[{"series_title":"Proceedings of Machine Learning Research","publication_status":"published","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."}],"conference":{"location":"Valencia, SPAIN","name":"27th International Conference on Artificial Intelligence and Statistics (AISTATS)","start_date":"2024-05-02"},"department":[{"_id":"DEP5000"},{"_id":"DEP5023"}],"publisher":"MLResearchPress ","type":"conference_editor_article","user_id":"83781","_id":"12815","title":"Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning","citation":{"ieee":"J. Tebbe, C. Zimmer, A. Steland, M. Lange-Hegermann, and F. Mies, <i>Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning</i>. MLResearchPress , 2024, pp. 1333–1341.","apa":"Tebbe, J., Zimmer, C., Steland, A., Lange-Hegermann, M., &#38; Mies, F. (2024). Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning. In <i>International Conference on Artificial Intelligence and Statistics (AISTATS), Vol. 238</i> (pp. 1333–1341). MLResearchPress .","chicago":"Tebbe, Jörn, Christoph Zimmer, Ansgar Steland, Markus Lange-Hegermann, and Fabian Mies. <i>Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning</i>. <i>International Conference on Artificial Intelligence and Statistics (AISTATS), Vol. 238</i>. Proceedings of Machine Learning Research. MLResearchPress , 2024.","ama":"Tebbe J, Zimmer C, Steland A, Lange-Hegermann M, Mies F. <i>Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning</i>. MLResearchPress ; 2024:1333-1341.","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).","mla":"Tebbe, Jörn, et al. “Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning.” <i>International Conference on Artificial Intelligence and Statistics (AISTATS), Vol. 238</i>, MLResearchPress , 2024, pp. 1333–41.","havard":"J. Tebbe, C. Zimmer, A. Steland, M. Lange-Hegermann, F. Mies, Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning, MLResearchPress , 2024.","bjps":"<b>Tebbe J <i>et al.</i></b> (2024) <i>Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning</i>. MLResearchPress .","ufg":"<b>Tebbe, Jörn u. a.</b>: Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning, o. O. 2024 (Proceedings of Machine Learning Research).","short":"J. Tebbe, C. Zimmer, A. Steland, M. Lange-Hegermann, F. Mies, Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning, MLResearchPress , 2024.","chicago-de":"Tebbe, Jörn, Christoph Zimmer, Ansgar Steland, Markus Lange-Hegermann und Fabian Mies. 2024. <i>Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning</i>. <i>International Conference on Artificial Intelligence and Statistics (AISTATS), Vol. 238</i>. Proceedings of Machine Learning Research. MLResearchPress .","din1505-2-1":"<span style=\"font-variant:small-caps;\">Tebbe, Jörn</span> ; <span style=\"font-variant:small-caps;\">Zimmer, Christoph</span> ; <span style=\"font-variant:small-caps;\">Steland, Ansgar</span> ; <span style=\"font-variant:small-caps;\">Lange-Hegermann, Markus</span> ; <span style=\"font-variant:small-caps;\">Mies, Fabian</span>: <i>Efficiently Computable Safety Bounds for Gaussian Processes in Active Learning</i>, <i>Proceedings of Machine Learning Research</i> : MLResearchPress , 2024"},"main_file_link":[{"url":"https://proceedings.mlr.press/v238/tebbe24a.html","open_access":"1"}],"author":[{"id":"85958","full_name":"Tebbe, Jörn","first_name":"Jörn","last_name":"Tebbe"},{"last_name":"Zimmer","first_name":"Christoph","full_name":"Zimmer, Christoph"},{"last_name":"Steland","full_name":"Steland, Ansgar","first_name":"Ansgar"},{"full_name":"Lange-Hegermann, Markus","first_name":"Markus","last_name":"Lange-Hegermann","id":"71761"},{"full_name":"Mies, Fabian","last_name":"Mies","first_name":"Fabian"}],"year":"2024","status":"public","page":"1333-1341","publication":"International Conference on Artificial Intelligence and Statistics (AISTATS), Vol. 238","publication_identifier":{"issn":["2640-3498"]},"date_created":"2025-04-17T07:58:19Z","date_updated":"2025-06-25T12:47:19Z","oa":"1","language":[{"iso":"eng"}]},{"title":"Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients","series_title":"Proceedings of machine learning research : PMLR","volume":202,"citation":{"din1505-2-1":"<span style=\"font-variant:small-caps;\">Härkönen, Marc </span> ; <span style=\"font-variant:small-caps;\">Lange-Hegermann, Markus</span> ; <span style=\"font-variant:small-caps;\"> Raiţă, Bogdan</span>: <i>Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients</i>, <i>Proceedings of machine learning research : PMLR</i>. Bd. 202 : MLResearchPress , 2023","chicago-de":"Härkönen, Marc , Markus Lange-Hegermann und Bogdan  Raiţă. 2023. <i>Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients</i>. <i>40th International Conference on Machine Learning</i>. Bd. 202. Proceedings of machine learning research : PMLR. MLResearchPress .","short":"M. Härkönen, M. Lange-Hegermann, B.  Raiţă, Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients, MLResearchPress , 2023.","chicago":"Härkönen, Marc , Markus Lange-Hegermann, and Bogdan  Raiţă. <i>Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients</i>. <i>40th International Conference on Machine Learning</i>. Vol. 202. Proceedings of Machine Learning Research : PMLR. MLResearchPress , 2023.","apa":"Härkönen, M., Lange-Hegermann, M., &#38;  Raiţă, B. (2023). Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients. In <i>40th International Conference on Machine Learning</i> (Vol. 202). MLResearchPress .","bjps":"<b>Härkönen M, Lange-Hegermann M and  Raiţă B</b> (2023) <i>Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients</i>. MLResearchPress .","mla":"Härkönen, Marc, et al. “Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients.” <i>40th International Conference on Machine Learning</i>, vol. 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.","ufg":"<b>Härkönen, Marc/Lange-Hegermann, Markus/ Raiţă, Bogdan</b>: 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).","ieee":"M. Härkönen, M. Lange-Hegermann, and B.  Raiţă, <i>Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients</i>, vol. 202. MLResearchPress , 2023.","ama":"Härkönen M, Lange-Hegermann M,  Raiţă B. <i>Gaussian Process Priors for Systems of Linear Partial Differential Equations with Constant Coefficients</i>. Vol 202. MLResearchPress ; 2023."},"publication_status":"published","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."}],"year":"2023","author":[{"full_name":"Härkönen, Marc ","last_name":"Härkönen","first_name":"Marc "},{"id":"71761","full_name":"Lange-Hegermann, Markus","last_name":"Lange-Hegermann","first_name":"Markus"},{"full_name":" Raiţă, Bogdan","last_name":" Raiţă","first_name":"Bogdan"}],"department":[{"_id":"DEP5023"}],"conference":{"start_date":"2023-07-23","end_date":"2023-07-29","location":"Honolulu, HI","name":"40th International Conference on Machine Learning"},"publisher":"MLResearchPress ","status":"public","publication":"40th International Conference on Machine Learning","publication_identifier":{"issn":["2640-3498"]},"date_created":"2025-04-22T14:14:42Z","type":"conference_editor_article","intvolume":"       202","user_id":"83781","date_updated":"2025-06-26T07:56:15Z","_id":"12828","language":[{"iso":"eng"}]},{"publication_identifier":{"issn":["2640-3498"]},"publication":"24th International Conference on Artificial Intelligence and Statistics (AISTATS)","date_created":"2025-04-14T13:58:16Z","date_updated":"2025-06-26T13:42:36Z","editor":[{"last_name":"Banerjee","full_name":"Banerjee, A.","first_name":"A."},{"last_name":"Fukumizu","first_name":"K.","full_name":"Fukumizu, K."}],"keyword":["FUNCTIONAL REGRESSION","PREDICTION","ALGORITHMS","COMPLEXITY","MODELS"],"language":[{"iso":"eng"}],"title":"Linearly Constrained Gaussian Processes with Boundary Conditions","volume":130,"citation":{"ieee":"M. Lange-Hegermann, <i>Linearly Constrained Gaussian Processes with Boundary Conditions</i>, vol. 130. MLResearchPress , 2021.","ama":"Lange-Hegermann M. <i>Linearly Constrained Gaussian Processes with Boundary Conditions</i>. Vol 130. (Banerjee A, Fukumizu K, eds.). MLResearchPress ; 2021.","chicago-de":"Lange-Hegermann, Markus. 2021. <i>Linearly Constrained Gaussian Processes with Boundary Conditions</i>. Hg. von A. Banerjee und K. Fukumizu. <i>24th International Conference on Artificial Intelligence and Statistics (AISTATS)</i>. Bd. 130. Proceedings of machine learning research : PMLR . MLResearchPress .","din1505-2-1":"<span style=\"font-variant:small-caps;\">Lange-Hegermann, Markus</span> ; <span style=\"font-variant:small-caps;\">Banerjee, A.</span> ; <span style=\"font-variant:small-caps;\">Fukumizu, K.</span> (Hrsg.): <i>Linearly Constrained Gaussian Processes with Boundary Conditions</i>, <i>Proceedings of machine learning research : PMLR </i>. Bd. 130 : MLResearchPress , 2021","short":"M. Lange-Hegermann, Linearly Constrained Gaussian Processes with Boundary Conditions, MLResearchPress , 2021.","chicago":"Lange-Hegermann, Markus. <i>Linearly Constrained Gaussian Processes with Boundary Conditions</i>. Edited by A. Banerjee and K. Fukumizu. <i>24th International Conference on Artificial Intelligence and Statistics (AISTATS)</i>. Vol. 130. Proceedings of Machine Learning Research : PMLR . MLResearchPress , 2021.","apa":"Lange-Hegermann, M. (2021). Linearly Constrained Gaussian Processes with Boundary Conditions. In A. Banerjee &#38; K. Fukumizu (Eds.), <i>24th International Conference on Artificial Intelligence and Statistics (AISTATS)</i> (Vol. 130). MLResearchPress .","havard":"M. Lange-Hegermann, Linearly Constrained Gaussian Processes with Boundary Conditions, MLResearchPress , 2021.","bjps":"<b>Lange-Hegermann M</b> (2021) <i>Linearly Constrained Gaussian Processes with Boundary Conditions</i>, Banerjee A and Fukumizu K (eds). MLResearchPress .","mla":"Lange-Hegermann, Markus. “Linearly Constrained Gaussian Processes with Boundary Conditions.” <i>24th International Conference on Artificial Intelligence and Statistics (AISTATS)</i>, edited by A. Banerjee and K. Fukumizu, vol. 130, MLResearchPress , 2021.","ufg":"<b>Lange-Hegermann, Markus</b>: 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)."},"author":[{"first_name":"Markus","full_name":"Lange-Hegermann, Markus","last_name":"Lange-Hegermann","id":"71761"}],"year":"2021","status":"public","type":"conference_editor_article","user_id":"83781","intvolume":"       130","_id":"12786","series_title":"Proceedings of machine learning research : PMLR ","publication_status":"published","conference":{"name":"24th International Conference on Artificial Intelligence and Statistics (AISTATS)","location":"Virtual","end_date":"2021-04-15","start_date":"2021-04-13"},"department":[{"_id":"DEP5000"},{"_id":"DEP5023"}],"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."}],"publisher":"MLResearchPress ","quality_controlled":"1"}]