Safely Learning Dynamical Systems.
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| Title: | Safely Learning Dynamical Systems. |
|---|---|
| Authors: | Ahmadi, Amir Ali1 (AUTHOR) aaa@princeton.edu, Chaudhry, Abraar2 (AUTHOR) achaudhry61@gatech.edu, Sindhwani, Vikas3 (AUTHOR) sindhwani@google.com, Tu, Stephen3 (AUTHOR) stephentu@google.com |
| Source: | Foundations of Computational Mathematics. Apr2026, Vol. 26 Issue 2, p999-1068. 70p. |
| Subjects: | Dynamical systems, Semidefinite programming, Uncertain systems, Mathematical optimization, Nonlinear mechanics, Linear programming, Feedback control systems |
| Abstract: | A fundamental challenge in learning an unknown dynamical system is to reduce model uncertainty by making measurements while maintaining safety. In this work, we formulate a mathematical definition of what it means to safely learn a dynamical system by sequentially deciding where to initialize the next trajectory. In our framework, the state of the system is required to stay within a safety region for a horizon of T time steps under the action of all dynamical systems that (i) belong to a given initial uncertainty set, and (ii) are consistent with the information gathered so far. For our first set of results, we consider the setting of safely learning a linear dynamical system involving n states. For the case T = 1 , we present a linear programming-based algorithm that either safely recovers the true dynamics from at most n trajectories, or certifies that safe learning is impossible. For T = 2 , we give a semidefinite representation of the set of safe initial conditions and show that ⌈ n / 2 ⌉ trajectories generically suffice for safe learning. For T = ∞ , we provide semidefinite representable inner approximations of the set of safe initial conditions and show that one trajectory generically suffices for safe learning. Finally, we extend a number of our results to the cases where the initial uncertainty set contains sparse, low-rank, or permutation matrices, or when the dynamical system involves a control input. Our second set of results concerns the problem of safely learning a general class of nonlinear dynamical systems. For the case T = 1 , we give a second-order cone programming based representation of the set of safe initial conditions. For T = ∞ , we provide semidefinite representable inner approximations to the set of safe initial conditions. We show how one can safely collect trajectories and fit a polynomial model of the nonlinear dynamics that is consistent with the initial uncertainty set and best agrees with the observations. We also present extensions of some of our results to the cases where the measurements are noisy or the dynamical system involves disturbances. [ABSTRACT FROM AUTHOR] |
| Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
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| Header | DbId: egs DbLabel: Engineering Source An: 193283910 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Safely Learning Dynamical Systems. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Ahmadi%2C+Amir+Ali%22">Ahmadi, Amir Ali</searchLink><relatesTo>1</relatesTo> (AUTHOR)<i> aaa@princeton.edu</i><br /><searchLink fieldCode="AR" term="%22Chaudhry%2C+Abraar%22">Chaudhry, Abraar</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> achaudhry61@gatech.edu</i><br /><searchLink fieldCode="AR" term="%22Sindhwani%2C+Vikas%22">Sindhwani, Vikas</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> sindhwani@google.com</i><br /><searchLink fieldCode="AR" term="%22Tu%2C+Stephen%22">Tu, Stephen</searchLink><relatesTo>3</relatesTo> (AUTHOR)<i> stephentu@google.com</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Foundations+of+Computational+Mathematics%22">Foundations of Computational Mathematics</searchLink>. Apr2026, Vol. 26 Issue 2, p999-1068. 70p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Dynamical+systems%22">Dynamical systems</searchLink><br /><searchLink fieldCode="DE" term="%22Semidefinite+programming%22">Semidefinite programming</searchLink><br /><searchLink fieldCode="DE" term="%22Uncertain+systems%22">Uncertain systems</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+optimization%22">Mathematical optimization</searchLink><br /><searchLink fieldCode="DE" term="%22Nonlinear+mechanics%22">Nonlinear mechanics</searchLink><br /><searchLink fieldCode="DE" term="%22Linear+programming%22">Linear programming</searchLink><br /><searchLink fieldCode="DE" term="%22Feedback+control+systems%22">Feedback control systems</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: A fundamental challenge in learning an unknown dynamical system is to reduce model uncertainty by making measurements while maintaining safety. In this work, we formulate a mathematical definition of what it means to safely learn a dynamical system by sequentially deciding where to initialize the next trajectory. In our framework, the state of the system is required to stay within a safety region for a horizon of T time steps under the action of all dynamical systems that (i) belong to a given initial uncertainty set, and (ii) are consistent with the information gathered so far. For our first set of results, we consider the setting of safely learning a linear dynamical system involving n states. For the case T = 1 , we present a linear programming-based algorithm that either safely recovers the true dynamics from at most n trajectories, or certifies that safe learning is impossible. For T = 2 , we give a semidefinite representation of the set of safe initial conditions and show that ⌈ n / 2 ⌉ trajectories generically suffice for safe learning. For T = ∞ , we provide semidefinite representable inner approximations of the set of safe initial conditions and show that one trajectory generically suffices for safe learning. Finally, we extend a number of our results to the cases where the initial uncertainty set contains sparse, low-rank, or permutation matrices, or when the dynamical system involves a control input. Our second set of results concerns the problem of safely learning a general class of nonlinear dynamical systems. For the case T = 1 , we give a second-order cone programming based representation of the set of safe initial conditions. For T = ∞ , we provide semidefinite representable inner approximations to the set of safe initial conditions. We show how one can safely collect trajectories and fit a polynomial model of the nonlinear dynamics that is consistent with the initial uncertainty set and best agrees with the observations. We also present extensions of some of our results to the cases where the measurements are noisy or the dynamical system involves disturbances. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Foundations of Computational Mathematics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1007/s10208-025-09689-8 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 70 StartPage: 999 Subjects: – SubjectFull: Dynamical systems Type: general – SubjectFull: Semidefinite programming Type: general – SubjectFull: Uncertain systems Type: general – SubjectFull: Mathematical optimization Type: general – SubjectFull: Nonlinear mechanics Type: general – SubjectFull: Linear programming Type: general – SubjectFull: Feedback control systems Type: general Titles: – TitleFull: Safely Learning Dynamical Systems. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Ahmadi, Amir Ali – PersonEntity: Name: NameFull: Chaudhry, Abraar – PersonEntity: Name: NameFull: Sindhwani, Vikas – PersonEntity: Name: NameFull: Tu, Stephen IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 04 Text: Apr2026 Type: published Y: 2026 Identifiers: – Type: issn-print Value: 16153375 Numbering: – Type: volume Value: 26 – Type: issue Value: 2 Titles: – TitleFull: Foundations of Computational Mathematics Type: main |
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