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 (18)

• 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
• Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledge
• Physics guided machine learning using simplified theories
• Nonparametric forecasting of low-dimensional dynamical systems
• Achieving Conservation of Energy in Neural Network Emulators for Climate Modeling
• Feature engineering and symbolic regression methods for detecting hidden physics from sparse sensors
• DR-RNN: A deep residual recurrent neural network for model reduction
• Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction
• Memory embedded non-intrusive reduced order modeling of non-ergodic flows
• Model fusion with physics-guided machine learning
• Optimal Nonlinear Eddy Viscosity in Galerkin Models of Turbulent Flows
• Reduced-space Gaussian Process Regression for Data-Driven Probabilistic Forecast of Chaotic Dynamical Systems
• Embedding Hard Physical Constraints in Neural Network Coarse-Graining of 3D Turbulence
• Intrusive and non-intrusive reduced order modeling of the rotating thermal shallow water equation
• Long-Time Reynolds Averaging of Reduced Order Models for Fluid Flows: Preliminary Results
• Reduced order modeling of fluid flows: Machine learning, Kolmogorov barrier, closure modeling, and partitioning
• 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