{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,27]],"date-time":"2026-06-27T16:49:01Z","timestamp":1782578941020,"version":"3.54.5"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T00:00:00Z","timestamp":1779148800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T00:00:00Z","timestamp":1779148800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["U24A2001"],"award-info":[{"award-number":["U24A2001"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12271108"],"award-info":[{"award-number":["12271108"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2026,7]]},"DOI":"10.1007\/s10915-026-03301-z","type":"journal-article","created":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T09:46:02Z","timestamp":1779183962000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Online Tensor-Based Dynamic Mode Decomposition for Time-Varying System"],"prefix":"10.1007","volume":"108","author":[{"given":"Wanli","family":"Ma","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chengdong","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6192-0546","authenticated-orcid":false,"given":"Yimin","family":"Wei","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,5,19]]},"reference":[{"key":"3301_CR1","doi-asserted-by":"publisher","first-page":"209","DOI":"10.3934\/jcd.2020009","volume":"7","author":"M Alfatlawi","year":"2020","unstructured":"Alfatlawi, M., Srivastava, V.: An incremental approach to online dynamic mode decomposition for time-varying systems with applications to EEG data modeling. J. Comput. Dyn. 7, 209\u2013241 (2020)","journal-title":"J. Comput. Dyn."},{"issue":"22","key":"3301_CR2","doi-asserted-by":"publisher","first-page":"12685","DOI":"10.1021\/acs.chemrev.9b00829","volume":"120","author":"B Bauer","year":"2020","unstructured":"Bauer, B., Bravyi, S., Motta, M., Chan, G.K.L.: Quantum algorithms for quantum chemistry and quantum materials science. Chem. Rev. 120(22), 12685\u201312717 (2020)","journal-title":"Chem. Rev."},{"key":"3301_CR3","doi-asserted-by":"crossref","unstructured":"Brand, M.: Incremental singular value decomposition of uncertain data with missing values. In: Computer Vision\u2013ECCV 2002: 7th European Conference on Computer Vision Copenhagen, Denmark, May 28\u201331, 2002 Proceedings, Part I 7, pp. 707\u2013720. Springer (2002)","DOI":"10.1007\/3-540-47969-4_47"},{"issue":"2","key":"3301_CR4","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1137\/21M1401243","volume":"64","author":"SL Brunton","year":"2022","unstructured":"Brunton, S.L., Budi\u0161i\u0107, M., Kaiser, E., Kutz, J.N.: Modern Koopman theory for dynamical systems. SIAM Rev. 64(2), 229\u2013340 (2022)","journal-title":"SIAM Rev."},{"issue":"7","key":"3301_CR5","first-page":"808","volume":"78","author":"A Chatterjee","year":"2000","unstructured":"Chatterjee, A.: An introduction to the proper orthogonal decomposition. Curr. Sci. 78(7), 808\u2013817 (2000)","journal-title":"Curr. Sci."},{"issue":"1","key":"3301_CR6","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1007\/s10444-018-9622-8","volume":"45","author":"M Che","year":"2019","unstructured":"Che, M., Wei, Y.: Randomized algorithms for the approximations of Tucker and the tensor train decompositions. Adv. Comput. Math. 45(1), 395\u2013428 (2019)","journal-title":"Adv. Comput. Math."},{"issue":"1","key":"3301_CR7","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1007\/s10915-026-03208-9","volume":"107","author":"M Che","year":"2026","unstructured":"Che, M., Wei, Y., Yan, H.: Efficient randomized algorithms for computing an approximation of the tensor train decomposition. J. Sci. Comput. 107(1), 2 (2026)","journal-title":"J. Sci. Comput."},{"key":"3301_CR8","doi-asserted-by":"publisher","first-page":"887","DOI":"10.1007\/s00332-012-9130-9","volume":"22","author":"KK Chen","year":"2012","unstructured":"Chen, K.K., Tu, J.H., Rowley, C.W.: Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. J. Nonlinear Sci. 22, 887\u2013915 (2012)","journal-title":"J. Nonlinear Sci."},{"key":"3301_CR9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2025.113996","volume":"533","author":"Y Chen","year":"2025","unstructured":"Chen, Y., Lin, Y., Sun, X., Yuan, C., Gao, Z.: Tensor decomposition-based neural operator with dynamic mode decomposition for parameterized time-dependent problems. J. Comput. Phys. 533, 113996 (2025)","journal-title":"J. Comput. Phys."},{"issue":"2","key":"3301_CR10","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1109\/MSP.2013.2297439","volume":"32","author":"A Cichocki","year":"2015","unstructured":"Cichocki, A., Mandic, D., De Lathauwer, L., Zhou, G., Zhao, Q., Caiafa, C., Phan, H.A.: Tensor decompositions for signal processing applications: from two-way to multiway component analysis. IEEE Signal Process. Mag. 32(2), 145\u2013163 (2015)","journal-title":"IEEE Signal Process. Mag."},{"issue":"3","key":"3301_CR11","doi-asserted-by":"publisher","first-page":"1084","DOI":"10.1137\/06066518X","volume":"30","author":"V De Silva","year":"2008","unstructured":"De Silva, V., Lim, L.H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30(3), 1084\u20131127 (2008)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"2","key":"3301_CR12","doi-asserted-by":"publisher","first-page":"A985","DOI":"10.1137\/21M1463665","volume":"45","author":"W Ding","year":"2023","unstructured":"Ding, W., Li, J.: Higher order extended dynamic mode decomposition based on the structured total least squares. SIAM J. Sci. Comput. 45(2), A985\u2013A1011 (2023)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"3301_CR13","doi-asserted-by":"publisher","first-page":"32","DOI":"10.1145\/3640012","volume":"50","author":"Z Drma\u010d","year":"2024","unstructured":"Drma\u010d, Z.: A LAPACK implementation of the dynamic mode decomposition. ACM Trans. Math. Softw. 50(1), 32 (2024)","journal-title":"ACM Trans. Math. Softw."},{"key":"3301_CR14","doi-asserted-by":"publisher","first-page":"40","DOI":"10.1016\/j.cviu.2016.02.005","volume":"146","author":"NB Erichson","year":"2016","unstructured":"Erichson, N.B., Donovan, C.: Randomized low-rank dynamic mode decomposition for motion detection. Comput. Vis. Image Underst. 146, 40\u201350 (2016)","journal-title":"Comput. Vis. Image Underst."},{"key":"3301_CR15","doi-asserted-by":"publisher","first-page":"1225","DOI":"10.1109\/LSP.2026.3673197","volume":"33","author":"Z He","year":"2026","unstructured":"He, Z., Hu, M., Lou, Y., Chen, C.: Tensor dynamic mode decomposition. IEEE Signal Process. Lett. 33, 1225\u20131229 (2026)","journal-title":"IEEE Signal Process. Lett."},{"issue":"11","key":"3301_CR16","doi-asserted-by":"publisher","DOI":"10.1063\/1.4901016","volume":"26","author":"MS Hemati","year":"2014","unstructured":"Hemati, M.S., Williams, M.O., Rowley, C.W.: Dynamic mode decomposition for large and streaming datasets. Phys. Fluids 26(11), 111701 (2014)","journal-title":"Phys. Fluids"},{"issue":"2","key":"3301_CR17","doi-asserted-by":"publisher","first-page":"A683","DOI":"10.1137\/100818893","volume":"34","author":"S Holtz","year":"2012","unstructured":"Holtz, S., Rohwedder, T., Schneider, R.: The alternating linear scheme for tensor optimization in the tensor train format. SIAM J. Sci. Comput. 34(2), A683\u2013A713 (2012)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"3301_CR18","doi-asserted-by":"publisher","DOI":"10.1088\/1367-2630\/12\/2\/025004","volume":"12","author":"R H\u00fcbener","year":"2010","unstructured":"H\u00fcbener, R., Nebendahl, V., D\u00fcr, W.: Concatenated tensor network states. New J. Phys. 12(2), 025004 (2010)","journal-title":"New J. Phys."},{"key":"3301_CR19","doi-asserted-by":"publisher","DOI":"10.1063\/1.4863670","volume":"26","author":"MR Jovanovi\u0107","year":"2014","unstructured":"Jovanovi\u0107, M.R., Schmid, P.J., Nichols, J.W.: Sparsity-promoting dynamic mode decomposition. Phys. Fluids 26, 024103 (2014)","journal-title":"Phys. Fluids"},{"issue":"7","key":"3301_CR20","doi-asserted-by":"publisher","first-page":"3359","DOI":"10.1088\/1361-6544\/aabc8f","volume":"31","author":"S Klus","year":"2018","unstructured":"Klus, S., Gel\u00df, P., Peitz, S., Sch\u00fctte, C.: Tensor-based dynamic mode decomposition. Nonlinearity 31(7), 3359 (2018)","journal-title":"Nonlinearity"},{"issue":"3","key":"3301_CR21","doi-asserted-by":"publisher","first-page":"455","DOI":"10.1137\/07070111X","volume":"51","author":"TG Kolda","year":"2009","unstructured":"Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455\u2013500 (2009)","journal-title":"SIAM Rev."},{"key":"3301_CR22","doi-asserted-by":"crossref","unstructured":"Kutz, J.N., Brunton, S.L., Brunton, B.W., Proctor, J.L.: Dynamic Mode Decomposition. Data-Driven Modeling of Complex Systems, vol. 149. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2016)","DOI":"10.1137\/1.9781611974508"},{"issue":"16","key":"3301_CR23","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2025.116952","volume":"474","author":"J Li","year":"2026","unstructured":"Li, J., Wang, X., Wang, K., Wei, Y.: Neural networks for solving least squares solution of polynomial systems with time-varying tensors. J. Comput. Appl. Math. 474(16), 116952 (2026)","journal-title":"J. Comput. Appl. Math."},{"key":"3301_CR24","doi-asserted-by":"crossref","unstructured":"Li, K.: Tensor train based higher-order dynamic mode decomposition-a new big data mining algorithm in energy networks. Ph.D. thesis, The University of Manchester (United Kingdom) (2023)","DOI":"10.3390\/math11081809"},{"issue":"2","key":"3301_CR25","doi-asserted-by":"publisher","first-page":"575","DOI":"10.4208\/cicp.OA-2023-0135","volume":"37","author":"Y Lin","year":"2025","unstructured":"Lin, Y., Sun, X., Nie, J., Chen, Y., Gao, Z.: An efficient reduced-order model based on dynamic mode decomposition for parameterized spatial high-dimensional pdes. Commun. Comput. Phys. 37(2), 575\u2013602 (2025)","journal-title":"Commun. Comput. Phys."},{"key":"3301_CR26","unstructured":"Matsumoto, D., Indinger, T.: On-the-fly algorithm for dynamic mode decomposition using incremental singular value decomposition and total least squares. Preprint at arXiv:1703.11004 (2017)"},{"key":"3301_CR27","doi-asserted-by":"crossref","unstructured":"Montangero, S., Montangero, E., Evenson: Introduction to Tensor Network Methods. Springer (2018)","DOI":"10.1007\/978-3-030-01409-4"},{"key":"3301_CR28","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1017\/S0022112003006694","volume":"497","author":"BR Noack","year":"2003","unstructured":"Noack, B.R., Afanasiev, K., Morzy\u0144ski, M., Tadmor, G., Thiele, F.: A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335\u2013363 (2003)","journal-title":"J. Fluid Mech."},{"issue":"5","key":"3301_CR29","doi-asserted-by":"publisher","first-page":"2295","DOI":"10.1137\/090752286","volume":"33","author":"IV Oseledets","year":"2011","unstructured":"Oseledets, I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295\u20132317 (2011)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"3301_CR30","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1137\/15M1013857","volume":"15","author":"JL Proctor","year":"2016","unstructured":"Proctor, J.L., Brunton, S.L., Kutz, J.N.: Dynamic mode decomposition with control. SIAM J. Appl. Dyn. Syst. 15(1), 142\u2013161 (2016)","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"3301_CR31","unstructured":"Qu, G., Shi, Y., Lale, S., Anandkumar, A., Wierman, A.: Stable online control of linear time-varying systems. In: Learning for Dynamics and Control, pp. 742\u2013753. PMLR (2021)"},{"issue":"03","key":"3301_CR32","doi-asserted-by":"publisher","first-page":"997","DOI":"10.1142\/S0218127405012429","volume":"15","author":"CW Rowley","year":"2005","unstructured":"Rowley, C.W.: Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifurc. Chaos 15(03), 997\u20131013 (2005)","journal-title":"Int. J. Bifurc. Chaos"},{"key":"3301_CR33","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1017\/S0022112009992059","volume":"641","author":"CW Rowley","year":"2009","unstructured":"Rowley, C.W., Mezi\u0107, I., Bagheri, S., Schlatter, P., Henningson, D.S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115\u2013127 (2009)","journal-title":"J. Fluid Mech."},{"key":"3301_CR34","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1017\/S0022112010001217","volume":"656","author":"PJ Schmid","year":"2010","unstructured":"Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5\u201328 (2010)","journal-title":"J. Fluid Mech."},{"key":"3301_CR35","doi-asserted-by":"crossref","unstructured":"Tjandra, A., Sakti, S., Nakamura, S.: Compressing recurrent neural network with tensor train. In: 2017 International Joint Conference on Neural Networks (IJCNN), pp. 4451\u20134458. IEEE (2017)","DOI":"10.1109\/IJCNN.2017.7966420"},{"issue":"2","key":"3301_CR36","doi-asserted-by":"publisher","first-page":"391","DOI":"10.3934\/jcd.2014.1.391","volume":"1","author":"JH Tu","year":"2014","unstructured":"Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: On dynamic mode decomposition: theory and applications. J. Comput. Dyn. 1(2), 391\u2013421 (2014)","journal-title":"J. Comput. Dyn."},{"issue":"1","key":"3301_CR37","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1137\/1038003","volume":"38","author":"L Vandenberghe","year":"1996","unstructured":"Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Rev. 38(1), 49\u201395 (1996)","journal-title":"SIAM Rev."},{"key":"3301_CR38","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1016\/j.neucom.2021.11.108","volume":"472","author":"X Wang","year":"2022","unstructured":"Wang, X., Mo, C., Qiao, S., Wei, Y.: Predefined-time convergent neural networks for solving the time-varying nonsingular multi-linear tensor equations. Neurocomputing 472, 68\u201384 (2022)","journal-title":"Neurocomputing"},{"key":"3301_CR39","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1007\/s40314-024-03070-1","volume":"44","author":"X Wang","year":"2025","unstructured":"Wang, X., Shan, J., Wei, Y.: Neural networks for total least squares solution of the time-varying linear systems. Comput. Appl. Math. 44, 106 (2025)","journal-title":"Comput. Appl. Math."},{"key":"3301_CR40","doi-asserted-by":"publisher","first-page":"16079","DOI":"10.1109\/TASE.2025.3575246","volume":"22","author":"X Wang","year":"2025","unstructured":"Wang, X., Stanimirovi\u0107, P.S., Wei, Y.: Dynamic approaches for finding least squares solution of time-varying multi-linear systems. IEEE Trans. Autom. Sci. Eng. 22, 16079\u201316090 (2025)","journal-title":"IEEE Trans. Autom. Sci. Eng."},{"key":"3301_CR41","doi-asserted-by":"publisher","DOI":"10.1142\/10950","volume-title":"Numerical and Symbolic Computations of Generalized Inverses","author":"Y Wei","year":"2018","unstructured":"Wei, Y., Stanimirovi\u0107, P., Petkovi\u0107, M.: Numerical and Symbolic Computations of Generalized Inverses. World Scientific, Hackensack, NJ (2018)"},{"key":"3301_CR42","doi-asserted-by":"publisher","first-page":"1307","DOI":"10.1007\/s00332-015-9258-5","volume":"25","author":"MO Williams","year":"2015","unstructured":"Williams, M.O., Kevrekidis, I.G., Rowley, C.W.: A data-driven approximation of the Koopman operator: extending dynamic mode decomposition. J. Nonlinear Sci. 25, 1307\u20131346 (2015)","journal-title":"J. Nonlinear Sci."},{"key":"3301_CR43","doi-asserted-by":"publisher","first-page":"20210686","DOI":"10.1098\/rsif.2021.0686","volume":"18","author":"Z Wu","year":"2021","unstructured":"Wu, Z., Brunton, S.L., Revzen, S.: Challenges in dynamic mode decomposition. J. R. Soc. Interface 18, 20210686 (2021)","journal-title":"J. R. Soc. Interface"},{"key":"3301_CR44","doi-asserted-by":"crossref","unstructured":"Xie, S., Miura, A., Ono, K.: Error-bounded scalable parallel tensor train decomposition. In: 2023 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW), pp. 345\u2013353. IEEE (2023)","DOI":"10.1109\/IPDPSW59300.2023.00064"},{"key":"3301_CR45","unstructured":"Yang, Y., Krompass, D., Tresp, V.: Tensor-train recurrent neural networks for video classification. In: International Conference on Machine Learning, pp. 3891\u20133900. PMLR (2017)"},{"key":"3301_CR46","first-page":"1","volume":"74","author":"Y Yin","year":"2025","unstructured":"Yin, Y., Yuan, R., Lv, Y., Wu, H., Li, H., Zhu, W.: Low-rank tensor train dynamic mode decomposition: an enhanced multivariate signal processing method for mechanical fault diagnosis. IEEE Trans. Instrum. Meas. 74, 1\u201316 (2025)","journal-title":"IEEE Trans. Instrum. Meas."},{"issue":"3","key":"3301_CR47","doi-asserted-by":"publisher","first-page":"1586","DOI":"10.1137\/18M1192329","volume":"18","author":"H Zhang","year":"2019","unstructured":"Zhang, H., Rowley, C.W., Deem, E.A., Cattafesta, L.N.: Online dynamic mode decomposition for time-varying systems. SIAM J. Appl. Dyn. Syst. 18(3), 1586\u20131609 (2019)","journal-title":"SIAM J. Appl. Dyn. Syst."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03301-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-026-03301-z","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-026-03301-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,27]],"date-time":"2026-06-27T16:31:42Z","timestamp":1782577902000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-026-03301-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,19]]},"references-count":47,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,7]]}},"alternative-id":["3301"],"URL":"https:\/\/doi.org\/10.1007\/s10915-026-03301-z","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,5,19]]},"assertion":[{"value":"3 February 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 April 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 April 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 May 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"3"}}