Hybrid analysis and modeling, eclecticism, and multifidelity computing toward digital twin revolution

Most modeling approaches lie in either of the two categories: physics-based or data-driven. Recently, a third approach which is a combination of these deterministic and statistical models is emerging for scientific applications. To leverage these developments, our aim in this perspective paper is centered around exploring numerous principle concepts to address the challenges of (i) trustworthiness and generalizability in developing data-driven models to shed light on understanding the fundamental trade-offs in their accuracy and efficiency, and (ii) seamless integration of interface learning and multifidelity coupling approaches that transfer and represent information between different entities, particularly when different scales are governed by different physics, each operating on a different level of abstraction. Addressing these challenges could enable the revolution of digital twin technologies for scientific and engineering applications.

[1]  Kate Burningham,et al.  Imagined publics and engagement around renewable energy technologies in the UK , 2012 .

[2]  Francis X. Giraldo,et al.  A Conservative Discontinuous Galerkin Semi-Implicit Formulation for the Navier-Stokes Equations in Nonhydrostatic Mesoscale Modeling , 2009, SIAM J. Sci. Comput..

[3]  J. D. M. James Quadratic Volume-Preserving Maps: (Un)stable Manifolds, Hyperbolic Dynamics, and Vortex-Bubble Bifurcations , 2013 .

[4]  Omer San,et al.  COLREG-Compliant Collision Avoidance for Unmanned Surface Vehicle Using Deep Reinforcement Learning , 2020, IEEE Access.

[5]  Ionel M. Navon,et al.  An Optimal Nudging Data Assimilation Scheme Using Parameter Estimation , 1992 .

[6]  Yue Yu,et al.  An Asymptotically Compatible Formulation for Local-to-Nonlocal Coupling Problems without Overlapping Regions , 2019, ArXiv.

[7]  W. Hsieh,et al.  Coupling Neural Networks to Incomplete Dynamical Systems via Variational Data Assimilation , 2001 .

[8]  Ahmed K. Noor,et al.  Reduced Basis Technique for Nonlinear Analysis of Structures , 1980 .

[9]  Jamey D. Jacob,et al.  Sub-grid scale model classification and blending through deep learning , 2018, Journal of Fluid Mechanics.

[10]  Peter Bauer,et al.  A digital twin of Earth for the green transition , 2021, Nature Climate Change.

[11]  F. Chinesta,et al.  A Short Review in Model Order Reduction Based on Proper Generalized Decomposition , 2018 .

[12]  Pierre Sagaut,et al.  Multiscale and Multiresolution Approaches in Turbulence - Les, Des and Hybrid Rans/Les Methods: Applications and Guidelines , 2013 .

[13]  W. Wall,et al.  Fixed-point fluid–structure interaction solvers with dynamic relaxation , 2008 .

[14]  Hans von Storch,et al.  Dynamical downscaling: Assessment of model system dependent retained and added variability for two different regional climate models , 2008 .

[15]  K. Kendrick,et al.  Partial Granger causality—Eliminating exogenous inputs and latent variables , 2008, Journal of Neuroscience Methods.

[16]  Petros Koumoutsakos,et al.  Data-assisted reduced-order modeling of extreme events in complex dynamical systems , 2018, PloS one.

[17]  T. A. Porsching,et al.  Estimation of the error in the reduced basis method solution of nonlinear equations , 1985 .

[18]  L. Agostini Exploration and prediction of fluid dynamical systems using auto-encoder technology , 2020 .

[19]  Marc G. D. Geers,et al.  A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials , 2017, J. Comput. Phys..

[20]  Bruce R. Rosen,et al.  Image reconstruction by domain-transform manifold learning , 2017, Nature.

[21]  M. Sahimi,et al.  Machine learning in geo- and environmental sciences: From small to large scale , 2020, Advances in Water Resources.

[22]  Alain Chedin,et al.  A Neural Network Approach for a Fast and Accurate Computation of a Longwave Radiative Budget , 1998 .

[23]  Remi Manceau,et al.  A SEAMLESS HYBRID RANS-LES MODEL BASED ON TRANSPORT EQUATIONS FOR THE SUBGRID STRESSES AND ELLIPTIC BLENDING , 2010, Proceeding of Fifth International Symposium on Turbulence and Shear Flow Phenomena.

[24]  Bo Ren,et al.  Fluid directed rigid body control using deep reinforcement learning , 2018, ACM Trans. Graph..

[25]  Jan M. Nordbotten,et al.  Robust Discretization of Flow in Fractured Porous Media , 2016, SIAM J. Numer. Anal..

[26]  Omer San,et al.  Feature engineering and symbolic regression methods for detecting hidden physics from sparse sensor observation data , 2019 .

[27]  Grigorios A. Pavliotis,et al.  Multiscale Methods: Averaging and Homogenization , 2008 .

[28]  Jürgen Schmidhuber,et al.  Multi-column deep neural networks for image classification , 2012, 2012 IEEE Conference on Computer Vision and Pattern Recognition.

[29]  Emil M. Constantinescu,et al.  Multiphysics simulations , 2013, HiPC 2013.

[30]  P. Beran,et al.  Reduced-order modeling: new approaches for computational physics , 2004 .

[31]  Odile Papini,et al.  Information Fusion , 2014, Computer Vision, A Reference Guide.

[32]  Jean-François Mahfouf,et al.  Evaluation of the Jacobians of Infrared Radiation Models for Variational Data Assimilation , 2001 .

[33]  R. Ganguli,et al.  The digital twin of discrete dynamic systems: Initial approaches and future challenges , 2020, Applied Mathematical Modelling.

[34]  Petros Koumoutsakos,et al.  Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks , 2018, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[35]  Ke Li,et al.  D3M: A Deep Domain Decomposition Method for Partial Differential Equations , 2020, IEEE Access.

[36]  William W. Hsieh,et al.  Nonlinear principal component analysis by neural networks , 2001 .

[37]  Jaideep Pathak,et al.  Combining Machine Learning with Knowledge-Based Modeling for Scalable Forecasting and Subgrid-Scale Closure of Large, Complex, Spatiotemporal Systems , 2020, Chaos.

[38]  Jun-Ho Oh,et al.  Hybrid Learning of Mapping and its Jacobian in Multilayer Neural Networks , 1996, Neural Computation.

[39]  N. K. Sinha,et al.  A review of some model reduction techniques , 1981, Canadian Electrical Engineering Journal.

[40]  Jack P. C. Kleijnen,et al.  Kriging Metamodeling in Simulation: A Review , 2007, Eur. J. Oper. Res..

[41]  K. Giannakoglou,et al.  Advances in Evolutionary and Deterministic Methods for Design, Optimization and Control in Engineering and Sciences , 2016, Computational Methods in Applied Sciences.

[42]  E Weinan,et al.  Heterogeneous multiscale method: A general methodology for multiscale modeling , 2003 .

[43]  Francisco Chinesta,et al.  A Simulation App based on reduced order modeling for manufacturing optimization of composite outlet guide vanes , 2017, Advanced Modeling and Simulation in Engineering Sciences.

[44]  Scott T. M. Dawson,et al.  Model Reduction for Flow Analysis and Control , 2017 .

[45]  Bram van Ginneken,et al.  A survey on deep learning in medical image analysis , 2017, Medical Image Anal..

[46]  Eric Blayo,et al.  Towards Optimized Schwarz Methods for the Navier–Stokes Equations , 2016, J. Sci. Comput..

[47]  Prakash Vedula,et al.  Subgrid modelling for two-dimensional turbulence using neural networks , 2018, Journal of Fluid Mechanics.

[48]  Shane Legg,et al.  Human-level control through deep reinforcement learning , 2015, Nature.

[49]  R. Grigoriev,et al.  Forecasting Fluid Flows Using the Geometry of Turbulence. , 2016, Physical review letters.

[50]  Omer San,et al.  Data-driven recovery of hidden physics in reduced order modeling of fluid flows , 2020, Physics of Fluids.

[51]  Axel Klawonn,et al.  Machine Learning in Adaptive Domain Decomposition Methods - Predicting the Geometric Location of Constraints , 2019, SIAM J. Sci. Comput..

[52]  J. D. Mireles James Quadratic Volume-Preserving Maps: (Un)stable Manifolds, Hyperbolic Dynamics, and Vortex-Bubble Bifurcations , 2013 .

[53]  Constantinos Theodoropoulos,et al.  Optimisation and Linear Control of Large Scale Nonlinear Systems: A Review and a Suite of Model Reduction-Based Techniques , 2011 .

[54]  B. Haasdonk,et al.  Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition , 2011 .

[55]  J. Marsden,et al.  Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows , 2005 .

[56]  Gianluigi Rozza,et al.  Model Order Reduction in Fluid Dynamics: Challenges and Perspectives , 2014 .

[57]  H. Storch,et al.  Optimal Spectral Nudging for Global Dynamic Downscaling , 2017 .

[58]  Akira Oyama,et al.  Closed-Loop Flow Separation Control Using the Deep Q Network over Airfoil , 2020 .

[59]  F. Waleffe Exact coherent structures in channel flow , 2001, Journal of Fluid Mechanics.

[60]  Omer San,et al.  Interface learning of multiphysics and multiscale systems , 2020, Physical review. E.

[61]  Axel Klawonn,et al.  Monolithic Overlapping Schwarz Domain Decomposition Methods with GDSW Coarse Spaces for Incompressible Fluid Flow Problems , 2019, SIAM J. Sci. Comput..

[62]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

[63]  Tayfun E. Tezduyar,et al.  Modelling of fluid–structure interactions with the space–time finite elements: Solution techniques , 2007 .

[64]  Prabhat,et al.  Physics-informed machine learning: case studies for weather and climate modelling , 2021, Philosophical Transactions of the Royal Society A.

[65]  Victorita Dolean,et al.  An introduction to domain decomposition methods - algorithms, theory, and parallel implementation , 2015 .

[66]  Pierre Ladevèze,et al.  Error Estimate Procedure in the Finite Element Method and Applications , 1983 .

[67]  Fernando Lau,et al.  Model order reduction in aerodynamics: Review and applications , 2019, Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering.

[68]  Roger A. Sauer,et al.  A monolithic fluid-structure interaction formulation for solid and liquid membranes including free-surface contact , 2017, Computer Methods in Applied Mechanics and Engineering.

[69]  Theodoros Damoulas,et al.  Variational Autoencoding of PDE Inverse Problems , 2020, ArXiv.

[70]  Siddhartha Verma,et al.  Restoring Chaos Using Deep Reinforcement Learning , 2020, Chaos.

[71]  I. Babuska,et al.  A‐posteriori error estimates for the finite element method , 1978 .

[72]  M. Marques,et al.  Recent advances and applications of machine learning in solid-state materials science , 2019, npj Computational Materials.

[73]  Charbel Farhat,et al.  A stochastic projection-based hyperreduced order model for model-form uncertainties in vibration analysis , 2018 .

[74]  Alexandre Allauzen,et al.  Control of chaotic systems by deep reinforcement learning , 2019, Proceedings of the Royal Society A.

[75]  Julien Yvonnet,et al.  Homogenization methods and multiscale modeling : Nonlinear problems , 2017 .

[76]  Sebastian Scher,et al.  Weather and climate forecasting with neural networks: using general circulation models (GCMs) with different complexity as a study ground , 2019, Geoscientific Model Development.

[77]  D. Stauffer,et al.  Optimal determination of nudging coefficients using the adjoint equations , 1993 .

[78]  J. Oden,et al.  A Posteriori Error Estimation in Finite Element Analysis: Oden/A Posteriori , 2000 .

[79]  Petros Koumoutsakos,et al.  Efficient collective swimming by harnessing vortices through deep reinforcement learning , 2018, Proceedings of the National Academy of Sciences.

[80]  Jean Rabault,et al.  Artificial neural networks trained through deep reinforcement learning discover control strategies for active flow control , 2018, Journal of Fluid Mechanics.

[81]  M. P. Brenner,et al.  Perspective on machine learning for advancing fluid mechanics , 2019, Physical Review Fluids.

[82]  Ruth N. Bolton,et al.  Customer experience challenges: bringing together digital, physical and social realms , 2018, Journal of Service Management.

[83]  Mohammad Abid Bazaz,et al.  A review of parametric model order reduction techniques , 2012, 2012 IEEE International Conference on Signal Processing, Computing and Control.

[84]  A. Chatterjee An introduction to the proper orthogonal decomposition , 2000 .

[85]  Harald Martens,et al.  Quantitative Big Data: where chemometrics can contribute , 2015 .

[86]  Trond Kvamsdal,et al.  Numerical investigation of modeling frameworks and geometric approximations on NREL 5 MW wind turbine , 2019, Renewable Energy.

[87]  Richard A. Anthes,et al.  Data Assimilation and Initialization of Hurricane Prediction Models , 1974 .

[88]  David Greiner,et al.  Advances in Evolutionary and Deterministic Methods for Design, Optimization and Control in Engineering and Sciences , 2016 .

[89]  Harbir Antil,et al.  Bilevel optimization, deep learning and fractional Laplacian regularization with applications in tomography , 2019, Inverse Problems.

[90]  Stefan Steinerberger,et al.  Clustering with t-SNE, provably , 2017, SIAM J. Math. Data Sci..

[91]  Xin Wang,et al.  A smart agriculture IoT system based on deep reinforcement learning , 2019, Future Gener. Comput. Syst..

[92]  Icíar Alfaro,et al.  Reduced order modeling for physically-based augmented reality , 2018, Computer Methods in Applied Mechanics and Engineering.

[93]  James Demmel,et al.  Matrix factorizations at scale: A comparison of scientific data analytics in spark and C+MPI using three case studies , 2016, 2016 IEEE International Conference on Big Data (Big Data).

[94]  Jing Li,et al.  A FETI-DP Type Domain Decomposition Algorithm for Three-Dimensional Incompressible Stokes Equations , 2014, SIAM J. Numer. Anal..

[95]  J. C. Cajas,et al.  Domain Decomposition Methods for Domain Composition Purpose: Chimera, Overset, Gluing and Sliding Mesh Methods , 2016, Archives of Computational Methods in Engineering.

[96]  Juan J. Alonso,et al.  Airfoil design optimization using reduced order models based on proper orthogonal decomposition , 2000 .

[97]  Stephen R. Marsland,et al.  A self-organising network that grows when required , 2002, Neural Networks.

[98]  A. Keane,et al.  Evolutionary Optimization of Computationally Expensive Problems via Surrogate Modeling , 2003 .

[99]  William D. Collins,et al.  Parameterization of Generalized Cloud Overlap for Radiative Calculations in General Circulation Models , 2001 .

[100]  C. Turc,et al.  Schur complement Domain Decomposition Methods for the solution of multiple scattering problems , 2016, 1608.00034.

[101]  Barry Smith,et al.  Domain Decomposition Methods for Partial Differential Equations , 1997 .

[102]  Yao Zhang,et al.  Application of Convolutional Neural Network to Predict Airfoil Lift Coefficient , 2017, ArXiv.

[103]  Clarence W. Rowley,et al.  Linearly-Recurrent Autoencoder Networks for Learning Dynamics , 2017, SIAM J. Appl. Dyn. Syst..

[104]  Vassilios Theofilis,et al.  Modal Analysis of Fluid Flows: An Overview , 2017, 1702.01453.

[105]  Omer San,et al.  Deep Reinforcement Learning Controller for 3D Path Following and Collision Avoidance by Autonomous Underwater Vehicles , 2020, Frontiers in Robotics and AI.

[106]  Vladimir M. Krasnopolsky,et al.  A new synergetic paradigm in environmental numerical modeling: Hybrid models combining deterministic and machine learning components , 2006 .

[107]  Gecheng Zha,et al.  High-Performance Airfoil Using Coflow Jet Flow Control , 2007 .

[108]  D. B. P. Huynh,et al.  Data‐driven physics‐based digital twins via a library of component‐based reduced‐order models , 2020, International Journal for Numerical Methods in Engineering.

[109]  J. Doyle,et al.  Numerical Modeling and Predictability of Mountain Wave-Induced Turbulence and Rotors , 2016 .

[110]  T. Chan,et al.  Domain decomposition algorithms , 1994, Acta Numerica.

[111]  Omer San,et al.  Long short-term memory embedded nudging schemes for nonlinear data assimilation of geophysical flows , 2020 .

[112]  Mohammad Bagher Menhaj,et al.  Training feedforward networks with the Marquardt algorithm , 1994, IEEE Trans. Neural Networks.

[113]  Matthew F. Barone,et al.  On the stability and convergence of a Galerkin reduced order model (ROM) of compressible flow with solid wall and far‐field boundary treatment , 2010 .

[114]  Trond Kvamsdal,et al.  Fast divergence-conforming reduced basis methods for steady Navier–Stokes flow , 2018, Computer Methods in Applied Mechanics and Engineering.

[115]  Kjell Magne Mathisen,et al.  Superconvergent Patch Recovery for plate problems using statically admissible stress resultant fields , 1999 .

[116]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[117]  Jacques Periaux,et al.  On Domain Decomposition Methods , 1988 .

[118]  E. Knobloch,et al.  Exact coherent structures in an asymptotically reduced description of parallel shear flows , 2015 .

[119]  Rami M. Younis,et al.  Localized Linear Systems in Sequential Implicit Simulation of Two-Phase Flow and Transport , 2017 .

[120]  E Weinan,et al.  Heterogeneous multiscale methods: A review , 2007 .

[121]  S. Lakshmivarahan,et al.  Nudging Methods: A Critical Overview , 2013 .

[122]  Traian Iliescu,et al.  A long short-term memory embedding for hybrid uplifted reduced order models , 2019, Physica A: Statistical Mechanics and its Applications.

[123]  Endre Süli,et al.  Adaptive finite element methods for differential equations , 2003, Lectures in mathematics.

[124]  Andrei Draganescu,et al.  Multigrid Preconditioners for the Newton-Krylov Method in the Optimal Control of the Stationary Navier-Stokes Equations , 2018, SIAM J. Numer. Anal..

[125]  Babak Hejazialhosseini,et al.  Reinforcement Learning and Wavelet Adapted Vortex Methods for Simulations of Self-propelled Swimmers , 2014, SIAM J. Sci. Comput..

[126]  John Harlim,et al.  Bridging data science and dynamical systems theory , 2020 .

[127]  Aditya Konduri,et al.  Asynchronous finite-difference schemes for partial differential equations , 2014, J. Comput. Phys..

[128]  Claes Johnson,et al.  Adaptive finite element methods in computational mechanics , 1992 .

[129]  Hang Yu,et al.  Deep reinforcement learning with its application for lung cancer detection in medical Internet of Things , 2019, Future Gener. Comput. Syst..

[130]  Karthik Duraisamy,et al.  Prediction of aerodynamic flow fields using convolutional neural networks , 2019, Computational Mechanics.

[131]  G. Haller Lagrangian Coherent Structures , 2015 .

[132]  Gianluigi Rozza,et al.  A reduced order variational multiscale approach for turbulent flows , 2018, Advances in Computational Mathematics.

[133]  Sharath Girimaji,et al.  Proxy-equation paradigm: A strategy for massively parallel asynchronous computations. , 2017, Physical review. E.

[134]  Claus-Dieter Munz,et al.  Deep Neural Networks for Data-Driven Turbulence Models , 2018, J. Comput. Phys..

[135]  G. Grell,et al.  A North American Hourly Assimilation and Model Forecast Cycle: The Rapid Refresh , 2016 .

[136]  Hongkai Zhao,et al.  Absorbing boundary conditions for domain decomposition , 1998 .

[137]  He Zhang,et al.  Digital Twin in Industry: State-of-the-Art , 2019, IEEE Transactions on Industrial Informatics.

[138]  Andrea Toselli,et al.  Recent developments in domain decomposition methods , 2002 .

[139]  Alessio Fumagalli,et al.  Unified approach to discretization of flow in fractured porous media , 2018, Computational Geosciences.

[140]  K. M. Okstad,et al.  Numerical benchmarking of fluid–structure interaction: An isogeometric finite element approach , 2016 .

[141]  E. S. Politis,et al.  Modeling wake effects in large wind farms in complex terrain: the problem, the methods and the issues , 2012 .

[142]  B. R. Noack,et al.  Closed-Loop Turbulence Control: Progress and Challenges , 2015 .

[143]  Weiwei Zhang,et al.  Deep neural network for unsteady aerodynamic and aeroelastic modeling across multiple Mach numbers , 2019, Nonlinear Dynamics.

[144]  Omer San,et al.  Memory embedded non-intrusive reduced order modeling of non-ergodic flows , 2019 .

[145]  Senlin Zhu,et al.  River/stream water temperature forecasting using artificial intelligence models: a systematic review , 2020, Acta Geophysica.

[146]  Jens L. Eftang,et al.  An hp certified reduced basis method for parametrized parabolic partial differential equations , 2011 .

[147]  Guillaume Houzeaux,et al.  A Review of Domain Decomposition Methods for Simulation of Fluid Flows: Concepts, Algorithms, and Applications , 2020 .

[148]  Pan He,et al.  Adversarial Examples: Attacks and Defenses for Deep Learning , 2017, IEEE Transactions on Neural Networks and Learning Systems.

[149]  O. Widlund,et al.  Iterative Methods for the Solution of Elliptic Problems on Regions, Partitioned Into Substructures , 2015 .

[150]  Amir Younan,et al.  Model Reduction Methods for Rotor Dynamic Analysis: A Survey and Review , 2010 .

[151]  R. Todling,et al.  Data Assimilation in the Presence of Forecast Bias: The GEOS Moisture Analysis , 2000 .

[152]  Siamak Niroomandi,et al.  Accounting for large deformations in real-time simulations of soft tissues based on reduced-order models , 2012, Comput. Methods Programs Biomed..

[153]  John D. Horel,et al.  Sensitivity of a Spectrally Filtered and Nudged Limited-Area Model to Outer Model Options , 1996 .

[154]  Mohamed Gad-el-Hak,et al.  Modern developments in flow control , 1996 .

[155]  O. Zienkiewicz,et al.  Analysis of the Zienkiewicz–Zhu a‐posteriori error estimator in the finite element method , 1989 .

[156]  Omer San,et al.  Comparative study of sequential data assimilation methods for the Kuramoto-Sivashinsky equation , 2020, AIAA Scitech 2021 Forum.

[157]  Omer San,et al.  A nudged hybrid analysis and modeling approach for realtime wake-vortex transport and decay prediction , 2020, Computers & Fluids.

[158]  Georgiy L. Stenchikov,et al.  Spectral nudging to eliminate the effects of domain position and geometry in regional climate model simulations , 2004 .

[159]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .

[160]  Blake Barker,et al.  Parameterization Method for Unstable Manifolds of Standing Waves on the Line , 2020, SIAM J. Appl. Dyn. Syst..

[161]  Vladimir M. Krasnopolsky,et al.  Complex hybrid models combining deterministic and machine learning components for numerical climate modeling and weather prediction , 2006, Neural Networks.

[162]  Arlindo da Silva,et al.  Data assimilation in the presence of forecast bias , 1998 .

[163]  T. Hughes,et al.  Large Eddy Simulation and the variational multiscale method , 2000 .

[164]  Abdulmajeed A. Mohamad,et al.  A review of the development of hybrid atomistic–continuum methods for dense fluids , 2010 .

[165]  Ronald A. DeVore,et al.  Tree approximation of the long wave radiation parameterization in the NCAR CAM global climate model , 2011, J. Comput. Appl. Math..

[166]  Traian Iliescu,et al.  A non-intrusive reduced order modeling framework for quasi-geostrophic turbulence , 2019, Physical review. E.

[167]  Raphael T. Haftka,et al.  Surrogate-based Analysis and Optimization , 2005 .

[168]  C. Farhat,et al.  A method of finite element tearing and interconnecting and its parallel solution algorithm , 1991 .

[169]  R. Freund Model reduction methods based on Krylov subspaces , 2003, Acta Numerica.

[170]  A. Rasheed,et al.  Implementation and comparison of three isogeometric Navier–Stokes solvers applied to simulation of flow past a fixed 2D NACA0012 airfoil at high Reynolds number , 2015 .

[171]  Trond Kvamsdal,et al.  Simple a posteriori error estimators in adaptive isogeometric analysis , 2015, Comput. Math. Appl..

[172]  Volker Mehrmann,et al.  The Shifted Proper Orthogonal Decomposition: A Mode Decomposition for Multiple Transport Phenomena , 2015, SIAM J. Sci. Comput..

[173]  S. Nepomnyaschikh,et al.  Domain Decomposition Methods , 2007 .

[174]  Peter A. G. Watson,et al.  Applying Machine Learning to Improve Simulations of a Chaotic Dynamical System Using Empirical Error Correction , 2019, Journal of advances in modeling earth systems.

[175]  James B. Rawlings,et al.  Model predictive control with linear models , 1993 .

[176]  Daniel A. Keim,et al.  What you see is what you can change: Human-centered machine learning by interactive visualization , 2017, Neurocomputing.

[177]  Michael Schneier,et al.  Diagnostics for Eddy Viscosity Models of Turbulence Including Data-Driven/Neural Network Based Parameterizations , 2019, Results in Applied Mathematics.

[178]  T. Coroller,et al.  Deep Learning Predicts Lung Cancer Treatment Response from Serial Medical Imaging , 2019, Clinical Cancer Research.

[179]  Karthik Duraisamy,et al.  Modal Analysis of Fluid Flows: Applications and Outlook , 2019, AIAA Journal.

[180]  Omer San,et al.  Taming an Autonomous Surface Vehicle for Path Following and Collision Avoidance Using Deep Reinforcement Learning , 2019, IEEE Access.

[181]  Thomas J. R. Hughes,et al.  The multiscale formulation of large eddy simulation: Decay of homogeneous isotropic turbulence , 2001 .

[182]  Trond Kvamsdal,et al.  Goal oriented error estimators for Stokes equations based on variationally consistent postprocessing , 2003 .

[183]  John D. Macpherson Technology Focus: Drilling Management and Automation (September 2015) , 2015 .

[184]  O'Connell,et al.  Molecular dynamics-continuum hybrid computations: A tool for studying complex fluid flows. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[185]  Sergei V. Kalinin,et al.  Big-deep-smart data in imaging for guiding materials design. , 2015, Nature materials.

[186]  Giovanni Samaey,et al.  Equation-free multiscale computation: algorithms and applications. , 2009, Annual review of physical chemistry.

[187]  S. M. Rahman,et al.  A Hybrid Approach for Model Order Reduction of Barotropic Quasi-Geostrophic Turbulence , 2018, Fluids.

[188]  Henry D. I. Abarbanel,et al.  Machine Learning: Deepest Learning as Statistical Data Assimilation Problems , 2017, Neural Computation.

[189]  A. Patera,et al.  Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations , 2007 .

[190]  David Q. Mayne,et al.  Model predictive control: Recent developments and future promise , 2014, Autom..

[191]  A. Piacentini,et al.  Determination of optimal nudging coefficients , 2003 .

[192]  Karthik Duraisamy,et al.  Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamics , 2019, Computer Methods in Applied Mechanics and Engineering.

[193]  Francisco Herrera,et al.  Enabling Smart Data: Noise filtering in Big Data classification , 2017, Inf. Sci..

[195]  Mary F. Wheeler,et al.  Physical and Computational Domain Decompositions for Modeling Subsurface Flows , 2007 .

[196]  J. Peraire,et al.  A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations , 1997 .

[197]  Jacob Fish,et al.  Multiscale Methods: Bridging the Scales in Science and Engineering , 2009 .

[198]  Liu Yang,et al.  Reinforcement learning for bluff body active flow control in experiments and simulations , 2020, Proceedings of the National Academy of Sciences.

[199]  Shady E. Ahmed,et al.  Reduced order modeling of fluid flows: Machine learning, Kolmogorov barrier, closure modeling, and partitioning , 2020, 2005.14246.

[200]  H. S. Tang,et al.  An exploratory study on machine learning to couple numerical solutions of partial differential equations , 2020, ArXiv.

[201]  Andrea Toselli,et al.  Domain decomposition methods : algorithms and theory , 2005 .

[202]  I. Kevrekidis,et al.  Equation-free/Galerkin-free POD-assisted computation of incompressible flows , 2005 .

[203]  Trond Kvamsdal,et al.  Superconvergent patch recovery and a posteriori error estimation technique in adaptive isogeometric analysis , 2017 .

[204]  M. Kramer Nonlinear principal component analysis using autoassociative neural networks , 1991 .

[205]  Azeddine Kourta,et al.  Closed-loop separation control over a sharp edge ramp using genetic programming , 2015, 1508.05268.

[206]  Anthony T. Patera,et al.  Certified reduced basis model validation: A frequentistic uncertainty framework , 2012 .

[207]  Trond Kvamsdal,et al.  Isogeometric analysis using LR B-splines , 2014 .

[208]  Chao Yan,et al.  Non-intrusive reduced-order modeling for fluid problems: A brief review , 2019, Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering.

[209]  Heung-Il Suk,et al.  Deep Learning in Medical Image Analysis. , 2017, Annual review of biomedical engineering.

[210]  Z. Bai Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems , 2002 .

[211]  P. Stern,et al.  Automatic choice of global shape functions in structural analysis , 1978 .

[212]  A. A. Mishra,et al.  Optimization under turbulence model uncertainty for aerospace design , 2019, Physics of Fluids.

[213]  Martin J. Gander,et al.  Heterogeneous Optimized Schwarz Methods for Second Order Elliptic PDEs , 2019, SIAM J. Sci. Comput..

[214]  Knut Steinar Bjorkevoll,et al.  Use of High Fidelity Models for Real Time Status Detection with Field Examples from Automated MPD Operations in the North Sea , 2015 .

[215]  Martin T. Hagan,et al.  Neural network design , 1995 .

[216]  Noëlle A. Scott,et al.  The "weight smoothing" regularization of MLP for Jacobian stabilization , 1999, IEEE Trans. Neural Networks.

[217]  Raducanu Razvan,et al.  MATHEMATICAL MODELS and METHODS in APPLIED SCIENCES , 2012 .

[218]  Fernando Iafrate From Big Data to Smart Data: Iafrate/From Big Data to Smart Data , 2015 .

[219]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[220]  Xiaoming He,et al.  Parallel, non-iterative, multi-physics domain decomposition methods for time-dependent Stokes-Darcy systems , 2014, Math. Comput..

[221]  Hui Tang,et al.  Active control of vortex-induced vibration of a circular cylinder using machine learning , 2019, Physics of Fluids.

[222]  Peter R. Oke,et al.  A deterministic formulation of the ensemble Kalman filter: an alternative to ensemble square root filters , 2008 .

[223]  Anthony T. Patera,et al.  A component-based hybrid reduced basis/finite element method for solid mechanics with local nonlinearities , 2018 .

[224]  Cheng Huang,et al.  Reduced-Order Modeling Framework for Combustor Instabilities Using Truncated Domain Training , 2018, AIAA Journal.

[225]  P. Gosselet,et al.  A monolithic strategy based on an hybrid domain decomposition method for multiphysic problems , 2004 .

[226]  Martin J. Gander,et al.  Techniques for Locally Adaptive Time Stepping Developed over the Last Two Decades , 2013, Domain Decomposition Methods in Science and Engineering XX.

[227]  C. Meneveau Big wind power: seven questions for turbulence research , 2019, Journal of Turbulence.

[228]  Anthony T. Patera,et al.  High-Fidelity Real-Time Simulation on Deployed Platforms , 2011 .

[229]  Omer San,et al.  PyDA: A Hands-On Introduction to Dynamical Data Assimilation with Python , 2020, Fluids.

[230]  Omer San,et al.  Principal interval decomposition framework for POD reduced‐order modeling of convective Boussinesq flows , 2015 .

[231]  Kookjin Lee,et al.  Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders , 2018, J. Comput. Phys..

[232]  Olof B. Widlund,et al.  FETI‐DP, BDDC, and block Cholesky methods , 2006 .

[233]  Charles Meneveau,et al.  Modelling yawed wind turbine wakes: a lifting line approach , 2018, Journal of Fluid Mechanics.

[234]  E Kaiser,et al.  Sparse identification of nonlinear dynamics for model predictive control in the low-data limit , 2017, Proceedings of the Royal Society A.

[235]  Alfio Quarteroni,et al.  The Interface Control Domain Decomposition (ICDD) Method for Elliptic Problems , 2013, SIAM J. Control. Optim..

[236]  O. C. Zienkiewicz,et al.  A simple error estimator and adaptive procedure for practical engineerng analysis , 1987 .

[237]  Bryan A. Tolson,et al.  Review of surrogate modeling in water resources , 2012 .

[238]  Michael G. Kapteyn,et al.  A probabilistic graphical model foundation for enabling predictive digital twins at scale , 2020, Nature Computational Science.

[239]  James Demmel,et al.  Matrix Factorization at Scale: a Comparison of Scientific Data Analytics in Spark and C+MPI Using Three Case Studies , 2016, ArXiv.

[240]  W. Wall,et al.  Truly monolithic algebraic multigrid for fluid–structure interaction , 2011 .

[241]  Trond Kvamsdal,et al.  Simulation of airflow past a 2D NACA0015 airfoil using an isogeometric incompressible Navier-Stokes solver with the Spalart-Allmaras turbulence model , 2015 .

[242]  Aris Tsangrassoulis,et al.  Algorithms for optimization of building design: A review , 2014 .

[243]  Prakash Vedula,et al.  Data-driven deconvolution for large eddy simulations of Kraichnan turbulence , 2018, Physics of Fluids.

[244]  H. P. Lee,et al.  PROPER ORTHOGONAL DECOMPOSITION AND ITS APPLICATIONS—PART I: THEORY , 2002 .

[245]  A. Telea,et al.  Computing and Visualization in Science , 2022 .

[246]  S. Brunton,et al.  Discovering governing equations from data by sparse identification of nonlinear dynamical systems , 2015, Proceedings of the National Academy of Sciences.

[247]  Sondipon Adhikari,et al.  Machine learning based digital twin for dynamical systems with multiple time-scales , 2020, ArXiv.

[248]  Richard S. Sutton,et al.  Reinforcement Learning: An Introduction , 1998, IEEE Trans. Neural Networks.

[249]  Omer San,et al.  Spatiotemporally dynamic implicit large eddy simulation using machine learning classifiers , 2020 .

[250]  H. Storch,et al.  A Spectral Nudging Technique for Dynamical Downscaling Purposes , 2000 .

[251]  T. Wick,et al.  To appear in Fluid-Structure Interactions . Modeling , Adaptive Discretization and Solvers , 2017 .

[252]  Hui Tang,et al.  Applying deep reinforcement learning to active flow control in turbulent conditions , 2020, 2006.10683.

[253]  Michelle Girvan,et al.  Hybrid Forecasting of Chaotic Processes: Using Machine Learning in Conjunction with a Knowledge-Based Model , 2018, Chaos.

[254]  Gilead Tadmor,et al.  Reduced-Order Modelling for Flow Control , 2013 .

[255]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[256]  Francisco Chinesta,et al.  A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids - Part II: Transient simulation using space-time separated representations , 2007 .

[257]  Omer San,et al.  Digital Twin: Values, Challenges and Enablers From a Modeling Perspective , 2019, IEEE Access.

[258]  William Gropp,et al.  DOMAIN DECOMPOSITION METHODS IN COMPUTATIONAL FLUID DYNAMICS , 1991 .

[259]  M. Ainsworth,et al.  A posteriori error estimators in the finite element method , 1991 .

[260]  Yoshua Bengio,et al.  Generative Adversarial Nets , 2014, NIPS.

[261]  D. Chalikov,et al.  New Approach to Calculation of Atmospheric Model Physics: Accurate and Fast Neural Network Emulation of Longwave Radiation in a Climate Model , 2005 .

[262]  Prakash Vedula,et al.  A priori analysis on deep learning of subgrid-scale parameterizations for Kraichnan turbulence , 2019, Theoretical and Computational Fluid Dynamics.

[263]  Lili Lei,et al.  Nudging, Ensemble, and Nudging Ensembles for Data Assimilation in the Presence of Model Error , 2015 .

[264]  Pieter Abbeel,et al.  Benchmarking Deep Reinforcement Learning for Continuous Control , 2016, ICML.

[265]  Omer San,et al.  Neural network closures for nonlinear model order reduction , 2017, Adv. Comput. Math..

[266]  F. Porté-Agel,et al.  A new analytical model for wind-turbine wakes , 2013 .

[267]  Duncan A. Mellichamp,et al.  A unified derivation and critical review of modal approaches to model reduction , 1982 .

[268]  Francisco Chinesta,et al.  Virtual, Digital and Hybrid Twins: A New Paradigm in Data-Based Engineering and Engineered Data , 2018, Archives of Computational Methods in Engineering.

[269]  Karen Willcox,et al.  A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems , 2015, SIAM Rev..

[270]  Suraj Pawar,et al.  An Evolve-Then-Correct Reduced Order Model for Hidden Fluid Dynamics , 2019, Mathematics.

[271]  Long Cu Ngo,et al.  A comparative study between partitioned and monolithic methods for the problems with 3D fluid-structure interaction of blood vessels , 2017 .

[272]  M. Alford,et al.  Application of a model of internal hydraulic jumps , 2017, Journal of Fluid Mechanics.

[273]  Martin J. Gander,et al.  Schwarz Methods over the Course of Time , 2008 .

[274]  Pierre Ladevèze Editorial – Advanced modeling and simulation in engineering sciences , 2014, Adv. Model. Simul. Eng. Sci..

[275]  A. Quarteroni,et al.  On the coupling of 1D and 3D diffusion-reaction equations. Applications to tissue perfusion problems , 2008 .

[276]  William W. Hsieh,et al.  Applying Neural Network Models to Prediction and Data Analysis in Meteorology and Oceanography. , 1998 .

[277]  I. Mezić,et al.  Analysis of Fluid Flows via Spectral Properties of the Koopman Operator , 2013 .

[278]  Atanas Stefanov,et al.  Periodic Traveling Waves of the Regularized Short Pulse and Ostrovsky Equations: Existence and Stability , 2017, SIAM J. Math. Anal..

[279]  J. Blum,et al.  Back and forth nudging algorithm for data assimilation problems , 2005 .

[280]  Dirk Roose,et al.  Coping with complexity : model reduction and data analysis , 2011 .

[281]  T. Hughes Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods , 1995 .

[282]  Kevin B. Flores,et al.  Hybrid modeling and prediction of dynamical systems , 2017, PLoS Comput. Biol..

[283]  Martin J. Gander,et al.  Optimized Schwarz methods for a diffusion problem with discontinuous coefficient , 2014, Numerical Algorithms.

[284]  Joachim Denzler,et al.  Deep learning and process understanding for data-driven Earth system science , 2019, Nature.

[285]  Geoffrey E. Hinton,et al.  Visualizing Data using t-SNE , 2008 .

[286]  B. Moore Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[287]  C. W. Gear,et al.  Equation-Free, Coarse-Grained Multiscale Computation: Enabling Mocroscopic Simulators to Perform System-Level Analysis , 2003 .

[288]  P. Spalart,et al.  A hybrid RANS-LES approach with delayed-DES and wall-modelled LES capabilities , 2008 .

[289]  M. Ani Hsieh,et al.  Small and Adrift with Self-Control: Using the Environment to Improve Autonomy , 2015, ISRR.

[290]  Jennifer R. Lukes,et al.  A hybrid atomistic-continuum model for fluid flow using LAMMPS and OpenFOAM , 2013, Comput. Phys. Commun..

[291]  Roger L. Davis,et al.  A multi-code-coupling interface for combustor/turbomachinery simulations , 2001 .

[292]  Geoffrey E. Hinton,et al.  Stochastic Neighbor Embedding , 2002, NIPS.

[293]  Raluca Radu,et al.  Spectral nudging in a spectral regional climate model , 2008 .

[294]  Anthony J. Jakeman,et al.  A review of surrogate models and their application to groundwater modeling , 2015 .

[295]  Lukas Hewing,et al.  Learning-Based Model Predictive Control: Toward Safe Learning in Control , 2020, Annu. Rev. Control. Robotics Auton. Syst..

[296]  T. McAvoy,et al.  Nonlinear principal component analysis—Based on principal curves and neural networks , 1996 .

[297]  Alfio Quarteroni,et al.  A 3D/1D geometrical multiscale model of cerebral vasculature , 2009 .

[298]  Jiliang Tang,et al.  Adversarial Attacks and Defenses in Images, Graphs and Text: A Review , 2019, International Journal of Automation and Computing.

[299]  Charbel Farhat,et al.  Learning constitutive relations from indirect observations using deep neural networks , 2020, J. Comput. Phys..

[300]  R. Pinnau Model Reduction via Proper Orthogonal Decomposition , 2008 .

[301]  Christopher J. Roy,et al.  Review of code and solution verification procedures for computational simulation , 2005 .

[302]  Anthony T. Patera,et al.  Output bounds for reduced-order approximations of elliptic partial differential equations , 2001 .

[303]  B. Eckhardt,et al.  Small scale exact coherent structures at large Reynolds numbers in plane Couette flow , 2017, 1710.00741.

[304]  S. De,et al.  Integrating machine learning and multiscale modeling—perspectives, challenges, and opportunities in the biological, biomedical, and behavioral sciences , 2019, npj Digital Medicine.

[305]  Variable Selection in Nonlinear Principal Component Analysis , 2016 .

[306]  Dimitris Drikakis,et al.  Machine-Learning Methods for Computational Science and Engineering , 2020, Comput..

[307]  J. Peinke,et al.  Grand challenges in the science of wind energy , 2019, Science.

[308]  Alfio Quarteroni,et al.  Geometric multiscale modeling of the cardiovascular system, between theory and practice , 2016 .

[309]  Ming Zhong,et al.  Nonparametric inference of interaction laws in systems of agents from trajectory data , 2018, Proceedings of the National Academy of Sciences.

[310]  A. Soliman,et al.  Author Biography , 2018, Understanding Language Use in the Classroom.

[311]  Knut Morten Okstad,et al.  Error estimation based on Superconvergent Patch Recovery using statically admissible stress fields , 1998 .

[312]  E Weinan,et al.  Domain decomposition interface preconditioners for fourth-order elliptic problems , 1991 .

[313]  Yogendra Joshi,et al.  Reduced order thermal modeling of data centers via proper orthogonal decomposition: a review , 2010 .

[314]  Paris Perdikaris,et al.  Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data , 2019, J. Comput. Phys..

[315]  M. Tokarev,et al.  Deep Reinforcement Learning Control of Cylinder Flow Using Rotary Oscillations at Low Reynolds Number , 2020, Energies.

[316]  Alfio Quarteroni,et al.  Analysis of a Geometrical Multiscale Model Based on the Coupling of ODE and PDE for Blood Flow Simulations , 2003, Multiscale Model. Simul..

[317]  Omer San,et al.  Reduced order modeling of fluid flows: Machine learning, Kolmogorov barrier, closure modeling, and partitioning (Invited) , 2020 .

[318]  Masahiro Kuroda,et al.  Nonlinear Principal Component Analysis and Its Applications , 2016 .

[319]  Francisco Charte,et al.  A practical tutorial on autoencoders for nonlinear feature fusion: Taxonomy, models, software and guidelines , 2018, Inf. Fusion.

[320]  Thomas Richter,et al.  5. Recent development of robust monolithic fluid-structure interaction solvers , 2017 .

[321]  Adam H. Monahan,et al.  Nonlinear Principal Component Analysis by Neural Networks: Theory and Application to the Lorenz System , 2000 .

[322]  Gianluigi Rozza,et al.  Reduced Order Methods for Modeling and Computational Reduction , 2013 .

[323]  Carlos A. Felippa,et al.  Staggered transient analysis procedures for coupled mechanical systems: Formulation , 1980 .

[324]  Dirk Hartmann,et al.  Model Order Reduction a Key Technology for Digital Twins , 2018 .

[325]  Olivier P. Le Maître,et al.  Polynomial chaos expansion for sensitivity analysis , 2009, Reliab. Eng. Syst. Saf..

[326]  Xiao-Chuan Cai,et al.  Scalable parallel methods for monolithic coupling in fluid-structure interaction with application to blood flow modeling , 2010, J. Comput. Phys..

[327]  S. Michael Spottswood,et al.  A review of indirect/non-intrusive reduced order modeling of nonlinear geometric structures , 2013 .

[328]  C. Caldwell Mathematics of Computation , 1999 .

[329]  Karthik Duraisamy,et al.  Physics-Informed Probabilistic Learning of Linear Embeddings of Nonlinear Dynamics with Guaranteed Stability , 2019, SIAM J. Appl. Dyn. Syst..

[330]  N. Nguyen,et al.  REDUCED BASIS APPROXIMATION AND A POSTERIORI ERROR ESTIMATION FOR THE PARAMETRIZED UNSTEADY BOUSSINESQ EQUATIONS , 2011 .

[331]  A. Geer,et al.  Learning earth system models from observations: machine learning or data assimilation? , 2021, Philosophical Transactions of the Royal Society A.

[332]  Paris Perdikaris,et al.  Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , 2019, J. Comput. Phys..

[333]  Burak Aksoylu,et al.  Physics guided machine learning using simplified theories , 2020, Physics of Fluids.

[334]  Kurt Hornik,et al.  Multilayer feedforward networks are universal approximators , 1989, Neural Networks.

[335]  Hod Lipson,et al.  Distilling Free-Form Natural Laws from Experimental Data , 2009, Science.

[336]  S. Pawar,et al.  A deep learning enabler for nonintrusive reduced order modeling of fluid flows , 2019, Physics of Fluids.

[337]  Charbel Farhat,et al.  Partitioned analysis of coupled mechanical systems , 2001 .

[338]  Ajmal Mian,et al.  Threat of Adversarial Attacks on Deep Learning in Computer Vision: A Survey , 2018, IEEE Access.

[339]  Anthony T. Patera,et al.  A Two-Step Certified Reduced Basis Method , 2012, J. Sci. Comput..

[340]  Symplectic homology and periodic orbits near symplectic submanifolds , 2002, math/0210468.

[341]  W. Musial,et al.  A systems engineering vision for floating offshore wind cost optimization , 2020 .

[342]  Omer San,et al.  Stabilized principal interval decomposition method for model reduction of nonlinear convective systems with moving shocks , 2018, Computational and Applied Mathematics.

[343]  van Eh Harald Brummelen,et al.  Partitioned iterative solution methods for fluid–structure interaction , 2011 .

[344]  Siegfried Wahl,et al.  Leveraging uncertainty information from deep neural networks for disease detection , 2016, Scientific Reports.

[345]  Vladimir M. Krasnopolsky,et al.  A neural network technique to improve computational efficiency of numerical oceanic models , 2002 .

[346]  Gautam Reddy,et al.  Learning to soar in turbulent environments , 2016, Proceedings of the National Academy of Sciences.

[347]  Omer San,et al.  A neural network approach for the blind deconvolution of turbulent flows , 2017, Journal of Fluid Mechanics.

[348]  Geoffrey E. Hinton,et al.  Reducing the Dimensionality of Data with Neural Networks , 2006, Science.

[349]  G. Martin,et al.  Nonlinear model predictive control , 1999, Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251).

[350]  Jean-Luc Aider,et al.  Closed-loop separation control using machine learning , 2014, Journal of Fluid Mechanics.

[351]  Petros Koumoutsakos,et al.  Machine Learning for Fluid Mechanics , 2019, Annual Review of Fluid Mechanics.

[352]  Manfred Morari,et al.  Model predictive control: Theory and practice - A survey , 1989, Autom..