllermeier E, Held S, Neiger R. Evaluating tests in medical
diagnosis-Combining machine learning with game-theoretical concepts. In: In
Proceedings IPMU 16th International Conference on Information Processing and Management
of Uncertainty in Knowledge-Based Systems, Part 1, Eindhoven, The Netherlands.
Springer; 2016:450-461.'
apa: Pfannschmidt, K., Hüllermeier, E., Held, S., & Neiger, R. (2016). Evaluating
tests in medical diagnosis-Combining machine learning with game-theoretical concepts.
In In Proceedings IPMU 16th International Conference on Information Processing
and Management of Uncertainty in Knowledge-Based Systems, Part 1, Eindhoven,
The Netherlands (pp. 450–461). Springer.
bibtex: '@inproceedings{Pfannschmidt_Hüllermeier_Held_Neiger_2016, title={Evaluating
tests in medical diagnosis-Combining machine learning with game-theoretical concepts},
booktitle={In Proceedings IPMU 16th International Conference on Information Processing
and Management of Uncertainty in Knowledge-Based Systems, Part 1, Eindhoven,
The Netherlands}, publisher={Springer}, author={Pfannschmidt, Karlson and Hüllermeier,
Eyke and Held, S. and Neiger, R.}, year={2016}, pages={450–461} }'
chicago: Pfannschmidt, Karlson, Eyke Hüllermeier, S. Held, and R. Neiger. “Evaluating
Tests in Medical Diagnosis-Combining Machine Learning with Game-Theoretical Concepts.”
In In Proceedings IPMU 16th International Conference on Information Processing
and Management of Uncertainty in Knowledge-Based Systems, Part 1, Eindhoven,
The Netherlands, 450–61. Springer, 2016.
ieee: K. Pfannschmidt, E. Hüllermeier, S. Held, and R. Neiger, “Evaluating tests
in medical diagnosis-Combining machine learning with game-theoretical concepts,”
in In Proceedings IPMU 16th International Conference on Information Processing
and Management of Uncertainty in Knowledge-Based Systems, Part 1, Eindhoven,
The Netherlands, 2016, pp. 450–461.
mla: Pfannschmidt, Karlson, et al. “Evaluating Tests in Medical Diagnosis-Combining
Machine Learning with Game-Theoretical Concepts.” In Proceedings IPMU 16th
International Conference on Information Processing and Management of Uncertainty
in Knowledge-Based Systems, Part 1, Eindhoven, The Netherlands, Springer,
2016, pp. 450–61.
short: 'K. Pfannschmidt, E. Hüllermeier, S. Held, R. Neiger, in: In Proceedings
IPMU 16th International Conference on Information Processing and Management of
Uncertainty in Knowledge-Based Systems, Part 1, Eindhoven, The Netherlands, Springer,
2016, pp. 450–461.'
date_created: 2019-11-21T16:42:47Z
date_updated: 2022-01-06T06:52:15Z
department:
- _id: '34'
- _id: '7'
- _id: '355'
language:
- iso: eng
page: 450-461
publication: In Proceedings IPMU 16th International Conference on Information Processing
and Management of Uncertainty in Knowledge-Based Systems, Part 1, Eindhoven, The
Netherlands
publisher: Springer
status: public
title: Evaluating tests in medical diagnosis-Combining machine learning with game-theoretical
concepts
type: conference
user_id: '49109'
year: '2016'
...
---
_id: '10226'
author:
- first_name: Karlson
full_name: Pfannschmidt, Karlson
id: '13472'
last_name: Pfannschmidt
orcid: 0000-0001-9407-7903
- first_name: Eyke
full_name: Hüllermeier, Eyke
id: '48129'
last_name: Hüllermeier
- first_name: S.
full_name: Held, S.
last_name: Held
- first_name: R.
full_name: Neiger, R.
last_name: Neiger
citation:
ama: 'Pfannschmidt K, Hüllermeier E, Held S, Neiger R. Evaluating tests in medical
diagnosis-Combining machine learning with game-theoretical concepts. In: In
Proceedings IPMU 16th International Conference on Information Processing and Management
of Uncertainty in Knowledge-Based Systems, Part 1, Eindhoven, The Netherlands.
Springer; 2016:450-461.'
apa: Pfannschmidt, K., Hüllermeier, E., Held, S., & Neiger, R. (2016). Evaluating
tests in medical diagnosis-Combining machine learning with game-theoretical concepts.
In In Proceedings IPMU 16th International Conference on Information Processing
and Management of Uncertainty in Knowledge-Based Systems, Part 1, Eindhoven,
The Netherlands (pp. 450–461). Springer.
bibtex: '@inproceedings{Pfannschmidt_Hüllermeier_Held_Neiger_2016, title={Evaluating
tests in medical diagnosis-Combining machine learning with game-theoretical concepts},
booktitle={In Proceedings IPMU 16th International Conference on Information Processing
and Management of Uncertainty in Knowledge-Based Systems, Part 1, Eindhoven,
The Netherlands}, publisher={Springer}, author={Pfannschmidt, Karlson and Hüllermeier,
Eyke and Held, S. and Neiger, R.}, year={2016}, pages={450–461} }'
chicago: Pfannschmidt, Karlson, Eyke Hüllermeier, S. Held, and R. Neiger. “Evaluating
Tests in Medical Diagnosis-Combining Machine Learning with Game-Theoretical Concepts.”
In In Proceedings IPMU 16th International Conference on Information Processing
and Management of Uncertainty in Knowledge-Based Systems, Part 1, Eindhoven,
The Netherlands, 450–61. Springer, 2016.
ieee: K. Pfannschmidt, E. Hüllermeier, S. Held, and R. Neiger, “Evaluating tests
in medical diagnosis-Combining machine learning with game-theoretical concepts,”
in In Proceedings IPMU 16th International Conference on Information Processing
and Management of Uncertainty in Knowledge-Based Systems, Part 1, Eindhoven,
The Netherlands, 2016, pp. 450–461.
mla: Pfannschmidt, Karlson, et al. “Evaluating Tests in Medical Diagnosis-Combining
Machine Learning with Game-Theoretical Concepts.” In Proceedings IPMU 16th
International Conference on Information Processing and Management of Uncertainty
in Knowledge-Based Systems, Part 1, Eindhoven, The Netherlands, Springer,
2016, pp. 450–61.
short: 'K. Pfannschmidt, E. Hüllermeier, S. Held, R. Neiger, in: In Proceedings
IPMU 16th International Conference on Information Processing and Management of
Uncertainty in Knowledge-Based Systems, Part 1, Eindhoven, The Netherlands, Springer,
2016, pp. 450–461.'
date_created: 2019-06-11T15:11:54Z
date_updated: 2022-01-06T06:50:32Z
department:
- _id: '34'
- _id: '7'
- _id: '355'
language:
- iso: eng
page: 450-461
publication: In Proceedings IPMU 16th International Conference on Information Processing
and Management of Uncertainty in Knowledge-Based Systems, Part 1, Eindhoven, The
Netherlands
publisher: Springer
status: public
title: Evaluating tests in medical diagnosis-Combining machine learning with game-theoretical
concepts
type: conference
user_id: '49109'
year: '2016'
...