{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T19:11:55Z","timestamp":1757617915628,"version":"3.44.0"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,5,6]],"date-time":"2025-05-06T00:00:00Z","timestamp":1746489600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,5,6]],"date-time":"2025-05-06T00:00:00Z","timestamp":1746489600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100000185","name":"Defense Advanced Research Projects Agency","doi-asserted-by":"publisher","award":["FA8750-19-2-0502"],"award-info":[{"award-number":["FA8750-19-2-0502"]}],"id":[{"id":"10.13039\/100000185","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100011661","name":"Pacific Northwest National Laboratory","doi-asserted-by":"publisher","award":["528678"],"award-info":[{"award-number":["528678"]}],"id":[{"id":"10.13039\/100011661","id-type":"DOI","asserted-by":"publisher"}]},{"name":"ICB","award":["W911NF-19-D-0001"],"award-info":[{"award-number":["W911NF-19-D-0001"]}]},{"name":"ARO Young Investigator Grant","award":["W911NF-20-1- 0165"],"award-info":[{"award-number":["W911NF-20-1- 0165"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2025,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Koopman operators model nonlinear dynamics as a linear dynamic system acting on a nonlinear function as the state. This nonstandard state is often called a Koopman observable and is usually approximated numerically by a superposition of functions drawn from a <jats:italic>dictionary<\/jats:italic>. In a widely used algorithm, <jats:italic>extended dynamic mode decomposition<\/jats:italic> (EDMD), the dictionary functions are drawn from a fixed class of functions. Deep learning combined with EDMD has been used to learn novel dictionary functions in an algorithm called deep dynamic mode decomposition (deepDMD). The learned representation both (1) accurately models and (2) scales well with the dimension of the original nonlinear system. In this paper, we analyze the learned dictionaries from deepDMD and explore the theoretical basis for their strong performance. We explore State-Inclusive Logistic Lifting (SILL) dictionary functions to approximate Koopman observables. Error analysis of these dictionary functions show they satisfy a property of subspace approximation, which we define as uniform finite approximate closure. Typically, a Koopman dictionary\u2019s nonlinear functions are homogeneous. In this paper, we discover that structured mixing of heterogeneous dictionary functions drawn from different classes of nonlinear functions achieve the same accuracy and dimensional scaling as the deep-learning-based deepDMD algorithm Yeung et al. ( In: 2019 American Control Conference (ACC), 2019). We specifically show this by building a heterogeneous dictionary comprised of SILL functions and conjunctive radial basis functions (RBFs). This mixed dictionary achieves similar accuracy and dimensional scaling to deepDMD with an order of magnitude reduction in parameters, while maintaining geometric interpretability. These results strengthen the viability of dictionary-based Koopman models to solving high-dimensional nonlinear learning problems.<\/jats:p>","DOI":"10.1007\/s00332-025-10159-2","type":"journal-article","created":{"date-parts":[[2025,5,6]],"date-time":"2025-05-06T14:03:41Z","timestamp":1746540221000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Heterogeneous Mixtures of Dictionary Functions to Approximate Subspace Invariance in Koopman Operators: Why Deep Koopman Operators Work"],"prefix":"10.1007","volume":"35","author":[{"given":"Charles A.","family":"Johnson","sequence":"first","affiliation":[]},{"given":"Shara","family":"Balakrishnan","sequence":"additional","affiliation":[]},{"given":"Enoch","family":"Yeung","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,5,6]]},"reference":[{"key":"10159_CR1","volume-title":"Chaos, fractals, and noise: stochastic aspects of dynamics","author":"L Andrzej","year":"1998","unstructured":"Andrzej, L., Mackey, M.C.: Chaos, fractals, and noise: stochastic aspects of dynamics, vol. 97. Springer Science & Business Media, Cham (1998)"},{"issue":"4","key":"10159_CR2","doi-asserted-by":"publisher","first-page":"2096","DOI":"10.1137\/17M1125236","volume":"16","author":"H. Arbabi","year":"2017","unstructured":"Arbabi, H.., Mezic, I..: Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the Koopman operator. SIAM J. Appl. Dyn. Syst. 16(4), 2096\u20132126 (2017)","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"10159_CR3","doi-asserted-by":"publisher","first-page":"930","DOI":"10.1109\/18.256500","volume":"39","author":"A.R. Barron","year":"1993","unstructured":"Barron, A..R..: Universal approximation bounds for superpositions of a sigmoidal function. IEEE Trans. Inf. Theory 39, 930\u2013945 (1993)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"10159_CR4","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41467-017-02088-w","volume":"9","author":"J Bethany Lusch","year":"2018","unstructured":"Bethany Lusch, J., Kutz, Nathan, Brunton, Steven L.: Deep learning for universal linear embeddings of nonlinear dynamics. Nat. Commun. 9(1), 1\u201310 (2018)","journal-title":"Nat. Commun."},{"issue":"2","key":"10159_CR5","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0150171","volume":"11","author":"SL Brunton","year":"2016","unstructured":"Brunton, S.L., Brunton, B.W., Proctor, J.L., Kutz, J.N.: Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control. PloS one 11(2), e0150171 (2016)","journal-title":"PloS one"},{"issue":"15","key":"10159_CR6","doi-asserted-by":"publisher","first-page":"3932","DOI":"10.1073\/pnas.1517384113","volume":"113","author":"SL Brunton","year":"2016","unstructured":"Brunton, S.L., Proctor, J.L., Kutz, J.N.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences 113(15), 3932\u20133937 (2016)","journal-title":"Proceedings of the National Academy of Sciences"},{"issue":"4","key":"10159_CR7","doi-asserted-by":"publisher","DOI":"10.1063\/1.4772195","volume":"22","author":"Marko Budi\u0161i\u0107","year":"2012","unstructured":"Budi\u0161i\u0107, Marko, Mohr, Ryan, Mezi\u0107, Igor: Applied Koopmanism. Chaos: Interdiscip. J. Nonlinear Sci. 22(4), 047510 (2012)","journal-title":"Chaos: Interdiscip. J. Nonlinear Sci."},{"key":"10159_CR8","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511543241","volume-title":"Radial basis functions: theory and implementations","author":"MD Buhmann","year":"2003","unstructured":"Buhmann, M.D.: Radial basis functions: theory and implementations, vol. 12. Cambridge University Press, Cambridge (2003)"},{"issue":"3","key":"10159_CR9","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pone.0119821","volume":"10","author":"Bryan C Daniels","year":"2015","unstructured":"Daniels, Bryan C., Nemenman, Ilya: Efficient inference of parsimonious phenomenological models of cellular dynamics using s-systems and alternating regression. PLoS ONE 10(3), e0119821 (2015)","journal-title":"PLoS ONE"},{"key":"10159_CR10","unstructured":"Diederik, P., Kingma, Ba, J.: Adam: a method for stochastic optimization. arXiv preprint arXiv:1412.6980, (2014)"},{"key":"10159_CR11","unstructured":"Duchi, J., Hazan, E., Singer, Y.: Adaptive subgradient methods for online learning and stochastic optimization.  Journal of machine learning research, 12(7), (2011)"},{"issue":"7","key":"10159_CR12","doi-asserted-by":"publisher","first-page":"3442","DOI":"10.1109\/TAC.2021.3105318","volume":"67","author":"Masih Haseli","year":"2021","unstructured":"Haseli, Masih, Cort\u00e9s, Jorge: Learning Koopman eigenfunctions and invariant subspaces from data: symmetric subspace decomposition. IEEE Transactions on Automatic Control 67(7), 3442\u20133457 (2021)","journal-title":"IEEE Transactions on Automatic Control"},{"key":"10159_CR13","doi-asserted-by":"crossref","unstructured":"Hasnain, A., Boddupalli, N., Balakrishnan, S., Yeung, E.: Steady state programming of controlled nonlinear systems via deep dynamic mode decomposition. In  2020 American Control Conference (ACC), pages 4245\u20134251. IEEE, (2020)","DOI":"10.23919\/ACC45564.2020.9147218"},{"issue":"1","key":"10159_CR14","doi-asserted-by":"publisher","first-page":"3148","DOI":"10.1038\/s41467-023-37897-9","volume":"14","author":"Aqib Hasnain","year":"2023","unstructured":"Hasnain, Aqib, Balakrishnan, Shara, Joshy, Dennis M., Smith, Jen, Haase, Steven B., Yeung, Enoch: Learning perturbation-inducible cell states from observability analysis of transcriptome dynamics. Nat. Commun. 14(1), 3148 (2023)","journal-title":"Nat. Commun."},{"issue":"2","key":"10159_CR15","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1016\/0893-6080(91)90009-T","volume":"4","author":"Kurt Hornik","year":"1991","unstructured":"Hornik, Kurt: Approximation capabilities of multilayer feedforward networks. Neural Netw. 4(2), 251\u2013257 (1991)","journal-title":"Neural Netw."},{"key":"10159_CR16","doi-asserted-by":"crossref","unstructured":"Johnson, C. A., Yeung, E.: A class of logistic functions for approximating state-inclusive Koopman operators. In  2018 Annual American Control Conference (ACC), pages 4803\u20134810. IEEE, (2018)","DOI":"10.23919\/ACC.2018.8431525"},{"issue":"2","key":"10159_CR17","doi-asserted-by":"publisher","first-page":"687","DOI":"10.1007\/s00332-017-9423-0","volume":"28","author":"Milan Korda","year":"2018","unstructured":"Korda, Milan, Mezi\u0107, Igor: On convergence of extended dynamic mode decomposition to the Koopman operator. J. Nonlinear Sci. 28(2), 687\u2013710 (2018)","journal-title":"J. Nonlinear Sci."},{"issue":"12","key":"10159_CR18","doi-asserted-by":"publisher","first-page":"3397","DOI":"10.1109\/78.258082","volume":"41","author":"St\u00e9phane G Mallat","year":"1993","unstructured":"Mallat, St\u00e9phane. G., Zhang, Zhifeng: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397\u20133415 (1993)","journal-title":"IEEE Trans. Signal Process."},{"key":"10159_CR19","unstructured":"Matthew, O., Williams, Rowley, C. W., Kevrekidis, I. G.: A kernel-based approach to data-driven Koopman spectral analysis. arXiv preprint arXiv:1411.2260, (2014)"},{"issue":"6","key":"10159_CR20","doi-asserted-by":"publisher","first-page":"2550","DOI":"10.1109\/TAC.2019.2941433","volume":"65","author":"Alexandre Mauroy","year":"2019","unstructured":"Mauroy, Alexandre, Goncalves, Jorge: Koopman-based lifting techniques for nonlinear systems identification. IEEE Trans. Autom. Control 65(6), 2550\u20132565 (2019)","journal-title":"IEEE Trans. Autom. Control"},{"key":"10159_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-35713-9","volume-title":"The Koopman operator in systems and control","author":"Alexandre Mauroy","year":"2020","unstructured":"Mauroy, Alexandre, Susuki, Y., Mezi\u0107, I.: The Koopman operator in systems and control. Springer, Berlin (2020)"},{"issue":"1","key":"10159_CR22","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/s11071-005-2824-x","volume":"41","author":"Igor Mezi\u0107","year":"2005","unstructured":"Mezi\u0107, Igor: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41(1), 309\u2013325 (2005)","journal-title":"Nonlinear Dyn."},{"key":"10159_CR23","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1146\/annurev-fluid-011212-140652","volume":"45","author":"Igor Mezi\u0107","year":"2013","unstructured":"Mezi\u0107, Igor: Analysis of fluid flows via spectral properties of the Koopman operator. Annu. Rev. Fluid Mech. 45, 357\u2013378 (2013)","journal-title":"Annu. Rev. Fluid Mech."},{"key":"10159_CR24","doi-asserted-by":"crossref","unstructured":"Nandanoori, S. P., Kundu, S., Pal, S., Agarwal, K., Choudhury, S.: Model-agnostic algorithm for real-time attack identification in power grid using Koopman modes. In  2020 IEEE International Conference on Communications, Control, and Computing Technologies for Smart Grids (SmartGridComm), pages 1\u20136. IEEE, (2020)","DOI":"10.1109\/SmartGridComm47815.2020.9303022"},{"issue":"1","key":"10159_CR25","doi-asserted-by":"publisher","first-page":"558","DOI":"10.1137\/18M1177846","volume":"18","author":"Samuel E Otto","year":"2019","unstructured":"Otto, Samuel E., Rowley, Clarence W.: Linearly recurrent autoencoder networks for learning dynamics. SIAM J. Appl. Dyn. Syst. 18(1), 558\u2013593 (2019)","journal-title":"SIAM J. Appl. Dyn. Syst."},{"issue":"1","key":"10159_CR26","doi-asserted-by":"publisher","first-page":"909","DOI":"10.1137\/16M1062296","volume":"17","author":"JL Proctor","year":"2018","unstructured":"Proctor, J.L., Brunton, S.L., Kutz, J.N.: Generalizing Koopman theory to allow for inputs and control. SIAM J. Appl. Dyn. Syst. 17(1), 909\u2013930 (2018)","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"10159_CR27","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1017\/S0022112009992059","volume":"641","author":"Clarence W Rowley","year":"2009","unstructured":"Rowley, Clarence W., Mezi\u0107, Igor, Bagheri, Shervin, Schlatter, Philipp, Henningson, Dan S.: Pectral analysis of nonlinear flows. J. Fluid Mech. 641, 115\u2013127 (2009)","journal-title":"J. Fluid Mech."},{"key":"10159_CR28","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1017\/S0022112010001217","volume":"656","author":"Peter J Schmid","year":"2010","unstructured":"Schmid, Peter J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5\u201328 (2010)","journal-title":"J. Fluid Mech."},{"key":"10159_CR29","doi-asserted-by":"crossref","unstructured":"Sinha, S., Huang, B. Vaidya, U.: Robust approximation of Koopman operator and prediction in random dynamical systems. In  2018 Annual American Control Conference (ACC), pages 5491\u20135496. IEEE, (2018)","DOI":"10.23919\/ACC.2018.8431015"},{"key":"10159_CR30","doi-asserted-by":"crossref","unstructured":"Takeishi, N., Kawahara, Y., Yairi, T.: Learning Koopman invariant subspaces for dynamic mode decomposition. arXiv preprint arXiv:1710.04340, (2017)","DOI":"10.24963\/ijcai.2017\/392"},{"issue":"24","key":"10159_CR31","doi-asserted-by":"publisher","DOI":"10.1063\/1.5011399","volume":"148","author":"Christoph Wehmeyer","year":"2018","unstructured":"Wehmeyer, Christoph, No\u00e9, Frank: Time-lagged autoencoders: deep learning of slow collective variables for molecular kinetics. J. Chem. Phys. 148(24), 241703 (2018)","journal-title":"J. Chem. Phys."},{"issue":"6","key":"10159_CR32","doi-asserted-by":"publisher","first-page":"1307","DOI":"10.1007\/s00332-015-9258-5","volume":"25","author":"Matthew O Williams","year":"2015","unstructured":"Williams, Matthew O., Kevrekidis, Ioannis G., Rowley, Clarence W.: A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 25(6), 1307\u20131346 (2015)","journal-title":"J. Nonlinear Sci."},{"issue":"18","key":"10159_CR33","doi-asserted-by":"publisher","first-page":"704","DOI":"10.1016\/j.ifacol.2016.10.248","volume":"49","author":"Matthew O Williams","year":"2016","unstructured":"Williams, Matthew O., Hemati, Maziar S., Dawson, Scott TM., Kevrekidis, Ioannis G., Rowley, Clarence W.: Extending data-driven Koopman analysis to actuated systems. IFAC-PapersOnLine 49(18), 704\u2013709 (2016)","journal-title":"IFAC-PapersOnLine"},{"key":"10159_CR34","doi-asserted-by":"crossref","unstructured":"Yeung, E., Kundu, S., Hodas, N.: Learning deep neural network representations for Koopman operators of nonlinear dynamical systems. In  2019 American Control Conference (ACC), pages 4832\u20134839. IEEE, (2019)","DOI":"10.23919\/ACC.2019.8815339"},{"key":"10159_CR35","doi-asserted-by":"crossref","unstructured":"Zhang, Christophe, Zuazua, Enrique: A quantitative analysis of Koopman operator methods for system identification and predictions. Comptes Rendus. M\u00e9canique 351(S1), 1\u201331 (2023)","DOI":"10.5802\/crmeca.138"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-025-10159-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00332-025-10159-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-025-10159-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,6]],"date-time":"2025-09-06T13:43:50Z","timestamp":1757166230000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00332-025-10159-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,6]]},"references-count":35,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["10159"],"URL":"https:\/\/doi.org\/10.1007\/s00332-025-10159-2","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"type":"print","value":"0938-8974"},{"type":"electronic","value":"1432-1467"}],"subject":[],"published":{"date-parts":[[2025,5,6]]},"assertion":[{"value":"23 June 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 March 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 May 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"68"}}