{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,21]],"date-time":"2026-10-21T19:45:18Z","timestamp":1792611918700,"version":"4.5.0"},"reference-count":61,"publisher":"AIP Publishing","issue":"10","license":[{"start":{"date-parts":[[2026,10,19]],"date-time":"2026-10-19T00:00:00Z","timestamp":1792368000000},"content-version":"vor","delay-in-days":18,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2026,10,1]]},"abstract":"<jats:p>We show how the recent extension of spectral submanifold (SSM) theory to delay differential equations (DDEs) enables data-driven model reduction of nonlinear delay systems. First, using a scalar DDE with a single discrete delay, we compare equation-based and data-driven SSM reductions, to illustrate the need for the latter. We then use the same algorithm to obtain purely data-driven, SSM-reduced, delay-free ordinary differential equation models for several nonlinear delayed systems. Our approach requires no information about the form of the underlying DDE, or about the number and magnitude of the delays it contains. Our SSM-reduced, low-dimensional models remain predictive even for chaotic dynamics. We also illustrate the use of parametric SSM reduction to capture bifurcations in systems with both distributed and discrete delays. Finally, we extend the theoretical underpinning of delayed SSM reductions to non-autonomous systems with periodic delays and apply these results to experimental data from a control system with feedback delay and quantization.<\/jats:p>","DOI":"10.1063\/5.0344712","type":"journal-article","created":{"date-parts":[[2026,10,20]],"date-time":"2026-10-20T16:23:56Z","timestamp":1792513436000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":1,"title":["Data-driven reduced modeling of delayed dynamical systems via spectral submanifolds"],"prefix":"10.1063","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-9580-0938","authenticated-orcid":false,"given":"Giacomo","family":"Abbasciano","sequence":"first","affiliation":[{"name":"Department of Mechanical and Process Engineering, ETH Zurich , 8092 Zurich,","place":["Switzerland"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2009-705X","authenticated-orcid":false,"given":"Gergely","family":"Buza","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Process Engineering, ETH Zurich , 8092 Zurich,","place":["Switzerland"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1260-877X","authenticated-orcid":false,"given":"George","family":"Haller","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Process Engineering, ETH Zurich , 8092 Zurich,","place":["Switzerland"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,10,20]]},"reference":[{"key":"2026102012235465700_c1","doi-asserted-by":"publisher","first-page":"19435","DOI":"10.1038\/s41598-022-23822-5","article-title":"A time-delayed model for the spread of COVID-19 with vaccination","volume":"12","year":"2022","journal-title":"Sci. Rep."},{"issue":"27","key":"2026102012235465700_c2","doi-asserted-by":"publisher","first-page":"67","DOI":"10.1016\/j.ifacol.2024.10.301","article-title":"Reduction to spectral submanifolds in guided car-following with time delay","volume":"58","year":"2024","journal-title":"IFAC-PapersOnLine"},{"issue":"13","key":"2026102012235465700_c3","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1016\/j.ifacol.2025.10.029","article-title":"Discretization based spectral submanifolds in guided car-following with time delay","volume":"59","year":"2025","journal-title":"IFAC-PapersOnLine"},{"key":"2026102012235465700_c4"},{"issue":"11","key":"2026102012235465700_c5","doi-asserted-by":"publisher","first-page":"114306","DOI":"10.1063\/1.5006777","article-title":"Quantization improves stabilization of dynamical systems with delayed feedback","volume":"27","year":"2017","journal-title":"Chaos"},{"key":"2026102012235465700_c6","volume-title":"Semi-Discretization for Time-Delay Systems","year":"2011"},{"key":"2026102012235465700_c7","doi-asserted-by":"publisher","first-page":"588","DOI":"10.1177\/10775463221148893","article-title":"Digital stability of the furuta pendulum based on angle detection","volume":"30","year":"2024","journal-title":"J. Vib. Control"},{"key":"2026102012235465700_c8","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/j.ifacol.2015.09.359","article-title":"Semi-discretization and the time-delayed PDA feedback control of human balance","volume":"48","year":"2015","journal-title":"IFAC-Pap. OnLine"},{"key":"2026102012235465700_c9","doi-asserted-by":"crossref","unstructured":"G.\n            Buza\n           and G.Haller, \u201cExistence of spectral submanifolds in time delay systems,\u201d J. Nonlin. Sci. (in press) (2025).10.48550\/arXiv.2512.22062","DOI":"10.1007\/s00332-026-10327-y"},{"issue":"4300","key":"2026102012235465700_c10","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1126\/science.267326","article-title":"Oscillation and chaos in physiological control systems","volume":"197","year":"1977","journal-title":"Science"},{"key":"2026102012235465700_c11"},{"issue":"3","key":"2026102012235465700_c12","doi-asserted-by":"publisher","first-page":"1493","DOI":"10.1007\/s11071-016-2974-z","article-title":"Nonlinear normal modes and spectral submanifolds: Existence, uniqueness and use in model reduction","volume":"86","year":"2016","journal-title":"Nonlinear Dyn."},{"issue":"1","key":"2026102012235465700_c13","doi-asserted-by":"publisher","first-page":"872","DOI":"10.1038\/s41467-022-28518-y","article-title":"Data-driven modeling and prediction of non-linearizable dynamics via spectral submanifolds","volume":"13","year":"2022","journal-title":"Nat. Commun."},{"key":"2026102012235465700_c14","doi-asserted-by":"publisher","first-page":"A48","DOI":"10.1017\/jfm.2023.956","article-title":"Capturing the edge of chaos as a spectral submanifold in pipe flows","volume":"979","year":"2024","journal-title":"J. Fluid Mech."},{"key":"2026102012235465700_c15","doi-asserted-by":"publisher","first-page":"A1","DOI":"10.1017\/jfm.2025.10795","article-title":"Joint reduced model for the laminar and chaotic attractors in plane couette flow","volume":"1023","year":"2025","journal-title":"J. Fluid Mech."},{"issue":"1","key":"2026102012235465700_c16","doi-asserted-by":"publisher","first-page":"25991","DOI":"10.1038\/s41598-024-77639-5","article-title":"Data-driven modeling of subharmonic forced response due to nonlinear resonance","volume":"14","year":"2024","journal-title":"Sci. Rep."},{"key":"2026102012235465700_c17","doi-asserted-by":"publisher","first-page":"104753","DOI":"10.1016\/j.ijnonlinmec.2024.104753","article-title":"Model reduction to spectral submanifolds in piecewise smooth dynamical systems","volume":"163","year":"2024","journal-title":"Int. J. Nonlinear Mech."},{"key":"2026102012235465700_c18","doi-asserted-by":"publisher","first-page":"24539","DOI":"10.1007\/s11071-025-11378-8","article-title":"Data-driven reduction of the finite-element model of a tribomechadynamics benchmark problem","volume":"113","year":"2025","journal-title":"Nonlinear Dyn."},{"key":"2026102012235465700_c19","doi-asserted-by":"publisher","first-page":"119660","DOI":"10.1016\/j.jsv.2026.119660","article-title":"Data-driven modeling of global bifurcations and chaos in a mechanical system under delayed and quantized control","volume":"628","year":"2026","journal-title":"J. Sound Vib."},{"key":"2026102012235465700_c20"},{"issue":"2","key":"2026102012235465700_c21","doi-asserted-by":"publisher","first-page":"4074","DOI":"10.1016\/j.ifacol.2023.10.1734","article-title":"Practical deployment of spectral submanifold reduction for optimal control of high-dimensional systems","volume":"56","year":"2023","journal-title":"IFAC-PapersOnLine"},{"key":"2026102012235465700_c22","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1038\/s44182-025-00021-8","article-title":"Discovering dominant dynamics for nonlinear continuum robot control","volume":"3","year":"2025","journal-title":"npj Robot."},{"key":"2026102012235465700_c23","doi-asserted-by":"crossref","unstructured":"R. S.\n            Kaundinya\n          , J. I.Alora, J. G.Matt, L. A.Pabon, M.Pavone, and G.Haller, \u201cData-driven soft robot control via adiabatic spectral submanifolds,\u201d arXiv:2503.10919 (2025).","DOI":"10.1177\/02783649261461630"},{"key":"2026102012235465700_c24","doi-asserted-by":"crossref","unstructured":"L.\n            Bettini\n          , A.Kazemipour, R. K.Katzschmann, and G.Haller, \u201cNonlinear spectral modeling and control of soft-robotic muscles from data,\u201d arXiv:2601.03247 (2026).","DOI":"10.1038\/s41467-026-77664-0"},{"key":"2026102012235465700_c25","doi-asserted-by":"crossref","unstructured":"J.\n            King\n          , B.Kasz\u00e1s, G.Buza, W.Jussiau, and G.Haller, \u201cParametric spectral submanifolds across hopf bifurcations with applications to fluid dynamics,\u201d arXiv:2603.19079 (2026).","DOI":"10.1063\/5.0334774"},{"issue":"4","key":"2026102012235465700_c26","doi-asserted-by":"publisher","first-page":"513","DOI":"10.1007\/BF00045711","article-title":"Quasiperiodic oscillations in robot dynamics","volume":"8","year":"1995","journal-title":"Nonlinear Dyn."},{"key":"2026102012235465700_c27"},{"key":"2026102012235465700_c28","doi-asserted-by":"publisher","first-page":"14265","DOI":"10.1007\/s11071-025-10902-0","article-title":"Spectral submanifolds in time delay systems","volume":"113","year":"2025","journal-title":"Nonlinear Dyn."},{"issue":"15","key":"2026102012235465700_c29","doi-asserted-by":"publisher","first-page":"3932","DOI":"10.1073\/pnas.1517384113","article-title":"Discovering governing equations from data by sparse identification of nonlinear dynamical systems","volume":"113","year":"2016","journal-title":"Proc. Natl. Acad. Sci."},{"issue":"36","key":"2026102012235465700_c30","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1016\/j.ifacol.2022.11.337","article-title":"Learning time delay systems with neural ordinary differential equations","volume":"55","year":"2022","journal-title":"IFAC-PapersOnLine"},{"issue":"5","key":"2026102012235465700_c31","doi-asserted-by":"publisher","first-page":"3505","DOI":"10.1007\/s11071-023-09206-y","article-title":"Learn from one and predict all: Single trajectory learning for time delay systems","volume":"112","year":"2024","journal-title":"Nonlinear Dyn."},{"key":"2026102012235465700_c32","unstructured":"D.\n            Breda\n          , X. A.Ji, G.Orosz, and M.Tanveer, \u201cData-driven methods for delay differential equations,\u201d arXiv:2512.04894 (2025)."},{"key":"2026102012235465700_c33","doi-asserted-by":"publisher","first-page":"116439","DOI":"10.1016\/j.cam.2024.116439","article-title":"Data-driven discovery of delay differential equations with discrete delays","volume":"461","year":"2025","journal-title":"J. Comput. Appl. Math."},{"key":"2026102012235465700_c34","doi-asserted-by":"publisher","first-page":"20220556","DOI":"10.1098\/rspa.2022.0556","article-title":"Sindy for delay-differential equations: Application to model bacterial zinc response","volume":"479","year":"2023","journal-title":"Proc. R. Soc. A"},{"key":"2026102012235465700_c35"},{"key":"2026102012235465700_c36","doi-asserted-by":"publisher","first-page":"2645","DOI":"10.1007\/s11071-020-05891-1","article-title":"Invariant spectral foliations with applications to model order reduction and synthesis","volume":"101","year":"2020","journal-title":"Nonlinear Dyn."},{"issue":"4","key":"2026102012235465700_c37","doi-asserted-by":"publisher","first-page":"043135","DOI":"10.1063\/5.0243849","article-title":"Data-driven nonlinear model reduction to spectral submanifolds via oblique projection","volume":"35","year":"2025","journal-title":"Chaos"},{"issue":"5","key":"2026102012235465700_c38","doi-asserted-by":"publisher","first-page":"053122","DOI":"10.1063\/5.0319241","article-title":"General oblique projections for model reduction via spectral submanifolds","volume":"36","year":"2026","journal-title":"Chaos"},{"key":"2026102012235465700_c39"},{"issue":"1-2","key":"2026102012235465700_c40","doi-asserted-by":"publisher","first-page":"189","DOI":"10.1016\/0167-2789(83)90298-1","article-title":"Measuring the strangeness of strange attractors","volume":"9","year":"1983","journal-title":"Phys. D"},{"issue":"3-4","key":"2026102012235465700_c41","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1007\/BF01053745","article-title":"Embedology","volume":"65","year":"1991","journal-title":"J. Stat. Phys."},{"issue":"9","key":"2026102012235465700_c42","doi-asserted-by":"publisher","first-page":"094606","DOI":"10.1103\/PhysRevFluids.2.094606","article-title":"Lyapunov exponent as a metric for assessing the dynamic content and predictability of large-eddy simulations","volume":"2","year":"2017","journal-title":"Phys. Rev. Fluids"},{"issue":"3","key":"2026102012235465700_c43","doi-asserted-by":"publisher","first-page":"033140","DOI":"10.1063\/5.0179741","article-title":"Data-driven modeling and forecasting of chaotic dynamics on inertial manifolds constructed as spectral submanifolds","volume":"34","year":"2024","journal-title":"Chaos"},{"key":"2026102012235465700_c44","doi-asserted-by":"publisher","first-page":"R7","DOI":"10.1017\/jfm.2024.411","article-title":"Data-driven modelling of the regular and chaotic dynamics of an inverted flag from experiments","volume":"987","year":"2024","journal-title":"J. Fluid Mech."},{"key":"2026102012235465700_c45","doi-asserted-by":"publisher","first-page":"22079","DOI":"10.1007\/s11071-023-08705-2","article-title":"Model reduction for nonlinearizable dynamics via delay-embedded spectral submanifolds","volume":"111","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"2026102012235465700_c46","doi-asserted-by":"publisher","first-page":"10881","DOI":"10.1007\/s11071-024-10687-8","article-title":"Modeling nonlinear dynamics from videos","volume":"113","year":"2025","journal-title":"Nonlinear Dyn."},{"issue":"4","key":"2026102012235465700_c47","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1111\/j.1749-6632.1948.tb39854.x","article-title":"Circular causal systems in ecology","volume":"50","year":"1948","journal-title":"Ann. N. Y. Acad. Sci."},{"key":"2026102012235465700_c48","doi-asserted-by":"publisher","first-page":"118923","DOI":"10.1016\/j.jsv.2024.118923","article-title":"Nonlinear model reduction to random spectral submanifolds in random vibrations","volume":"600","year":"2025","journal-title":"J. Sound Vib."},{"issue":"16","key":"2026102012235465700_c49","doi-asserted-by":"publisher","first-page":"272","DOI":"10.1016\/j.ifacol.2022.09.036","article-title":"TDS-CONTROL: A MATLAB package for the analysis and controller-design of time-delay systems","volume":"55","year":"2022","journal-title":"IFAC-PapersOnLine"},{"key":"2026102012235465700_c50","unstructured":"P.\n            Appeltans\n           and W.Michiels, \u201cAnalysis and controller-design of time-delay systems using TDS-CONTROL: A tutorial and manual,\u201d arXiv:2305.00341 (2023)."},{"issue":"3","key":"2026102012235465700_c51","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1007\/BF00280975","article-title":"Time delays in single species growth models","volume":"4","year":"1977","journal-title":"J. Math. Biol."},{"key":"2026102012235465700_c52","doi-asserted-by":"publisher","first-page":"415","DOI":"10.1007\/BF02440161","article-title":"Micro-chaos in digital control","volume":"6","year":"1996","journal-title":"J. Nonlinear Sci."},{"key":"2026102012235465700_c53"},{"issue":"9","key":"2026102012235465700_c54","doi-asserted-by":"publisher","first-page":"7941","DOI":"10.1007\/s11071-022-08014-0","article-title":"Fast data-driven model reduction for nonlinear dynamical systems","volume":"111","year":"2023","journal-title":"Nonlinear Dyn."},{"issue":"1","key":"2026102012235465700_c55","doi-asserted-by":"publisher","first-page":"815","DOI":"10.1007\/s10884-023-10289-9","article-title":"Periodic center manifolds for DDEs in the light of suns and stars","volume":"37","year":"2025","journal-title":"J. Dyn. Differ. Equ."},{"key":"2026102012235465700_c56"},{"key":"2026102012235465700_c57","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1006\/jdeq.1997.3255","article-title":"Invariant foliations for C1 semigroups in Banach spaces","volume":"139","year":"1997","journal-title":"J. Differ. Equ."},{"key":"2026102012235465700_c58"},{"key":"2026102012235465700_c59","doi-asserted-by":"publisher","first-page":"557","DOI":"10.1112\/jlms\/s2-21.3.557","article-title":"A new proof of the pseudostable manifold theorem","volume":"s2-21","year":"1980","journal-title":"J. Lond. Math. Soc."},{"key":"2026102012235465700_c60","doi-asserted-by":"publisher","first-page":"1052","DOI":"10.1137\/23M154858X","article-title":"Spectral submanifolds of the Navier\u2013Stokes equations","volume":"23","year":"2024","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"2026102012235465700_c61","year":"2022"}],"container-title":["Chaos: An Interdisciplinary Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0344712\/21175867\/103103_1_5.0344712.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/cha\/article-pdf\/doi\/10.1063\/5.0344712\/21175867\/103103_1_5.0344712.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,10,20]],"date-time":"2026-10-20T16:23:57Z","timestamp":1792513437000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/cha\/article\/36\/10\/103103\/3406793\/Data-driven-reduced-modeling-of-delayed-dynamical"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10,1]]},"references-count":61,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2026,10,1]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0344712","relation":{},"ISSN":["1054-1500","1089-7682"],"issn-type":[{"value":"1054-1500","type":"print"},{"value":"1089-7682","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,10]]},"published":{"date-parts":[[2026,10,1]]},"article-number":"103103"}}