On closures for reduced order models $-$ A spectrum of first-principle to machine-learned avenues

Abstract

For over a century, reduced order models (ROMs) have been a fundamental discipline of theoretical fluid mechanics. Early examples include Galerkin models inspired by the Orr-Sommerfeld stability equation and numerous vortex models, of which the von Kármán vortex street is one of the most prominent. Subsequent ROMs typically relied on first principles, like mathematical Galerkin models, weakly nonlinear stability theory, and two- and three-dimensional vortex models. Aubry et al. [N. Aubry, P. Holmes, J. Lumley, and E. Stone, Journal of Fluid Mechanics, 192, 115-173 (1988)] pioneered data-driven proper orthogonal decomposition (POD) modeling. In early POD modeling, available data was used to build an optimal basis, which was then utilized in a classical Galerkin procedure to construct the ROM. But data has made a profound impact on ROMs beyond the Galerkin expansion. In this paper, we take a modest step and illustrate the impact of data-driven modeling on one significant ROM area. Specifically, we focus on ROM closures, which are correction terms that are added to classical ROMs in order to model the effect of the discarded ROM modes in under-resolved simulations. Through simple examples, we illustrate the main modeling principles used to construct classical ROMs, motivate and introduce modern ROM closures, and show how data-driven modeling, artificial intelligence, and machine learning have changed the standard ROM methodology over the last two decades. Finally, we outline our vision on how state-of-the-art data-driven modeling can continue to reshape the field of reduced order modeling.

References in corpus (111)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• Machine Learning for Fluid Mechanics
• Gene Expression Programming: a New Adaptive Algorithm for Solving Problems
• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data
• On Dynamic Mode Decomposition: Theory and Applications
• Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis
• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data
• A roadmap for the computation of persistent homology
• The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows
• Spectral proper orthogonal decomposition
• Proper Orthogonal Decomposition Closure Models For Turbulent Flows: A Numerical Comparison
• Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks
• Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks
• A New Selection Operator for the Discrete Empirical Interpolation Method -- improved a priori error bound and extensions
• Cluster-based reduced-order modelling of a mixing layer
• Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems
• Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledge
• Sparse reduced-order modeling : Sensor-based dynamics to full-state estimation
• Data-assisted reduced-order modeling of extreme events in complex dynamical systems
• Data-Driven POD-Galerkin Reduced Order Model for Turbulent Flows
• A Novel Model Order Reduction Approach for Navier-Stokes Equations at High Reynolds Number
• Machine Learning for Stochastic Parameterization: Generative Adversarial Networks in the Lorenz '96 Model
• Convolutional neural network based hierarchical autoencoder for nonlinear mode decomposition of fluid field data
• Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: a case study with the Lorenz 96 model
• On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high Reynolds number flow over an Ahmed body
• A deep learning enabler for non-intrusive reduced order modeling of fluid flows
• Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamics
• Enforcing Statistical Constraints in Generative Adversarial Networks for Modeling Chaotic Dynamical Systems
• An artificial neural network framework for reduced order modeling of transient flows
• Multi-Scale Proper Orthogonal Decomposition of Complex Fluid Flows
• Data-Driven Model Reduction for the Bayesian Solution of Inverse Problems
• Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability
• Fluid flow dynamics under location uncertainty
• A Perspective on Machine Learning Methods in Turbulence Modelling
• Prediction of Dynamical Systems by Symbolic Regression
• Data-driven reduced-order models via regularized operator inference for a single-injector combustion process
• Physics guided machine learning using simplified theories
• A Deep Learning based Approach to Reduced Order Modeling for Turbulent Flow Control using LSTM Neural Networks
• Nonlinear Model Order Reduction via Lifting Transformations and Proper Orthogonal Decomposition
• Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms
• On the stability of projection-based model order reduction for convection-dominated laminar and turbulent flows
• Extreme learning machine for reduced order modeling of turbulent geophysical flows
• Nonparametric forecasting of low-dimensional dynamical systems
• Comparison of POD reduced order strategies for the nonlinear 2D Shallow Water Equations
• Achieving Conservation of Energy in Neural Network Emulators for Climate Modeling
• Learning physics-based reduced-order models for a single-injector combustion process
• Data-driven recovery of hidden physics in reduced order modeling of fluid flows
• Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization
• Physics-constrained, low-dimensional models for MHD: First-principles and data-driven approaches
• Formulating turbulence closures using sparse regression with embedded form invariance
• A discrete approach to stochastic parametrization and dimensional reduction in nonlinear dynamics
• A non-intrusive reduced order modeling framework for quasi-geostrophic turbulence
• Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier-Stokes equations
• Data-driven Discovery of Closure Models
• Data-Driven Variational Multiscale Reduced Order Models
• Persistent topological features of dynamical systems
• Feature engineering and symbolic regression methods for detecting hidden physics from sparse sensors
• Physically-Constrained Data-Driven, Filtered Reduced Order Modeling of Fluid Flows
• Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism
• DR-RNN: A deep residual recurrent neural network for model reduction
• Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction
• Stochastic parameterization identification using ensemble Kalman filtering combined with expectation-maximization and Newton-Raphson maximum likelihood methods
• A stabilized proper orthogonal decomposition reduced-order model for large scale quasigeostrophic ocean circulation
• Proper orthogonal decomposition closure models for fluid flows: Burgers equation
• Domain-decomposition least-squares Petrov-Galerkin (DD-LSPG) nonlinear model reduction
• A long short-term memory embedding for hybrid uplifted reduced order models
• A POD-Galerkin reduced order model for a LES filtering approach
• A minimization principle for the description of time-dependent modes associated with transient instabilities
• Cluster-based network model
• On a certified Smagorinsky reduced basis turbulence model
• Memory embedded non-intrusive reduced order modeling of non-ergodic flows
• A Priori Estimation Of Memory Effects In Coarse-Grained Nonlinear Systems Using The Mori-Zwanzig Formalism
• Model fusion with physics-guided machine learning
• Continuous Data Assimilation Reduced Order Models of Fluid Flow
• Optimal Nonlinear Eddy Viscosity in Galerkin Models of Turbulent Flows
• A Dynamic Subgrid Scale Model for Large Eddy Simulations Based on the Mori-Zwanzig Formalism
• Reduced-space Gaussian Process Regression for Data-Driven Probabilistic Forecast of Chaotic Dynamical Systems
• Exploring the Topology of Dynamical Reconstructions
• A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions
• Data-driven super-parameterization using deep learning: Experimentation with multi-scale Lorenz 96 systems and transfer-learning
• Embedding Hard Physical Constraints in Neural Network Coarse-Graining of 3D Turbulence
• Turbulence Enrichment using Physics-informed Generative Adversarial Networks
• On-the-fly Reduced Order Modeling of Passive and Reactive Species via Time-Dependent Manifolds
• Non-intrusive model reduction of large-scale, nonlinear dynamical systems using deep learning
• Dimensionality Reduction and Reduced Order Modeling for Traveling Wave Physics
• Latent-space time evolution of non-intrusive reduced-order models using Gaussian process emulation
• Online adaptive basis refinement and compression for reduced-order models via vector-space sieving
• Data-driven closures for stochastic dynamical systems
• Real-Time Reduced-Order Modeling of Stochastic Partial Differential Equations via Time-Dependent Subspaces
• Reduced order methods for parametric optimal flow control in coronary bypass grafts, towards patient-specific data assimilation
• Data-driven Modeling of Rotating Detonation Waves
• Data-Driven Correction Reduced Order Models for the Quasi-Geostrophic Equations: A Numerical Investigation
• Neural Closure Models for Dynamical Systems
• Stochastic modelling of a noise driven global instability in a turbulent swirling jet
• A Modular Regularized Variational Multiscale Proper Orthogonal Decomposition for Incompressible Flows
• Data-driven Identification of 2D Partial Differential Equations using extracted physical features
• Intrusive and non-intrusive reduced order modeling of the rotating thermal shallow water equation
• Structure Preserving Model Order Reduction of Shallow Water Equations
• An Efficient Proper Orthogonal Decomposition based Reduced-Order Model
• Learning Discrepancy Models From Experimental Data
• Calibration of projection-based reduced-order models for unsteady compressible flows
• Data-driven model reduction for stochastic Burgers equations
• Long-Time Reynolds Averaging of Reduced Order Models for Fluid Flows: Preliminary Results
• Efficient Construction of Local Parametric Reduced Order Models Using Machine Learning Techniques
• Forward sensitivity approach for estimating eddy viscosity closures in nonlinear model reduction
• A Unified Framework for Multiscale Modeling using the Mori-Zwanzig Formalism and the Variational Multiscale Method
• Reduced order modeling of fluid flows: Machine learning, Kolmogorov barrier, closure modeling, and partitioning
• Large Eddy Simulation of Turbulent Barotropic Flows in Spectral Space on the Sphere
• A Numerical Investigation of the Lengthscale in the Mixing-Length Reduced Order Model of the Turbulent Channel Flow

Cited by in corpus (40)

• Enhancing Computational Fluid Dynamics with Machine Learning
• Operator inference for non-intrusive model reduction with quadratic manifolds
• Digital Twins in Wind Energy: Emerging Technologies and Industry-Informed Future Directions
• Nonlinear proper orthogonal decomposition for convection-dominated flows
• Linear and Nonlinear Dimensionality Reduction from Fluid Mechanics to Machine Learning
• $\text{FastSVD-ML-ROM}$: A Reduced-Order Modeling Framework based on Machine Learning for Real-Time Applications
• Predictive Reduced Order Modeling of Chaotic Multi-scale Problems Using Adaptively Sampled Projections
• Equation-free surrogate modeling of geophysical flows at the intersection of machine learning and data assimilation
• Lagrangian operator inference enhanced with structure-preserving machine learning for nonintrusive model reduction of mechanical systems
• Trajectory-optimized cluster-based network model for the sphere wake
• Comparison of neural closure models for discretised PDEs
• Frame invariant neural network closures for Kraichnan turbulence
• A Multifidelity deep operator network approach to closure for multiscale systems
• Conditional Gaussian Nonlinear System: a Fast Preconditioner and a Cheap Surrogate Model For Complex Nonlinear Systems
• CD-ROM: Complemented Deep-Reduced Order Model
• A Causality-Based Learning Approach for Discovering the Underlying Dynamics of Complex Systems from Partial Observations with Stochastic Parameterization
• Full and Reduced Order Model Consistency of the Nonlinearity Discretization in Incompressible Flows
• Scientific Machine Learning Based Reduced-Order Models for Plasma Turbulence Simulations
• Pressure Data-Driven Variational Multiscale Reduced Order Models
• CEBoosting: Online Sparse Identification of Dynamical Systems with Regime Switching by Causation Entropy Boosting
• CGNSDE: Conditional Gaussian Neural Stochastic Differential Equation for Modeling Complex Systems and Data Assimilation
• Numerical assessments of a nonintrusive surrogate model based on recurrent neural networks and proper orthogonal decomposition: Rayleigh Benard convection
• Shock trace prediction by reduced models for a viscous stochastic Burgers equation
• Prospects of federated machine learning in fluid dynamics
• Non-intrusive Balancing Transformation of Highly Stiff Systems with Lightly-damped Impulse Response
• Minimal model of quasi-cyclic behaviour in turbulence driven by Taylor--Green forcing
• A meshless method to compute the proper orthogonal decomposition and its variants from scattered data
• A pressure-free long-time stable reduced-order model for two-dimensional Rayleigh-Bénard convection
• Efficient Adjoint Petrov-Galerkin Reduced Order Models for fluid flows governed by the incompressible Navier-Stokes equations
• Enhancing Dynamical System Modeling through Interpretable Machine Learning Augmentations: A Case Study in Cathodic Electrophoretic Deposition
• Physically consistent predictive reduced-order modeling by enhancing Operator Inference with state constraints
• Stochastic Subspace via Probabilistic Principal Component Analysis for Characterizing Model Error
• A Priori Error Bounds and Parameter Scalings for the Time Relaxation Reduced Order Model
• Data augmentation for the POD formulation of the parametric laminar incompressible Navier-Stokes equations
• Finding Closure Terms Directly from Coarse Data for 2D Turbulent Flow
• Nonintrusive reduced order modeling of convective Boussinesq flows
• Turbulence suppression in plane Couette flow using reduced-order models
• Leveraging Scale Separation and Stochastic Closure for Data-Driven Prediction of Chaotic Dynamics
• Data-driven quantification of model-form uncertainty in Reynolds-averaged simulations of wind farms
• A Martingale-Free Introduction to Conditional Gaussian Nonlinear Systems