Data-Driven Variational Multiscale Reduced Order Models

Abstract

We propose a new data-driven reduced order model (ROM) framework that centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. The VMS methodology is a natural fit for the hierarchical structure of the ROM basis: In the first step, we use the ROM projection to separate the scales into three categories: (i) resolved large scales, (ii) resolved small scales, and (iii) unresolved scales. In the second step, we explicitly identify the VMS-ROM closure terms, i.e., the terms representing the interactions among the three types of scales. In the third step, we use available data to model the VMS-ROM closure terms. Thus, instead of phenomenological models used in VMS for standard numerical discretizations (e.g., eddy viscosity models), we utilize available data to construct new structural VMS-ROM closure models. Specifically, we build ROM operators (vectors, matrices, and tensors) that are closest to the true ROM closure terms evaluated with the available data. We test the new data-driven VMS-ROM in the numerical simulation of four test cases: (i) the 1D Burgers equation with viscosity coefficient $\nu =10^{-3}$; (ii) a 2D flow past a circular cylinder at Reynolds numbers $Re=100$, $Re=500$, and $Re=1000$; (iii) the quasi-geostrophic equations at Reynolds number $Re=450$ and Rossby number $Ro=0.0036$; and (iv) a 2D flow over a backward facing step at Reynolds number $Re=1000$. The numerical results show that the data-driven VMS-ROM is significantly more accurate than standard ROMs.

References in corpus (21)

• Proper Orthogonal Decomposition Closure Models For Turbulent Flows: A Numerical Comparison
• Data-assisted reduced-order modeling of extreme events in complex dynamical systems
• Data-Driven POD-Galerkin Reduced Order Model for Turbulent Flows
• On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high Reynolds number flow over an Ahmed body
• Time-series learning of latent-space dynamics for reduced-order model closure
• 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
• A non-intrusive reduced order modeling framework for quasi-geostrophic turbulence
• A reduced order variational multiscale approach for turbulent flows
• POD/DEIM Reduced-Order Strategies for Efficient Four Dimensional Variational Data Assimilation
• Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier-Stokes equations
• Approximate deconvolution large eddy simulation of a barotropic ocean circulation model
• Physically-Constrained Data-Driven, Filtered Reduced Order Modeling of Fluid Flows
• Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism
• A stabilized proper orthogonal decomposition reduced-order model for large scale quasigeostrophic ocean circulation
• Machine Learning for Prediction with Missing Dynamics
• On a certified Smagorinsky reduced basis turbulence model
• A Priori Estimation Of Memory Effects In Coarse-Grained Nonlinear Systems Using The Mori-Zwanzig Formalism
• Variational approach to closure of nonlinear dynamical systems: Autonomous case
• A Modular Regularized Variational Multiscale Proper Orthogonal Decomposition for Incompressible Flows
• Statistical closure modeling for reduced-order models of stationary systems by the ROMES method

Cited by in corpus (24)

• On closures for reduced order models $-$ A spectrum of first-principle to machine-learned avenues
• Reduced order models for Lagrangian hydrodynamics
• $\text{FastSVD-ML-ROM}$: A Reduced-Order Modeling Framework based on Machine Learning for Real-Time Applications
• Deep neural network enabled corrective source term approach to hybrid analysis and modeling
• 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
• A Causality-Based Learning Approach for Discovering the Underlying Dynamics of Complex Systems from Partial Observations with Stochastic Parameterization
• BAMCAFE: A Bayesian Machine Learning Advanced Forecast Ensemble Method for Complex Turbulent Systems with Partial Observations
• A data driven reduced order model of fluid flow by Auto-Encoder and self-attention deep learning methods
• Estimating flow fields with Reduced Order Models
• Full and Reduced Order Model Consistency of the Nonlinearity Discretization in Incompressible Flows
• Pressure Data-Driven Variational Multiscale Reduced Order Models
• CGNSDE: Conditional Gaussian Neural Stochastic Differential Equation for Modeling Complex Systems and Data Assimilation
• Lagrangian Data-Driven Reduced Order Modeling of Finite Time Lyapunov Exponents
• Modeling Partially Observed Nonlinear Dynamical Systems and Efficient Data Assimilation via Discrete-Time Conditional Gaussian Koopman Network
• Forward sensitivity approach for estimating eddy viscosity closures in nonlinear model reduction
• Shock trace prediction by reduced models for a viscous stochastic Burgers equation
• Approximate Deconvolution Leray Reduced Order Model for Convection-Dominated Flows
• Efficient Adjoint Petrov-Galerkin Reduced Order Models for fluid flows governed by the incompressible Navier-Stokes equations
• Reduced Order Models for the Quasi-Geostrophic Equations: A Brief Survey
• Data-Driven Closure of Projection-Based Reduced Order Models for Unsteady Compressible Flows
• Consistency of the Full and Reduced Order Models for Evolve-Filter-Relax Regularization of Convection-Dominated, Marginally-Resolved Flows
• An Efficient Continuous Data Assimilation Algorithm for the Sabra Shell Model of Turbulence
• A Martingale-Free Introduction to Conditional Gaussian Nonlinear Systems