---
_id: '12815'
abstract:
- lang: eng
text: Active learning of physical systems must commonly respect practical safety
constraints, which restricts the exploration of the design space. Gaussian Processes
(GPs) and their calibrated uncertainty estimations are widely used for this purpose.
In many technical applications the design space is explored via continuous trajectories,
along which the safety needs to be assessed. This is particularly challenging
for strict safety requirements in GP methods, as it employs computationally expensive
Monte-Carlo sampling of high quantiles. We address these challenges by providing
provable safety bounds based on the adaptively sampled median of the supremum
of the posterior GP. Our method significantly reduces the number of samples required
for estimating high safety probabilities, resulting in faster evaluation without
sacrificing accuracy and exploration speed. The effectiveness of our safe active
learning approach is demonstrated through extensive simulations and validated
using a real-world engine example.
author:
- first_name: Jörn
full_name: Tebbe, Jörn
id: '85958'
last_name: Tebbe
- first_name: Christoph
full_name: Zimmer, Christoph
last_name: Zimmer
- first_name: Ansgar
full_name: Steland, Ansgar
last_name: Steland
- first_name: Markus
full_name: Lange-Hegermann, Markus
id: '71761'
last_name: Lange-Hegermann
- first_name: Fabian
full_name: Mies, Fabian
last_name: Mies
citation:
ama: Tebbe J, Zimmer C, Steland A, Lange-Hegermann M, Mies F. Efficiently Computable
Safety Bounds for Gaussian Processes in Active Learning. MLResearchPress ;
2024:1333-1341.
apa: Tebbe, J., Zimmer, C., Steland, A., Lange-Hegermann, M., & Mies, F. (2024).
Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning.
In International Conference on Artificial Intelligence and Statistics (AISTATS),
Vol. 238 (pp. 1333–1341). MLResearchPress .
bjps: Tebbe J et al. (2024) Efficiently Computable Safety Bounds
for Gaussian Processes in Active Learning. MLResearchPress .
chicago: Tebbe, Jörn, Christoph Zimmer, Ansgar Steland, Markus Lange-Hegermann,
and Fabian Mies. Efficiently Computable Safety Bounds for Gaussian Processes
in Active Learning. International Conference on Artificial Intelligence
and Statistics (AISTATS), Vol. 238. Proceedings of Machine Learning Research.
MLResearchPress , 2024.
chicago-de: Tebbe, Jörn, Christoph Zimmer, Ansgar Steland, Markus Lange-Hegermann
und Fabian Mies. 2024. Efficiently Computable Safety Bounds for Gaussian Processes
in Active Learning. International Conference on Artificial Intelligence
and Statistics (AISTATS), Vol. 238. Proceedings of Machine Learning Research.
MLResearchPress .
din1505-2-1: 'Tebbe, Jörn ; Zimmer, Christoph ; Steland,
Ansgar ; Lange-Hegermann, Markus
; Mies, Fabian: Efficiently Computable
Safety Bounds for Gaussian Processes in Active Learning, Proceedings of
Machine Learning Research : MLResearchPress , 2024'
havard: J. Tebbe, C. Zimmer, A. Steland, M. Lange-Hegermann, F. Mies, Efficiently
Computable Safety Bounds for Gaussian Processes in Active Learning, MLResearchPress
, 2024.
ieee: J. Tebbe, C. Zimmer, A. Steland, M. Lange-Hegermann, and F. Mies, Efficiently
Computable Safety Bounds for Gaussian Processes in Active Learning. MLResearchPress
, 2024, pp. 1333–1341.
mla: Tebbe, Jörn, et al. “Efficiently Computable Safety Bounds for Gaussian Processes
in Active Learning.” International Conference on Artificial Intelligence and
Statistics (AISTATS), Vol. 238, MLResearchPress , 2024, pp. 1333–41.
short: J. Tebbe, C. Zimmer, A. Steland, M. Lange-Hegermann, F. Mies, Efficiently
Computable Safety Bounds for Gaussian Processes in Active Learning, MLResearchPress
, 2024.
ufg: 'Tebbe, Jörn u. a.: Efficiently Computable Safety Bounds for Gaussian
Processes in Active Learning, o. O. 2024 (Proceedings of Machine Learning Research).'
van: Tebbe J, Zimmer C, Steland A, Lange-Hegermann M, Mies F. Efficiently Computable
Safety Bounds for Gaussian Processes in Active Learning. International Conference
on Artificial Intelligence and Statistics (AISTATS), Vol. 238. MLResearchPress
; 2024. (Proceedings of Machine Learning Research).
conference:
location: Valencia, SPAIN
name: 27th International Conference on Artificial Intelligence and Statistics (AISTATS)
start_date: 2024-05-02
date_created: 2025-04-17T07:58:19Z
date_updated: 2025-06-25T12:47:19Z
department:
- _id: DEP5000
- _id: DEP5023
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://proceedings.mlr.press/v238/tebbe24a.html
oa: '1'
page: 1333-1341
publication: International Conference on Artificial Intelligence and Statistics (AISTATS),
Vol. 238
publication_identifier:
issn:
- 2640-3498
publication_status: published
publisher: 'MLResearchPress '
series_title: Proceedings of Machine Learning Research
status: public
title: Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning
type: conference_editor_article
user_id: '83781'
year: '2024'
...
---
_id: '12828'
abstract:
- lang: eng
text: Partial differential equations (PDEs) are important tools to model physical
systems and including them into machine learning models is an important way of
incorporating physical knowledge. Given any system of linear PDEs with constant
coefficients, we propose a family of Gaussian process (GP) priors, which we call
EPGP, such that all realizations are exact solutions of this system. We apply
the Ehrenpreis-Palamodov fundamental principle, which works as a non-linear Fourier
transform, to construct GP kernels mirroring standard spectral methods for GPs.
Our approach can infer probable solutions of linear PDE systems from any data
such as noisy measurements, or pointwise defined initial and boundary conditions.
Constructing EPGP-priors is algorithmic, generally applicable, and comes with
a sparse version (S-EPGP) that learns the relevant spectral frequencies and works
better for big data sets. We demonstrate our approach on three families of systems
of PDEs, the heat equation, wave equation, and Maxwell's equations, where we improve
upon the state of the art in computation time and precision, in some experiments
by several orders of magnitude.
author:
- first_name: 'Marc '
full_name: 'Härkönen, Marc '
last_name: Härkönen
- first_name: Markus
full_name: Lange-Hegermann, Markus
id: '71761'
last_name: Lange-Hegermann
- first_name: Bogdan
full_name: ' Raiţă, Bogdan'
last_name: ' Raiţă'
citation:
ama: Härkönen M, Lange-Hegermann M, Raiţă B. Gaussian Process Priors for Systems
of Linear Partial Differential Equations with Constant Coefficients. Vol 202.
MLResearchPress ; 2023.
apa: Härkönen, M., Lange-Hegermann, M., & Raiţă, B. (2023). Gaussian Process
Priors for Systems of Linear Partial Differential Equations with Constant Coefficients.
In 40th International Conference on Machine Learning (Vol. 202). MLResearchPress
.
bjps: Härkönen M, Lange-Hegermann M and Raiţă B (2023) Gaussian Process
Priors for Systems of Linear Partial Differential Equations with Constant Coefficients.
MLResearchPress .
chicago: 'Härkönen, Marc , Markus Lange-Hegermann, and Bogdan Raiţă. Gaussian
Process Priors for Systems of Linear Partial Differential Equations with Constant
Coefficients. 40th International Conference on Machine Learning. Vol.
202. Proceedings of Machine Learning Research : PMLR. MLResearchPress , 2023.'
chicago-de: 'Härkönen, Marc , Markus Lange-Hegermann und Bogdan Raiţă. 2023. Gaussian
Process Priors for Systems of Linear Partial Differential Equations with Constant
Coefficients. 40th International Conference on Machine Learning. Bd.
202. Proceedings of machine learning research : PMLR. MLResearchPress .'
din1505-2-1: 'Härkönen, Marc ; Lange-Hegermann, Markus ;
Raiţă, Bogdan: Gaussian Process Priors for Systems of Linear Partial
Differential Equations with Constant Coefficients, Proceedings of machine
learning research : PMLR. Bd. 202 : MLResearchPress , 2023'
havard: M. Härkönen, M. Lange-Hegermann, B. Raiţă, Gaussian Process Priors for
Systems of Linear Partial Differential Equations with Constant Coefficients, MLResearchPress
, 2023.
ieee: M. Härkönen, M. Lange-Hegermann, and B. Raiţă, Gaussian Process Priors
for Systems of Linear Partial Differential Equations with Constant Coefficients,
vol. 202. MLResearchPress , 2023.
mla: Härkönen, Marc, et al. “Gaussian Process Priors for Systems of Linear Partial
Differential Equations with Constant Coefficients.” 40th International Conference
on Machine Learning, vol. 202, MLResearchPress , 2023.
short: M. Härkönen, M. Lange-Hegermann, B. Raiţă, Gaussian Process Priors for Systems
of Linear Partial Differential Equations with Constant Coefficients, MLResearchPress
, 2023.
ufg: 'Härkönen, Marc/Lange-Hegermann, Markus/ Raiţă, Bogdan: Gaussian Process
Priors for Systems of Linear Partial Differential Equations with Constant Coefficients,
Bd. 202, o. O. 2023 (Proceedings of machine learning research : PMLR).'
van: 'Härkönen M, Lange-Hegermann M, Raiţă B. Gaussian Process Priors for Systems
of Linear Partial Differential Equations with Constant Coefficients. 40th International
Conference on Machine Learning. MLResearchPress ; 2023. (Proceedings of machine
learning research : PMLR; vol. 202).'
conference:
end_date: 2023-07-29
location: Honolulu, HI
name: 40th International Conference on Machine Learning
start_date: 2023-07-23
date_created: 2025-04-22T14:14:42Z
date_updated: 2025-06-26T07:56:15Z
department:
- _id: DEP5023
intvolume: ' 202'
language:
- iso: eng
publication: 40th International Conference on Machine Learning
publication_identifier:
issn:
- 2640-3498
publication_status: published
publisher: 'MLResearchPress '
series_title: 'Proceedings of machine learning research : PMLR'
status: public
title: Gaussian Process Priors for Systems of Linear Partial Differential Equations
with Constant Coefficients
type: conference_editor_article
user_id: '83781'
volume: 202
year: '2023'
...
---
_id: '12786'
abstract:
- lang: eng
text: 'One goal in Bayesian machine learning is to encode prior knowledge into prior
distributions, to model data efficiently. We consider prior knowledge from systems
of linear partial differential equations together with their boundary conditions.
We construct multi-output Gaussian process priors with realizations in the solution
set of such systems, in particular only such solutions can be represented by Gaussian
process regression. The construction is fully algorithmic via Grobner bases and
it does not employ any approximation. It builds these priors combining two parametrizations
via a pullback: the first parametrizes the solutions for the system of differential
equations and the second parametrizes all functions adhering to the boundary conditions.'
author:
- first_name: Markus
full_name: Lange-Hegermann, Markus
id: '71761'
last_name: Lange-Hegermann
citation:
ama: Lange-Hegermann M. Linearly Constrained Gaussian Processes with Boundary
Conditions. Vol 130. (Banerjee A, Fukumizu K, eds.). MLResearchPress ; 2021.
apa: Lange-Hegermann, M. (2021). Linearly Constrained Gaussian Processes with Boundary
Conditions. In A. Banerjee & K. Fukumizu (Eds.), 24th International Conference
on Artificial Intelligence and Statistics (AISTATS) (Vol. 130). MLResearchPress
.
bjps: Lange-Hegermann M (2021) Linearly Constrained Gaussian Processes
with Boundary Conditions, Banerjee A and Fukumizu K (eds). MLResearchPress
.
chicago: 'Lange-Hegermann, Markus. Linearly Constrained Gaussian Processes with
Boundary Conditions. Edited by A. Banerjee and K. Fukumizu. 24th International
Conference on Artificial Intelligence and Statistics (AISTATS). Vol. 130.
Proceedings of Machine Learning Research : PMLR . MLResearchPress , 2021.'
chicago-de: 'Lange-Hegermann, Markus. 2021. Linearly Constrained Gaussian Processes
with Boundary Conditions. Hg. von A. Banerjee und K. Fukumizu. 24th International
Conference on Artificial Intelligence and Statistics (AISTATS). Bd. 130. Proceedings
of machine learning research : PMLR . MLResearchPress .'
din1505-2-1: 'Lange-Hegermann, Markus
; Banerjee, A. ; Fukumizu,
K. (Hrsg.): Linearly Constrained Gaussian Processes with Boundary Conditions,
Proceedings of machine learning research : PMLR . Bd. 130 : MLResearchPress
, 2021'
havard: M. Lange-Hegermann, Linearly Constrained Gaussian Processes with Boundary
Conditions, MLResearchPress , 2021.
ieee: M. Lange-Hegermann, Linearly Constrained Gaussian Processes with Boundary
Conditions, vol. 130. MLResearchPress , 2021.
mla: Lange-Hegermann, Markus. “Linearly Constrained Gaussian Processes with Boundary
Conditions.” 24th International Conference on Artificial Intelligence and Statistics
(AISTATS), edited by A. Banerjee and K. Fukumizu, vol. 130, MLResearchPress
, 2021.
short: M. Lange-Hegermann, Linearly Constrained Gaussian Processes with Boundary
Conditions, MLResearchPress , 2021.
ufg: 'Lange-Hegermann, Markus: Linearly Constrained Gaussian Processes with
Boundary Conditions, Bd. 130, hg. von Banerjee, A./Fukumizu, K., o. O. 2021 (Proceedings
of machine learning research : PMLR ).'
van: 'Lange-Hegermann M. Linearly Constrained Gaussian Processes with Boundary Conditions.
Banerjee A, Fukumizu K, editors. 24th International Conference on Artificial Intelligence
and Statistics (AISTATS). MLResearchPress ; 2021. (Proceedings of machine learning
research : PMLR ; vol. 130).'
conference:
end_date: 2021-04-15
location: Virtual
name: 24th International Conference on Artificial Intelligence and Statistics (AISTATS)
start_date: 2021-04-13
date_created: 2025-04-14T13:58:16Z
date_updated: 2025-06-26T13:42:36Z
department:
- _id: DEP5000
- _id: DEP5023
editor:
- first_name: A.
full_name: Banerjee, A.
last_name: Banerjee
- first_name: K.
full_name: Fukumizu, K.
last_name: Fukumizu
intvolume: ' 130'
keyword:
- FUNCTIONAL REGRESSION
- PREDICTION
- ALGORITHMS
- COMPLEXITY
- MODELS
language:
- iso: eng
publication: 24th International Conference on Artificial Intelligence and Statistics
(AISTATS)
publication_identifier:
issn:
- 2640-3498
publication_status: published
publisher: 'MLResearchPress '
quality_controlled: '1'
series_title: 'Proceedings of machine learning research : PMLR '
status: public
title: Linearly Constrained Gaussian Processes with Boundary Conditions
type: conference_editor_article
user_id: '83781'
volume: 130
year: '2021'
...