A Data Driven, Convex Optimization Approach to Learning Koopman Operators

Mario Sznaier
Proceedings of the 3rd Conference on Learning for Dynamics and Control, PMLR 144:436-446, 2021.

Abstract

Koopman operators provide tractable means of learning linear approximations of non-linear dynamics. Many approaches have been proposed to find these operators, typically based upon approximations using an a-priori fixed class of models. However, choosing appropriate models and bounding the approximation error is far from trivial. Motivated by these difficulties, in this paper we propose an optimization based approach to learning Koopman operators from data. Our results show that the Koopman operator, the associated Hilbert space of observables and a suitable dictionary can be obtained by solving two rank-constrained semi-definite programs (SDP). While in principle these problems are NP-hard, the use of standard relaxations of rank leads to convex SDPs.

Cite this Paper

BibTeX
@InProceedings{pmlr-v144-sznaier21a, title = {A Data Driven, Convex Optimization Approach to Learning Koopman Operators}, author = {Sznaier, Mario}, booktitle = {Proceedings of the 3rd Conference on Learning for Dynamics and Control}, pages = {436--446}, year = {2021}, editor = {Jadbabaie, Ali and Lygeros, John and Pappas, George J. and A. Parrilo, Pablo and Recht, Benjamin and Tomlin, Claire J. and Zeilinger, Melanie N.}, volume = {144}, series = {Proceedings of Machine Learning Research}, month = {07 -- 08 June}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v144/sznaier21a/sznaier21a.pdf}, url = {https://proceedings.mlr.press/v144/sznaier21a.html}, abstract = {Koopman operators provide tractable means of learning linear approximations of non-linear dynamics. Many approaches have been proposed to find these operators, typically based upon approximations using an a-priori fixed class of models. However, choosing appropriate models and bounding the approximation error is far from trivial. Motivated by these difficulties, in this paper we propose an optimization based approach to learning Koopman operators from data. Our results show that the Koopman operator, the associated Hilbert space of observables and a suitable dictionary can be obtained by solving two rank-constrained semi-definite programs (SDP). While in principle these problems are NP-hard, the use of standard relaxations of rank leads to convex SDPs.} }
Endnote
%0 Conference Paper %T A Data Driven, Convex Optimization Approach to Learning Koopman Operators %A Mario Sznaier %B Proceedings of the 3rd Conference on Learning for Dynamics and Control %C Proceedings of Machine Learning Research %D 2021 %E Ali Jadbabaie %E John Lygeros %E George J. Pappas %E Pablo A. Parrilo %E Benjamin Recht %E Claire J. Tomlin %E Melanie N. Zeilinger %F pmlr-v144-sznaier21a %I PMLR %P 436--446 %U https://proceedings.mlr.press/v144/sznaier21a.html %V 144 %X Koopman operators provide tractable means of learning linear approximations of non-linear dynamics. Many approaches have been proposed to find these operators, typically based upon approximations using an a-priori fixed class of models. However, choosing appropriate models and bounding the approximation error is far from trivial. Motivated by these difficulties, in this paper we propose an optimization based approach to learning Koopman operators from data. Our results show that the Koopman operator, the associated Hilbert space of observables and a suitable dictionary can be obtained by solving two rank-constrained semi-definite programs (SDP). While in principle these problems are NP-hard, the use of standard relaxations of rank leads to convex SDPs.
APA
Sznaier, M.. (2021). A Data Driven, Convex Optimization Approach to Learning Koopman Operators. Proceedings of the 3rd Conference on Learning for Dynamics and Control, in Proceedings of Machine Learning Research 144:436-446 Available from https://proceedings.mlr.press/v144/sznaier21a.html.

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