{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T06:05:04Z","timestamp":1780725904857,"version":"3.54.1"},"reference-count":104,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T00:00:00Z","timestamp":1790812800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,5,25]],"date-time":"2026-05-25T00:00:00Z","timestamp":1779667200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Communications in Nonlinear Science and Numerical Simulation"],"published-print":{"date-parts":[[2026,10]]},"DOI":"10.1016\/j.cnsns.2026.110137","type":"journal-article","created":{"date-parts":[[2026,5,25]],"date-time":"2026-05-25T16:15:15Z","timestamp":1779725715000},"page":"110137","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"P3","title":["Surrogate normal-forms for the numerical bifurcation and stability analysis of Navier-Stokes flows via machine learning"],"prefix":"10.1016","volume":"161","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2989-4397","authenticated-orcid":false,"given":"Alessandro","family":"Della Pia","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9840-0018","authenticated-orcid":false,"given":"Dimitrios G.","family":"Patsatzis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0810-8812","authenticated-orcid":false,"given":"Gianluigi","family":"Rozza","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4888-467X","authenticated-orcid":false,"given":"Lucia","family":"Russo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9568-3355","authenticated-orcid":false,"given":"Constantinos","family":"Siettos","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"key":"10.1016\/j.cnsns.2026.110137_bib0001","doi-asserted-by":"crossref","unstructured":"Haller G.. Modeling nonlinear dynamics from equations and data with applications to solids, fluids, and controls. 2025.","DOI":"10.1137\/1.9781611978353"},{"issue":"265-284","key":"10.1016\/j.cnsns.2026.110137_bib0002","first-page":"25","article-title":"Auto: a program for the automatic bifurcation analysis of autonomous systems","volume":"30","author":"Doedel","year":"1981","journal-title":"Congr Numer"},{"key":"10.1016\/j.cnsns.2026.110137_bib0003","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1145\/779359.779362","article-title":"Matcont: a matlab package for numerical bifurcation analysis of odes","volume":"29","author":"Dhooge","year":"2003","journal-title":"ACM Trans Math Softw"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110137_bib0004","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1137\/030600746","article-title":"Numerical continuation of bifurcations of limit cycles in MATLAB","volume":"27","author":"Govaerts","year":"2005","journal-title":"SIAM J Sci Comput"},{"issue":"1-3","key":"10.1016\/j.cnsns.2026.110137_bib0005","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.jnnfm.2007.11.002","article-title":"A timestepper approach for the systematic bifurcation and stability analysis of polymer extrusion dynamics","volume":"151","author":"Kavousanakis","year":"2008","journal-title":"J Non-Newton Fluid Mech"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110137_bib0006","doi-asserted-by":"crossref","first-page":"13","DOI":"10.1016\/j.jcp.2004.04.018","article-title":"Newton-Krylov continuation of periodic orbits for navier\u2013stokes flows","volume":"201","author":"S\u00e1nchez","year":"2004","journal-title":"J Comput Phys"},{"issue":"43-44","key":"10.1016\/j.cnsns.2026.110137_bib0007","doi-asserted-by":"crossref","first-page":"3480","DOI":"10.1016\/j.cma.2007.11.033","article-title":"Newton-Krylov solvers for the equation-free computation of coarse traveling waves","volume":"197","author":"Samaey","year":"2008","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110137_bib0008","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1006\/jcph.2002.7072","article-title":"A continuation and bifurcation technique for navier\u2013stokes flows","volume":"180","author":"Sanchez","year":"2002","journal-title":"J Comput Phys"},{"issue":"01","key":"10.1016\/j.cnsns.2026.110137_bib0009","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1142\/S0218127410025399","article-title":"On the multiple shooting continuation of periodic orbits by newton\u2013krylov methods","volume":"20","author":"S\u00e1nchez","year":"2010","journal-title":"Int J Bifurc Chaos"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110137_bib0010","doi-asserted-by":"crossref","first-page":"674","DOI":"10.1137\/140981010","article-title":"Continuation of bifurcations of periodic orbits for large-scale systems","volume":"14","author":"Net","year":"2015","journal-title":"SIAM J Appl Dyn Syst"},{"key":"10.1016\/j.cnsns.2026.110137_bib0011","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1016\/j.jcp.2012.12.034","article-title":"Matrix-free continuation of limit cycles for bifurcation analysis of large thermoacoustic systems","volume":"240","author":"Waugh","year":"2013","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110137_bib0012","series-title":"Navier-Stokes equations and nonlinear functional analysis","author":"Temam","year":"1995"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110137_bib0013","doi-asserted-by":"crossref","first-page":"540","DOI":"10.1016\/0022-247X(90)90061-J","article-title":"On approximate inertial manifolds to the navier-stokes equations","volume":"149","author":"Titi","year":"1990","journal-title":"J Math Anal Appl"},{"issue":"4","key":"10.1016\/j.cnsns.2026.110137_bib0014","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1002\/num.20249","article-title":"Numerical solution of parametrized navier\u2013stokes equations by reduced basis methods","volume":"23","author":"Quarteroni","year":"2007","journal-title":"Numer Methods Partial Differ Equ Int J"},{"key":"10.1016\/j.cnsns.2026.110137_bib0015","series-title":"Reduced order methods for modeling and computational reduction","volume":"vol. 9","author":"Quarteroni","year":"2014"},{"key":"10.1016\/j.cnsns.2026.110137_bib0016","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1038\/s43588-022-00264-7","article-title":"Enhancing computational fluid dynamics with machine learning","volume":"2","author":"Vinuesa","year":"2022","journal-title":"Nat Comput Sci"},{"key":"10.1016\/j.cnsns.2026.110137_bib0017","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2020.109513","article-title":"Data-driven POD-Galerkin reduced order model for turbulent flows","volume":"416","author":"Hijazi","year":"2020","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110137_bib0018","doi-asserted-by":"crossref","DOI":"10.1016\/j.compfluid.2023.105813","article-title":"An artificial neural network approach to bifurcating phenomena in computational fluid dynamics","volume":"254","author":"Pichi","year":"2023","journal-title":"Comput Fluids"},{"key":"10.1016\/j.cnsns.2026.110137_bib0019","series-title":"Practical bifurcation and stability analysis","volume":"vol. 5","author":"Seydel","year":"2009"},{"issue":"5","key":"10.1016\/j.cnsns.2026.110137_bib0020","doi-asserted-by":"crossref","first-page":"1015","DOI":"10.1090\/S0002-9904-1970-12537-X","article-title":"Invariant manifolds","volume":"76","author":"Hirsch","year":"1970","journal-title":"Bull Am Math Soc"},{"key":"10.1016\/j.cnsns.2026.110137_bib0021","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/s002050200200","article-title":"Invariant manifolds and the long-time asymptotics of the navier-stokes and vorticity equations on r2","volume":"163","author":"Gallay","year":"2002","journal-title":"Arch Ration Mech Anal"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110137_bib0022","doi-asserted-by":"crossref","first-page":"377","DOI":"10.3934\/jcd.2014.1.377","article-title":"Equation-free computation of coarse-grained center manifolds of microscopic simulators","volume":"1","author":"Siettos","year":"2014","journal-title":"J Comput Dyn"},{"key":"10.1016\/j.cnsns.2026.110137_bib0023","series-title":"Normally hyperbolic invariant manifolds in dynamical systems","volume":"vol. 105","author":"Wiggins","year":"2013"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110137_bib0024","doi-asserted-by":"crossref","first-page":"1335","DOI":"10.1007\/s11075-021-01155-0","article-title":"A numerical method for the approximation of stable and unstable manifolds of microscopic simulators","volume":"89","author":"Siettos","year":"2022","journal-title":"Numer Algorithms"},{"key":"10.1016\/j.cnsns.2026.110137_bib0025","doi-asserted-by":"crossref","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","author":"Haller","year":"2016","journal-title":"Nonlinear Dyn"},{"issue":"2213","key":"10.1016\/j.cnsns.2026.110137_bib0026","article-title":"Explicit backbone curves from spectral submanifolds of forced-damped nonlinear mechanical systems","volume":"474","author":"Breunung","year":"2018","journal-title":"Proc R Soc A Math Phys Eng Sci"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110137_bib0027","doi-asserted-by":"crossref","first-page":"1052","DOI":"10.1137\/23M154858X","article-title":"Spectral submanifolds of the navier\u2013stokes equations","volume":"23","author":"Buza","year":"2024","journal-title":"SIAM J Appl Dyn Syst"},{"key":"10.1016\/j.cnsns.2026.110137_bib0028","doi-asserted-by":"crossref","unstructured":"Colombo A., Vizzaccaro A., Touz\u00e9 C., de F. Stabile A., Pastur L., Frangi A.. Reduced order modelling of hopf bifurcations for the Navier\u2013Stokes equations through invariant manifolds. 2025, arXiv: 251026542.","DOI":"10.1103\/w6z2-m5wk"},{"key":"10.1016\/j.cnsns.2026.110137_bib0029","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/j.compfluid.2018.01.035","article-title":"Finite volume POD-Galerkin stabilised reduced order methods for the parametrised incompressible Navier\u2013Stokes equations","volume":"173","author":"Stabile","year":"2018","journal-title":"Comput Fluids"},{"key":"10.1016\/j.cnsns.2026.110137_bib0030","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.jcp.2018.02.037","article-title":"Non-intrusive reduced order modelling of nonlinear problems using neural networks","volume":"363","author":"Hesthaven","year":"2018","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110137_bib0031","doi-asserted-by":"crossref","DOI":"10.1016\/j.compfluid.2022.105536","article-title":"A POD-Galerkin reduced order model for the Navier\u2013Stokes equations in stream function-vorticity formulation","volume":"244","author":"Girfoglio","year":"2022","journal-title":"Comput Fluids"},{"issue":"10","key":"10.1016\/j.cnsns.2026.110137_bib0032","doi-asserted-by":"crossref","first-page":"2337","DOI":"10.1063\/1.857881","article-title":"Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders","volume":"3","author":"Deane","year":"1991","journal-title":"Phys Fluids A Fluid Dyn"},{"key":"10.1016\/j.cnsns.2026.110137_bib0033","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1017\/S0022112002007991","article-title":"A low-dimensional model for simulating three-dimensional cylinder flow","volume":"458","author":"Ma","year":"2002","journal-title":"J Fluid Mech"},{"issue":"03","key":"10.1016\/j.cnsns.2026.110137_bib0034","doi-asserted-by":"crossref","first-page":"997","DOI":"10.1142\/S0218127405012429","article-title":"Model reduction for fluids, using balanced proper orthogonal decomposition","volume":"15","author":"Rowley","year":"2005","journal-title":"Int J Bifurc Chaos"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110137_bib0035","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1146\/annurev-fluid-010719-060214","article-title":"Machine learning for fluid mechanics","volume":"52","author":"Brunton","year":"2020","journal-title":"Annu Rev Fluid Mech"},{"issue":"21","key":"10.1016\/j.cnsns.2026.110137_bib0036","doi-asserted-by":"crossref","first-page":"7426","DOI":"10.1073\/pnas.0500334102","article-title":"Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps","volume":"102","author":"Coifman","year":"2005","journal-title":"Proc Natl Acad Sci"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110137_bib0037","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.acha.2005.07.005","article-title":"Geometric harmonics: a novel tool for multiscale out-of-sample extension of empirical functions","volume":"21","author":"Coifman","year":"2006","journal-title":"Appl Comput Harmon Anal"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110137_bib0038","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.acha.2005.07.004","article-title":"Diffusion maps, spectral clustering and reaction coordinates of dynamical systems","volume":"21","author":"Nadler","year":"2006","journal-title":"Appl Comput Harmon Anal"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110137_bib0039","doi-asserted-by":"crossref","first-page":"759","DOI":"10.1016\/j.acha.2015.06.008","article-title":"Parsimonious representation of nonlinear dynamical systems through manifold learning: a chemotaxis case study","volume":"44","author":"Dsilva","year":"2018","journal-title":"Appl Comput Harmon Anal"},{"issue":"34","key":"10.1016\/j.cnsns.2026.110137_bib0040","first-page":"1","article-title":"Numerical bifurcation analysis of PDEs from lattice boltzmann model simulations: a parsimonious machine learning approach","volume":"92","author":"Galaris","year":"2022","journal-title":"J Sci Comput"},{"issue":"A41","key":"10.1016\/j.cnsns.2026.110137_bib0041","first-page":"1","article-title":"Physics-agnostic and physics-infused machine learning for thin films flows: modelling, and predictions from small data","volume":"975","author":"Martin-Linares","year":"2023","journal-title":"J Fluid Mech"},{"key":"10.1016\/j.cnsns.2026.110137_bib0042","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2023.111953","article-title":"Data-driven control of agent-based models: an equation\/variable-free machine learning approach","volume":"478","author":"Patsatzis","year":"2023","journal-title":"J Comput Phys"},{"issue":"105187","key":"10.1016\/j.cnsns.2026.110137_bib0043","first-page":"1","article-title":"Learning the latent dynamics of fluid flows from high-fidelity numerical simulations using parsimonious diffusion maps","volume":"36","author":"Della Pia","year":"2024","journal-title":"Phys Fluids"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110137_bib0044","doi-asserted-by":"crossref","first-page":"112","DOI":"10.3390\/pr2010112","article-title":"Reduced models in chemical kinetics via nonlinear data-mining","volume":"2","author":"Chiavazzo","year":"2014","journal-title":"Processes"},{"issue":"8","key":"10.1016\/j.cnsns.2026.110137_bib0045","doi-asserted-by":"crossref","DOI":"10.1063\/5.0094887","article-title":"Time-series forecasting using manifold learning, radial basis function interpolation, and geometric harmonics","volume":"32","author":"Papaioannou","year":"2022","journal-title":"Chaos"},{"key":"10.1016\/j.cnsns.2026.110137_bib0046","first-page":"3870","article-title":"Scalable gradients for stochastic differential equations","volume":"108","author":"Li","year":"2020","journal-title":"Int Conf Artif Intell Stat"},{"issue":"4","key":"10.1016\/j.cnsns.2026.110137_bib0047","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1038\/s42256-022-00464-w","article-title":"Multiscale simulations of complex systems by learning their effective dynamics","volume":"4","author":"Vlachas","year":"2022","journal-title":"Nat Mach Intell"},{"issue":"12","key":"10.1016\/j.cnsns.2026.110137_bib0048","doi-asserted-by":"crossref","first-page":"1113","DOI":"10.1038\/s42256-022-00575-4","article-title":"Data-driven discovery of intrinsic dynamics","volume":"4","author":"Floryan","year":"2022","journal-title":"Nat Mach Intell"},{"key":"10.1016\/j.cnsns.2026.110137_bib0049","doi-asserted-by":"crossref","DOI":"10.1016\/j.eswa.2022.117038","article-title":"Towards extraction of orthogonal and parsimonious non-linear modes from turbulent flows","volume":"202","author":"Eivazi","year":"2022","journal-title":"Expert Syst Appl"},{"key":"10.1016\/j.cnsns.2026.110137_bib0050","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2024.112910","article-title":"Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era","volume":"506","author":"Koronaki","year":"2024","journal-title":"J Comput Phys"},{"issue":"15","key":"10.1016\/j.cnsns.2026.110137_bib0051","doi-asserted-by":"crossref","first-page":"3932","DOI":"10.1073\/pnas.1517384113","article-title":"Discovering governing equations from data by sparse identification of nonlinear dynamical systems","volume":"113","author":"Brunton","year":"2016","journal-title":"Proc Natl Acad Sci"},{"key":"10.1016\/j.cnsns.2026.110137_bib0052","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1007\/s00162-020-00528-w","article-title":"Machine-learning-based reduced-order modeling for unsteady flows around bluff bodies of various shapes","volume":"34","author":"Hasegawa","year":"2020","journal-title":"Theor Comput Fluid Dyn"},{"key":"10.1016\/j.cnsns.2026.110137_bib0053","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1007\/s00162-020-00536-w","article-title":"Data-driven modeling of the chaotic thermal convection in an annular thermosyphon","volume":"34","author":"Loiseau","year":"2020","journal-title":"Theor Comput Fluid Dyn"},{"key":"10.1016\/j.cnsns.2026.110137_bib0054","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1126\/sciadv.abm4786","article-title":"An empirical mean-field model of symmetry-breaking in a turbulent wake","volume":"8","author":"Callaham","year":"2022","journal-title":"Sci Adv"},{"issue":"45","key":"10.1016\/j.cnsns.2026.110137_bib0055","doi-asserted-by":"crossref","first-page":"22445","DOI":"10.1073\/pnas.1906995116","article-title":"Data-driven discovery of coordinates and governing equations","volume":"116","author":"Champion","year":"2019","journal-title":"Proc Natl Acad Sci"},{"key":"10.1016\/j.cnsns.2026.110137_bib0056","doi-asserted-by":"crossref","DOI":"10.1016\/j.oceaneng.2024.118639","article-title":"Variational autoencoders and transformers for multivariate time-series generative modeling and forecasting: applications to vortex-induced vibrations","volume":"310","author":"Mentzelopoulos","year":"2024","journal-title":"Ocean Eng"},{"key":"10.1016\/j.cnsns.2026.110137_bib0057","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2023.116072","article-title":"Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions","volume":"411","author":"Conti","year":"2023","journal-title":"Comput Methods Appl Mech Eng"},{"key":"10.1016\/j.cnsns.2026.110137_bib0058","doi-asserted-by":"crossref","DOI":"10.1063\/1.5128231","article-title":"On learning hamiltonian systems from data","volume":"29","author":"Bertalan","year":"2019","journal-title":"Chaos: Interdiscip J Nonlinear Sci"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110137_bib0059","doi-asserted-by":"crossref","DOI":"10.1063\/1.5126869","article-title":"Coarse-scale PDEs from fine-scale observations via machine learning","volume":"30","author":"Lee","year":"2020","journal-title":"Chaos Interdiscip J Nonlinear Sci"},{"issue":"12","key":"10.1016\/j.cnsns.2026.110137_bib0060","doi-asserted-by":"crossref","first-page":"4444","DOI":"10.1007\/s11837-020-04399-8","article-title":"Linking machine learning with multiscale numerics: data-driven discovery of homogenized equations","volume":"72","author":"Arbabi","year":"2020","journal-title":"Jom"},{"issue":"15","key":"10.1016\/j.cnsns.2026.110137_bib0061","first-page":"1","article-title":"Learning black- and gray-box chemotactic PDEs\/closures from agent based Monte Carlo simulation data","volume":"87","author":"Lee","year":"2023","journal-title":"J Math Biol"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110137_bib0062","doi-asserted-by":"crossref","DOI":"10.1063\/5.0113632","article-title":"Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning","volume":"33","author":"Dietrich","year":"2023","journal-title":"Chaos Interdiscip J Nonlinear Sci"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110137_bib0063","doi-asserted-by":"crossref","first-page":"4117","DOI":"10.1038\/s41467-024-48024-7","article-title":"Task-oriented machine learning surrogates for tipping points of agent-based models","volume":"15","author":"Fabiani","year":"2024","journal-title":"Nat Commun"},{"key":"10.1016\/j.cnsns.2026.110137_bib0064","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevFluids.4.054603","article-title":"Predictions of turbulent shear flows using deep neural networks","volume":"4","author":"Srinivasan","year":"2019","journal-title":"Phys Rev Fluids"},{"issue":"A24","key":"10.1016\/j.cnsns.2026.110137_bib0065","first-page":"1","article-title":"Cluster-based hierarchical network model of the fluidic pinball \u2013 cartographing transient and post-transient, multi-frequency, multi-attractor behaviour","volume":"934","author":"Deng","year":"2022","journal-title":"J Fluid Mech"},{"key":"10.1016\/j.cnsns.2026.110137_bib0066","doi-asserted-by":"crossref","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","author":"Cenedese","year":"2022","journal-title":"Nat Commun"},{"key":"10.1016\/j.cnsns.2026.110137_bib0067","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.physd.2016.12.005","article-title":"Reduced-space gaussian process regression for data-driven probabilistic forecast of chaotic dynamical systems","volume":"345","author":"Wan","year":"2017","journal-title":"Phys D Nonlinear Phenom"},{"issue":"12","key":"10.1016\/j.cnsns.2026.110137_bib0068","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/s10404-018-2164-z","article-title":"Accelerating multiscale modelling of fluids with on-the-fly gaussian process regression","volume":"22","author":"Stephenson","year":"2018","journal-title":"Microfluid Nanofluidics"},{"key":"10.1016\/j.cnsns.2026.110137_bib0069","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2020.113495","article-title":"Data-driven nonintrusive reduced order modeling for dynamical systems with moving boundaries using gaussian process regression","volume":"373","author":"Ma","year":"2021","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110137_bib0070","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3934\/mine.2022021","article-title":"A gaussian process regression approach within a data-driven POD framework for engineering problems in fluid dynamics","volume":"4","author":"Ortali","year":"2022","journal-title":"Math Eng"},{"issue":"1361","key":"10.1016\/j.cnsns.2026.110137_bib0071","first-page":"1","article-title":"\u03b2-variational autoencoders and transformers for reduced-order modelling of fluid flows","volume":"15","author":"Solera-Rico","year":"2024","journal-title":"Nat Commun"},{"key":"10.1016\/j.cnsns.2026.110137_bib0072","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1016\/j.cma.2019.03.050","article-title":"A localized reduced-order modeling approach for PDEs with bifurcating solutions","volume":"351","author":"Hess","year":"2019","journal-title":"Comput Methods Appl Mech Eng"},{"key":"10.1016\/j.cnsns.2026.110137_bib0073","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10444-020-09827-6","article-title":"Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method","volume":"47","author":"Pintore","year":"2021","journal-title":"Adv Comput Math"},{"issue":"4","key":"10.1016\/j.cnsns.2026.110137_bib0074","doi-asserted-by":"crossref","first-page":"1361","DOI":"10.1051\/m2an\/2022044","article-title":"Driving bifurcating parametrized nonlinear PDEs by optimal control strategies: application to Navier-Stokes equations with model order reduction","volume":"56","author":"Pichi","year":"2022","journal-title":"ESAIM Math Model Numer Anal"},{"issue":"10","key":"10.1016\/j.cnsns.2026.110137_bib0075","doi-asserted-by":"crossref","first-page":"1611","DOI":"10.1002\/fld.5118","article-title":"Model order reduction for bifurcating phenomena in fluid-structure interaction problems","volume":"94","author":"Khamlich","year":"2022","journal-title":"Int J Numer Methods Fluids"},{"issue":"4","key":"10.1016\/j.cnsns.2026.110137_bib0076","doi-asserted-by":"crossref","first-page":"715","DOI":"10.4310\/CMS.2003.v1.n4.a5","article-title":"Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level analysis","volume":"1","author":"Kevrekidis","year":"2003","journal-title":"Commun Math Sci"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110137_bib0077","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.jnnfm.2006.10.001","article-title":"Reduced computations for nematic-liquid crystals: a timestepper approach for systems with continuous symmetries","volume":"146","author":"Russo","year":"2007","journal-title":"J Non-Newtonian Fluid Mech"},{"key":"10.1016\/j.cnsns.2026.110137_bib0078","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.110.014121","article-title":"Machine learning for the identification of phase transitions in interacting agent-based systems: a Desai-Zwanzig example","volume":"110","author":"Evangelou","year":"2024","journal-title":"Phys Rev E"},{"key":"10.1016\/j.cnsns.2026.110137_bib0079","series-title":"Proceedings of the 2003 ACM Symposium on Applied Computing (SAC \u201903)","first-page":"161","article-title":"MatCont: A continuation toolbox in MATLAB","author":"Dhooge","year":"2003"},{"key":"10.1016\/j.cnsns.2026.110137_bib0080","doi-asserted-by":"crossref","DOI":"10.1017\/jfm.2020.692","article-title":"Bifurcation scenario in the two-dimensional laminar flow past a rotating cylinder","volume":"905","author":"Sierra","year":"2020","journal-title":"J Fluid Mech"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110137_bib0081","doi-asserted-by":"crossref","first-page":"568","DOI":"10.1016\/j.jcp.2005.01.024","article-title":"Equation-free\/galerkin-free POD-assisted computation of incompressible flows","volume":"207","author":"Sirisup","year":"2005","journal-title":"J Comput Phys"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110137_bib0082","doi-asserted-by":"crossref","DOI":"10.1063\/5.0157881","article-title":"Data-driven modelling of brain activity using neural networks, diffusion maps, and the Koopman operator","volume":"34","author":"Gallos","year":"2024","journal-title":"Chaos Interdiscip J Nonlinear Sci"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110137_bib0083","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/0022-0396(88)90110-6","article-title":"Inertial manifolds for nonlinear evolutionary equations","volume":"73","author":"Foias","year":"1988","journal-title":"J Differ Equ"},{"key":"10.1016\/j.cnsns.2026.110137_bib0084","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/BF01048790","article-title":"Spectral barriers and inertial manifolds for dissipative partial differential equations","volume":"1","author":"Constantin","year":"1989","journal-title":"J Dyn Differ Equ"},{"key":"10.1016\/j.cnsns.2026.110137_bib0085","series-title":"Integral manifolds and inertial manifolds for dissipative partial differential equations","volume":"vol. 70","author":"Constantin","year":"2012"},{"key":"10.1016\/j.cnsns.2026.110137_bib0086","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2023.112072","article-title":"Double diffusion maps and their latent harmonics for scientific computations in latent space","volume":"485","author":"Evangelou","year":"2023","journal-title":"J Comput Phys"},{"issue":"Suppl 1","key":"10.1016\/j.cnsns.2026.110137_bib0087","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/s10884-021-10127-w","article-title":"Enabling equation-free modeling via diffusion maps","volume":"36","author":"Chin","year":"2024","journal-title":"J Dyn Differ Equ"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110137_bib0088","doi-asserted-by":"crossref","first-page":"572","DOI":"10.1016\/S0021-9991(03)00298-5","article-title":"Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries","volume":"190","author":"Popinet","year":"2003","journal-title":"J Comput Phys"},{"key":"10.1016\/j.cnsns.2026.110137_bib0089","article-title":"The structure of inhomogeneous turbulent flows","author":"Lumley","year":"1967"},{"issue":"3","key":"10.1016\/j.cnsns.2026.110137_bib0090","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1090\/qam\/910462","article-title":"Turbulence and the dynamics of coherent structures, parts I\u2013III","volume":"45","author":"Sirovich","year":"1987","journal-title":"Q Appl Math"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110137_bib0091","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1016\/j.jcp.2019.04.015","article-title":"Manifold learning for parameter reduction","volume":"392","author":"Holiday","year":"2019","journal-title":"J Comput Phys"},{"issue":"38","key":"10.1016\/j.cnsns.2026.110137_bib0092","doi-asserted-by":"crossref","first-page":"16090","DOI":"10.1073\/pnas.0905547106","article-title":"Detecting intrinsic slow variables in stochastic dynamical systems by anisotropic diffusion maps","volume":"106","author":"Singer","year":"2009","journal-title":"Proc Natl Acad Sci"},{"issue":"4","key":"10.1016\/j.cnsns.2026.110137_bib0093","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1007\/s11571-020-09645-y","article-title":"Construction of embedded fMRI resting-state functional connectivity networks using manifold learning","volume":"15","author":"Gallos","year":"2021","journal-title":"Cognit Neurodyn"},{"issue":"2","key":"10.1016\/j.cnsns.2026.110137_bib0094","doi-asserted-by":"crossref","first-page":"842","DOI":"10.1137\/070696325","article-title":"Diffusion maps, reduction coordinates, and low dimensional representation of stochastic systems","volume":"7","author":"Coifman","year":"2008","journal-title":"Multiscale Model Simul"},{"key":"10.1016\/j.cnsns.2026.110137_bib0095","series-title":"\u00dcber die praktische Aufl\u00f6sung von linearen Integralgleichungen mit Anwendungen auf Randwertaufgaben der Potentialtheorie","author":"Nystr\u00f6m","year":"1929"},{"key":"10.1016\/j.cnsns.2026.110137_bib0096","series-title":"Elements of applied bifurcation theory","volume":"vol. 112","author":"Kuznetsov","year":"1998"},{"key":"10.1016\/j.cnsns.2026.110137_bib0097","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1017\/S0022112007005654","article-title":"Structural sensitivity of the first instability of the cylinder wake","volume":"581","author":"Giannetti","year":"2007","journal-title":"J Fluid Mech"},{"key":"10.1016\/j.cnsns.2026.110137_bib0098","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1146\/annurev.fl.28.010196.002401","article-title":"Vortex dynamics in the cylinder wake","volume":"28","author":"Williamson","year":"1996","journal-title":"Annu Rev Fluid Mech"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110137_bib0099","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1063\/1.869174","article-title":"Bifurcation phenomena in incompressible suddenexpansionflows","volume":"9","author":"Drikakis","year":"1997","journal-title":"Phys Fluids"},{"issue":"1","key":"10.1016\/j.cnsns.2026.110137_bib0100","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1080\/10618562.2016.1144877","article-title":"Symmetry breaking and preliminary results about a hopf bifurcation for incompressible viscous flow in an expansion channel","volume":"30","author":"Quaini","year":"2016","journal-title":"Int J Comput Fluid Dyn"},{"key":"10.1016\/j.cnsns.2026.110137_bib0101","doi-asserted-by":"crossref","first-page":"801","DOI":"10.1017\/S0022112065001702","article-title":"Report on the first european mechanics colloquium on coanda effect","volume":"23","author":"Wille","year":"1965","journal-title":"J Fluid Mech"},{"key":"10.1016\/j.cnsns.2026.110137_bib0102","doi-asserted-by":"crossref","first-page":"99","DOI":"10.2514\/2.68","article-title":"Bifurcation of low reynolds number flows in symmetric channels","volume":"35","author":"Battaglia","year":"1997","journal-title":"AIAA J"},{"issue":"A37","key":"10.1016\/j.cnsns.2026.110137_bib0103","first-page":"1","article-title":"Low-order model for successive bifurcations of the fluidic pinball","volume":"884","author":"Deng","year":"2020","journal-title":"J Fluid Mech"},{"key":"10.1016\/j.cnsns.2026.110137_bib0104","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1017\/S0022112003006694","article-title":"A hierarchy of low-dimensional models for the transient and post-transient cylinder wake","volume":"497","author":"Noack","year":"2003","journal-title":"J Fluid Mech"}],"container-title":["Communications in Nonlinear Science and Numerical Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S100757042600496X?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S100757042600496X?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T05:35:48Z","timestamp":1780724148000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S100757042600496X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,10]]},"references-count":104,"alternative-id":["S100757042600496X"],"URL":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110137","relation":{},"ISSN":["1007-5704"],"issn-type":[{"value":"1007-5704","type":"print"}],"subject":[],"published":{"date-parts":[[2026,10]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Surrogate normal-forms for the numerical bifurcation and stability analysis of Navier-Stokes flows via machine learning","name":"articletitle","label":"Article Title"},{"value":"Communications in Nonlinear Science and Numerical Simulation","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cnsns.2026.110137","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 The Author(s). Published by Elsevier B.V.","name":"copyright","label":"Copyright"}],"article-number":"110137"}}