Abstract
This chapter presents an overview of the most popular reduced order models found in the approximation of partial differential equations and their connection with machine learning techniques. Although the presentation is applicable to many problems in science and engineering, the focus is first-order evolution problems in time and, more specifically, flow problems. Particular emphasis is put on the distinction between intrusive models, which make use of the physical problem being modeled, and non-intrusive models, purely designed from data using machine learning strategies. For the former, models based on proper orthogonal decomposition and Galerkin projection are described in detail, whereas alternatives are only mentioned. Likewise, some modifications that are crucial in the applications are detailed. The progressive incorporation of machine learning methods is described, yielding first hybrid formulations and ending with pure data-driven approaches. An effort has been made to include references with applications of the methods being described.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
Greedy algorithms are a class of algorithms that are based on choosing the option which produces the largest immediate reward with the expectation that the successive application of greedy sampling will lead to a global optimum. Greedy algorithms may use an error estimator to guide the sampling.
- 2.
A non-Markovian term implies that the future state depends not only on the current values, but also on the past value, i.e. such processes have memory effects of the past values.
References
Abadía-Heredia R et al (2022) A predictive hybrid reduced order model based on proper orthogonal decom position combined with deep learning architectures. Expert Syst Appl 187:115910
Ahmed HF et al (2021) Machine learning-based reduced-order modeling of hydrodynamic forces using pressure mode decomposition. Proc Inst Mech Eng, Part G: J Aerosp Eng 235(16):2517–2528
Ahmed SE et al (2020) A long short-term memory embedding for hybrid uplifted reduced order models. Phys D: Nonlinear Phenom 409:132471
Ahmed SE et al (2021) On closures for reduced order models-a spectrum of first-principle to machine-learned avenues. Phys Fluids 33(9):091301
Akhtar I, Borggaard J, Hay A (2010) Shape sensitivity analysis in flow models using a finite-difference approach. Math Probl Eng
Alla A, Kutz JN (2017) Nonlinear model order reduction via dynamic mode decomposition. SIAM J Sci Comput 39(5):B778–B796
Amsallem D, Farhat C (2011) An online method for interpolating linear parametric reduced-order models. SIAM J Sci Comput 33(5):2169–2198
Amsallem D, Farhat C (2012) Stabilization of projection-based reduced-order models. Int J Numer Methods Eng 91(4):358–377
An SS, Kim T, James DL (2008) Optimizing cubature for efficient integration of subspace deformations. ACM Trans Graph 27(5):65:1–165:10
Antil H, Heinkenschloss M, Sorensen DC (2014) Application of the discrete empirical interpolation method to reduced order modeling of nonlinear and parametric systems. In: Quarteroni A, Rozza G (eds) Reduced order methods for modeling and computational reduction. MS &A—Modeling, Simulation and Applications. Springer International Publishing, Cham, pp 101–136
Arian E, Fahl M, Sachs EW (2000) Trust-Region Proper Orthogonal Decomposition for Flow Control. Technical report. Institute for Computer Applications in Science and Engineering, Hampton VA
Astrid P et al (2008) Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans Autom Control 53(10):2237–2251
Azaïez M, Chacon Rebollo T, Rubino S (2021) A cure for instabilities due to advection-dominance in POD solution to advection-diffusion-reaction equations. J Comput Phys 425:109916
Baiges J et al (2020) A finite element reduced-order model based on adaptive mesh refinement and artificial neural networks. Int J Numer Methods Eng 121(4):588–601
Baiges J et al (2021) An adaptive finite element strategy for the numerical simulation of additive manufacturing processes. Addit Manuf 37:101650
Baiges J, Codina R, Idelsohn S (2015) Reduced-order subscales for POD models. Comput Methods Appl Mech Eng 291:173–196
Baiges J, Codina R (2013a) A variational multiscale method with subscales on the element boundaries for the helmholtz equation. Int J Numer Methods Eng 93(6):664–684
Baiges J, Codina R, Idelsohn S (2013b) A domain decomposition strategy for reduced order models. Application to the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 267:23–42
Baiges J, Codina R, Idelsohn S (2013c) Explicit reduced-order models for the stabilized finite element approximation of the incompressible Navier–Stokes equations. Int J Numer Methods Fluids 72(12):1219–1243
Baldi P, Hornik K (1989) Neural networks and principal component analysis: learning from examples without local minima. Neural Netw 2(1):53–58
Ballard DH (1987) Modular learning in neural networks. In: Proceedings of the sixth national conference on artificial intelligence, vol 1. AAAI’87. AAAI Press, Seattle, Washington, pp 279–284
Ballarin F et al (2015) Supremizer stabilization of POD-galerkin approximation of parametrized steady incom pressible Navier–Stokes equations. Int J Numer Methods Eng 102(5):1136–1161
Barrault M et al (2004) An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique 339(9):667–672
Benosman M, Chakrabarty A, Borggaard J (2020) Reinforcement learning-based model reduction for partial differential equations. IFAC-PapersOnLine. 21st IFAC World Congress 53(2):7704–7709
Bergmann M, Cordier L, Brancher J-P (2007) Drag minimization of the cylinder wake by trust-region proper orthogonal decomposition. In: Active flow control. Springer, Berlin, pp 309–324
Bertsimas D, Dunn J (2017) Optimal classification trees. Mach Learn 106(7):1039–1082
Brooks AN, Hughes TJR (1982) Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput Methods Appl Mech Eng 32(1):199–259
Brunton SL et al (2017) Chaos as an intermittently forced linear system. Nat Commun 8(1):19
Brunton SL, Proctor JL, Kutz JN (2016) Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc Natl Acad Sci 113(15):3932–3937
Bui-Thanh T, Willcox K, Ghattas O (2008) Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J Sci Comput 30(6):3270–3288
Buoso S et al (2022)Stabilized reduced-order models for unsteady incompressible flows in three-dimensional parametrized domains. Comput Fluids 246:105604
Burkardt J, Gunzburger M, Lee H-C (2006) POD and CVT-based reduced-order modeling of Navier–Stokes flows. Comput Methods Appl Mech Eng 196(1–3):337–355
Callaham JL et al (2022) An empirical mean-field model of symmetry-breaking in a turbulent wake. Sci Adv 8(19):eabm4786
Carlberg K, Barone M, Antil H (2017) Galerkin v. Least-Squares Petrov-Galerkin projection in nonlinear model reduction. J Comput Phys 330:693–734
Carlberg K, Bou-Mosleh C, Farhat C (2011) Efficient non-linear model reduction via a least-squares petrov-galerkin projection and compressive tensor approximations. Int J Numer Methods Eng 86(2):155–181
Champion K et al (2019) Data-driven discovery of coordinates and governing equations. Proc Natl Acad Sci 116(45):22445–22451
Chatterjee A (2000) An introduction to the proper orthogonal decomposition. Curr Sci 78(7):808–817
Chaturantabut S, Sorensen DC (2010) Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput 32(5):2737–2764
Chen KK, Tu JH, Rowley CW (2012) Variants of dynamic mode decomposition: boundary condition, koopman, and fourier analyses. J Nonlinear Sci 22(6):887–915
Chen W et al (2021) Physics-informed machine learning for reduced-order modeling of nonlinear problems. J Comput Phys 446:110666
Chen Z, Zhao Y, Huang R (2019) Parametric reduced-order modeling of unsteady aerodynamics for hyper sonic vehicles. Aerosp Sci Technol 87:1–14
Chinesta F, Ammar, A Cueto E (2010) Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models. Arch Comput Methods Eng 17(4):327–350
Chinesta F, Ladeveze P, Cueto E (2011) A short review on model order reduction based on proper generalized decomposition. Arch Comput Methods Eng 18(4):395
Codina R (2000a) On stabilized finite element methods for linear systems of convection-diffusion-reaction equations. Comput Methods Appl Mech Eng 188(1):61–82
Codina R (2000b) Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods. Comput Methods Appl Mech Eng 190(13–14):1579–1599
Codina R (2002) Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput Methods Appl Mech Eng 191(39–40):4295–4321
Codina R et al (2007) Time dependent subscales in the stabilized finite element approximation of incompressible flow problems. Comput Methods Appl Mech Eng 196(21–24):2413–2430
Codina R et al (2018) Variational multiscale methods in computational fluid dynamics. Encycl Comput Mech 1–28
Codina R, Baiges J (2011) Finite element approximation of transmission conditions in fluids and solids introducing boundary subgrid scales. Int J Numer Methods Eng 87(1–5):386–411
Codina R, Principe J, Baiges J (2009) Subscales on the element boundaries in the variational two-scale finite element method. Comput Methods Appl Mech Eng 198(5–8):838–852
Codina R, Reyes R, Baiges J (2021) A posteriori error estimates in a finite element vms-based reduced order model for the incompressible Navier–Stokes equations. Mech Res Commun. Special Issue Honoring G.I. Taylor Medalist Prof. Arif Masud 112:103599
Dal Santo N et al (2019) An algebraic least squares reduced basis method for the solution of nonaffinely parametrized stokes equations. Comput Methods Appl Mech Eng 344:186–208
Daniel T et al (2020) Model order reduction assisted by deep neural networks (ROM-net). Adv Model Simul Eng Sci 7(1):16
Dar Z, Baiges J, Codina R (2023) Artificial neural network based correction models for reduced order models in computational fluid mechanics. Comput Methods Appl Mech Eng 415:116232
Deng N et al (2020) Low-order model for successive bifurcations of the fluidic pinball. J Fluid Mech 884:A37
Dupuis R, Jouhaud J-C, Sagaut P (2018) Surrogate modeling of aerodynamic simulations for multiple operating conditions using machine learning. AIAA J 56(9):3622–3635
Eckart C, Young G (1936) The approximation of one matrix by another of lower rank. Psychometrika 1(3):211–218
Eivazi H et al (2022) Towards extraction of orthogonal and parsimonious non-linear modes from turbulent flows. Expert Syst Appl 202:117038
Everson R, Sirovich L (1664) Karhunen-Loeve procedure for Gappy data. JOSA A 12(8):1657–1664
Fabra A, Baiges J, Codina R (2022) Finite element approximation of wave problems with correcting terms based on training artificial neural networks with fine solutions. Comput Methods Appl Mech Eng 399:115280
Farhat C, Chapman T, Avery P (2015) Structure-preserving, stability, and accuracy properties of the energy conserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models. Int J Numer Methods Eng 102(5):1077–1110
Fresca S, Dede’ L, Manzoni A (2021) A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs. J Sci Comput 87(2):61
Fresca S, Manzoni A (2022) POD-DL-ROM: enhancing deep learning-based reduced order models for non linear parametrized PDEs by proper orthogonal decomposition. Comput Methods Appl Mech Eng 388:114181
Galletti B et al (2004) Low-order modelling of laminar flow regimes past a confined square cylinder. J Fluid Mech 503:161–170
García-Archilla B, Novo J, Rubino S (2022) Error analysis of proper orthogonal decomposition data assimilation schemes with grad-div stabilization for the Navier–Stokes equations. J Comput Appl Math 411:114246
Giere S et al (2015) SUPG reduced order models for convection-dominated convection-diffusion-reaction equations. Comput Methods Appl Mech Eng 289:454–474
Giere S, John V (2017) Towards physically admissible reduced-order solutions for convection-diffusion problems. Appl Math Lett 73:78–83
Glaz B, Liu L, Friedmann PP (2010) Reduced-order nonlinear unsteady aerodynamic modeling using a surrogate-based recurrence framework. AIAA J 48(10):2418–2429
Gonzalez FJ, Balajewicz M (2018) Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems. arXiv:1808.01346 [physics]
Graham WR, Peraire J, Tang KY (1999) Optimal control of vortex shedding using low-order models. Part I-open-loop model development. Int J Numer Methods Eng 44(7):945–972
Graves A, Schmidhuber J (2005) Framewise phoneme classification with bidirectional LSTM and other neural network architectures. Neural Netw. IJCNN 2005 18(5):602–610
Guan Y, Brunton SL, Novosselov I (2021) Sparse nonlinear models of chaotic electroconvection. R Soc Open Sci 8(8):202367
Guo M, Hesthaven JS (2019) Data-driven reduced order modeling for time-dependent problems. Comput Methods Appl Mech Eng 345:75–99
Guo M, Hesthaven JS (2018) Reduced order modeling for nonlinear structural analysis using gaussian process regression. Comput Methods Appl Mech Eng 341:807–826
Hesthaven JS, Rozza G, Stamm B (2016) Certified reduced basis methods for parametrized partial differential equations. SpringerBriefs in Mathematics. Springer International Publishing, Cham
Hesthaven JS, Ubbiali S (2018) Non-intrusive reduced order modeling of nonlinear problems using neural networks. J Comput Phys 363:55–78
Higgins I et al (2022) Beta-VAE: learning basic visual concepts with a constrained variational framework. In: International conference on learning representations
Hochreiter S, Schmidhuber J (1997) Long short-term memory. Neural Comput 9(8):1735–1780
Hughes TJR et al (1998) The variational multiscale method-a paradigm for computational mechanics. Comput Methods Appl Mech Engineering Adv Stab Methods Comput Mech 166(1):3–24
Hunter A et al (2019) Reduced-order modeling through machine learning and graph-theoretic approaches for brittle fracture applications. Comput Mater Sci 157:87–98
Hurst HE (1951) Long-term storage capacity of reservoirs. Trans Am Soc Civ Eng 116(1):770–799
John Leask L (1967) The structure of inhomogeneous turbulent flows. Struct Inhomogeneous Turbul Flows 166–178
John V, Moreau B, Novo J (2022) Error analysis of a SUPG-stabilized POD-ROM method for convection diffusion-reaction equations. Comput Math Appl 122:48–60
Juang J-N (1994) Applied system identification. Prentice Hall
Kaheman K, Brunton SL, Kutz JN (2022) Automatic differentiation to simultaneously identify nonlinear dynamics and extract noise probability distributions from data. Mach Learn: Sci Technol 3(1):015031
Kaiser E et al (2014) Cluster-based reduced-order modelling of a mixing layer. J Fluid Mech 754:365–414
Kalashnikova I, Barone M (2011) Stable and efficient galerkin reduced order models for non-linear fluid flow. In: 6th AIAA theoretical fluid mechanics conference, p 3110
Kapteyn MG, Knezevic DJ, Willcox K (2020) Toward predictive digital twins via component-based reduced-order models and interpretable machine learning. In: AIAA scitech 2020 forum. American Institute of Aeronautics and Astronautics
Kast M, Guo M, Hesthaven JS (2020) A non-intrusive multifidelity method for the reduced order modeling of nonlinear problems. Comput Methods Appl Mech Eng 364:112947
Kingma DP, Welling M (2013) Auto-encoding variational bayes. In: International conference on learning representations
Lee K, Carlberg KT (2020a) Deep conservation: a latent-dynamics model for exact satisfaction of physical conservation laws. arXiv:1909.09754 [physics]
Lee K, Carlberg KT (2020b) Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. J Comput Phys 404:108973
LeGresley P, Alonso J (2000) Airfoil design optimization using reduced order models based on proper orthogonal decomposition. In: Fluids 2000 conference and exhibit. American Institute of Aeronautics and Astronautics
Li J, Du X, Martins JRRA (2022) Machine learning in aerodynamic shape optimization. Prog Aerosp Sci 134:100849
Ljung L (1998) System identification: theory for the user, 2nd edn. Pearson, Upper Saddle River, NJ
Loiseau J-C (2020) Data-driven modeling of the chaotic thermal convection in an annular thermosyphon. Theor Comput Fluid Dyn 34:339–365
Loiseau J-C, Brunton SL (2018) Constrained sparse Galerkin regression. J Fluid Mech 838:42–67
Lucia DJ, Beran PS (2003) Projection methods for reduced order models of compressible flows. J Comput Phys 188(1):252–280
Lusch B, Kutz JN, Brunton SL (2018). Deep learning for universal linear embeddings of nonlinear dynamics. Nat Commun 9(1):4950
Maulik R et al (2021) Latent-space time evolution of non-intrusive reduced-order models using gaussian process emulation. Phys D: Nonlinear Phenom 416:132797
Ma C, Wang J (2019) Model reduction with memory and the machine learning of dynamical systems. Commun Comput Phys 25(4)
Milano M, Koumoutsakos P (2002) Neural network modeling for near wall turbulent flow. J Comput Phys 182(1):1–26
Mohan AT, Gaitonde DV (2018) A deep learning based approach to reduced order modeling for turbulent flow control using LSTM neural networks. arXiv:1804.09269 [physics]
Mohan AT et al (2020) Embedding hard physical constraints in neural network coarse-graining of 3D turbulence. arXiv:2002.00021 [physics]
Mohebujjaman M, Rebholz L, Iliescu T (2019) Physically constrained data-driven correction for reduced order modeling of fluid flows. Int J Numer Methods Fluids 89(3):103–122
Mori H (1965) Transport, collective motion, and brownian motion*). Prog Theor Phys 33(3):423–455
Mou C et al (2021) Data-driven variational multiscale reduced order models. Comput Methods Appl Mech Eng 373:113470
Murata T, Fukami K, Fukagata K (2020) Nonlinear mode decomposition with convolutional neural networks for fluid dynamics. J Fluid Mech 882:A13
Noack BR et al (2003) A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J Fluid Mech 497:335–363
Noack BR et al (2016) Recursive dynamic mode decomposition of transient and post-transient wake flows. J Fluid Mech 809:843–872
Noack BR et al (eds) (2011) Reduced-order modelling for flow control, vol 528. Springer, CISM International Centre for Mechanical Sciences. Vienna
Otto SE, Rowley CW (2019) Linearly recurrent autoencoder networks for learning dynamics. SIAM J Appl Dyn Syst 18(1):558–593
Pacciarini P, Rozza G (2014) Stabilized reduced basis method for parametrized advection-diffusion PDEs. Comput Methods Appl Mech Eng 274:1–18
Pawar S et al (2019) A deep learning enabler for nonintrusive reduced order modeling of fluid flows. Phys Fluids 31(8):085101
Pawar S et al (2021) Model fusion with physics-guided machine learning: projection-based reduced-order modeling. Phys Fluids 33(6):067123
Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: a deep learning frame work for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378:686–707
Rasmussen CE, Williams CKI (2005) Gaussian processes for machine learning
Reyes R et al (2018) Reduced order models for thermally coupled low mach flows. Adv Model Simul Eng Sci 5(1):1–20
Reyes R, Codina R (2020) Projection-based reduced order models for flow problems: a variational multiscale approach. Comput Methods Appl Mech Eng 363:112844
Rozza G, Huynh DBP, Manzoni A (2013) Reduced basis approximation and a posteriori error estimation for stokes flows in parametrized geometries: roles of the inf-sup stability constants. Rozza
Rozza G, Lassila T, Manzoni A (2011) Reduced basis approximation for shape optimization in thermal flows with a parametrized polynomial geometric map. In: Spectral and high order methods for partial differential equations. Springer, Berlin, pp 307–315
Sahba S et al (2022) Dynamic mode decomposition for aero-optic wavefront characterization. Opt Eng 61(1):013105
San O, Iliescu T (2015) A stabilized proper orthogonal decomposition reduced-order model for large scale quasigeostrophic ocean circulation. Adv Comput Math 41(5):1289–1319
San O, Maulik R (2018a) Extreme learning machine for reduced order modeling of turbulent geophysical flows. Phys Rev E 97(4):42322
San O, Maulik R (2018b) Neural network closures for nonlinear model order reduction. Adv Comput Math 44(6):1717–1750
San O, Pawar S, Rasheed A (2022) Variational multiscale reinforcement learning for discovering reduced order closure models of nonlinear spatiotemporal transport systems. arXiv:2207.12854 [physics]
Schmid PJ (2010) Dynamic mode decomposition of numerical and experimental data. J Fluid Mech 656:5–28
Schmid PJ, Violato D, Scarano F (2012) Decomposition of time-resolved tomographic PIV. Exp Fluids 52(6):1567–1579
Shah NV et al (2022) Finite element based model order reduction for parametrized one-way coupled steady state linear thermo-mechanical problems. Finite Elem Anal Des 212:103837
Srinivasan PA et al (2019) Predictions of turbulent shear flows using deep neural networks. Phys Rev Fluids 4(5):054603
Suykens JAK et al (2002) Least squares support vector machines. World Scientific
Takeishi N, Kawahara Y, Yairi T (2017) Learning koopman invariant subspaces for dynamic mode decom position. Proceedings of the 31st international conference on neural information processing systems. NIPS’17. Curran Associates Inc., Red Hook, NY, USA, pp 1130–1140
Tello A, Codina R (2021) Field-to-field coupled fluid structure interaction: a reduced order model study. Int J Numer Methods Eng 122(1):53–81
Tello A, Codina R, Baiges J (2020) Fluid structure interaction by means of variational multiscale reduced order models. Int J Numer Methods Eng 121(12):2601–2625
Tissot G et al (2014) Model reduction using dynamic mode decomposition. Comptes Rendus Mécanique. Flow Separation Control 342(6):410–416
Tu JH et al (2014) On dynamic mode decomposition: theory and applications. J Comput Dyn 1(2)(Mon Dec 01 01:00:00 CET 2014):391–421
Vlachas PR et al (2018) Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks. Proc R Soc A: Math, Phys Eng Sci 474(2213):20170844
Wan ZY, Sapsis TP (2017) Reduced-space gaussian process regression for data-driven probabilistic forecast of chaotic dynamical systems. Phys D: Nonlinear Phenom 345:40–55
Wang Z et al (2012) Proper orthogonal decomposition closure models for turbulent flows: a numerical comparison. Comput Methods Appl Mech Eng 237:10–26
Wang Q, Hesthaven JS, Ray D (2019) Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem. J Comput Phys 384:289–307
Wang Q, Ripamonti N, Hesthaven JS (2020) Recurrent neural network closure of parametric POD-Galerkin reduced-order models based on the mori-zwanzig formalism. J Comput Phys 410:109402
Wehmeyer C, Noe F (2018) Time-lagged autoencoders: deep learning of slow collective variables for molecular kinetics. J Chem Phys 148(24):241703
Williams MO, Kevrekidis IG, Rowley CW (2015) A data-driven approximation of the koopman operator: extending dynamic mode decomposition. J Nonlinear Sci 25(6):1307–1346
Xie X, Webster C, Iliescu T (2020) Closure learning for nonlinear model reduction using deep residual neural network. Fluids 5(1):39
Xu S et al (2013) Multi-output least-squares support vector regression machines. Pattern Recognit Lett 34(9):1078–1084
Yousif MZ, Lim H-C (2022) Reduced-order modeling for turbulent wake of a finite wall-mounted square cylinder based on artificial neural network. Phys Fluids 34(1):015116
Yvonnet J, He Q-C (2007) The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains. J Comput Phys 223(1):341–368
Zhao H (2021) A reduced order model based on machine learning for numerical analysis: an application to geomechanics. Eng Appl Artif Intell 100:104194
Zhu Q, Guo Y, Lin W (2021)Neural delay differential equations. In: The international conference on learning representations, p 20
Zwanzig R (1960)Ensemble method in the theory of irreversibility. J Chem Phys 33(5):1338–1341
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Dar, Z., Baiges, J., Codina, R. (2023). Reduced Order Modeling. In: Rabczuk, T., Bathe, KJ. (eds) Machine Learning in Modeling and Simulation. Computational Methods in Engineering & the Sciences. Springer, Cham. https://doi.org/10.1007/978-3-031-36644-4_8
Download citation
DOI: https://doi.org/10.1007/978-3-031-36644-4_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-36643-7
Online ISBN: 978-3-031-36644-4
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)