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1. Residual-based data-driven variational multiscale reduced order models for parameter-dependent problems

   https://doi.org/10.1007/s40314-025-03273-0  

   Koc, Birgul; Rubino, Samuele; Chacón_Rebollo, Tomás; Iliescu, Traian (September 2025, Computational and Applied Mathematics)

   Abstract In this paper, we propose a novel residual-based data-driven closure strategy for reduced order models (ROMs) of under-resolved, convection-dominated problems. The new ROM closure model is constructed in a variational multiscale (VMS) framework by using the available full order model data and a model form ansatz that depends on the ROM residual. We emphasize that this closure modeling strategy is fundamentally different from the current data-driven ROM closures, which generally depend on the ROM coefficients. We investigate the new residual-based data-driven VMS ROM closure strategy in the numerical simulation of three test problems: (i) a one-dimensional parameter-dependent advection-diffusion problem; (ii) a two-dimensional time-dependent advection-diffusion-reaction problem with a small diffusion coefficient ($$\varepsilon = 1e-4$$ $\epsilon =1e-4$); and (iii) a two-dimensional flow past a cylinder at Reynolds number$$Re=1000$$ $Re=1000$. Our numerical investigation shows that the new residual-based data-driven VMS-ROM is more accurate than the standard coefficient-based data-driven VMS-ROM. 

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2. Reduced Order Models for the Quasi-Geostrophic Equations: A Brief Survey

   https://doi.org/10.3390/fluids6010016  

   Mou, Changhong; Wang, Zhu; Wells, David R.; Xie, Xuping; Iliescu, Traian (January 2021, Fluids)

   null (Ed.)

   Reduced order models (ROMs) are computational models whose dimension is significantly lower than those obtained through classical numerical discretizations (e.g., finite element, finite difference, finite volume, or spectral methods). Thus, ROMs have been used to accelerate numerical simulations of many query problems, e.g., uncertainty quantification, control, and shape optimization. Projection-based ROMs have been particularly successful in the numerical simulation of fluid flows. In this brief survey, we summarize some recent ROM developments for the quasi-geostrophic equations (QGE) (also known as the barotropic vorticity equations), which are a simplified model for geophysical flows in which rotation plays a central role, such as wind-driven ocean circulation in mid-latitude ocean basins. Since the QGE represent a practical compromise between efficient numerical simulations of ocean flows and accurate representations of large scale ocean dynamics, these equations have often been used in the testing of new numerical methods for ocean flows. ROMs have also been tested on the QGE for various settings in order to understand their potential in efficient numerical simulations of ocean flows. In this paper, we survey the ROMs developed for the QGE in order to understand their potential in efficient numerical simulations of more complex ocean flows: We explain how classical numerical methods for the QGE are used to generate the ROM basis functions, we outline the main steps in the construction of projection-based ROMs (with a particular focus on the under-resolved regime, when the closure problem needs to be addressed), we illustrate the ROMs in the numerical simulation of the QGE for various settings, and we present several potential future research avenues in the ROM exploration of the QGE and more complex models of geophysical flows. 

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3. Data‐Driven Optimization for the Evolve‐Filter‐Relax Regularization of Convection‐Dominated Flows

   https://doi.org/10.1002/nme.70042  

   Ivagnes, Anna; Strazzullo, Maria; Girfoglio, Michele; Iliescu, Traian; Rozza, Gianluigi (May 2025, International Journal for Numerical Methods in Engineering)

   ABSTRACT Numerical stabilization techniques are often employed in under‐resolved simulations of convection‐dominated flows to improve accuracy and mitigate spurious oscillations. Specifically, the evolve–filter–relax (EFR) algorithm is a framework that consists of evolving the solution, applying a filtering step to remove high‐frequency noise, and relaxing through a convex combination of filtered and original solutions. The stability and accuracy of the EFR solution strongly depend on two parameters, the filter radius and the relaxation parameter . Standard choices for these parameters are usually fixed in time, and related to the full order model setting, that is, the grid size for and the time step for . The key novelties with respect to the standard EFR approach are: (i) time‐dependent parameters and , and (ii) data‐driven adaptive optimization of the parameters in time, considering a fully‐resolved simulation as reference. In particular, we propose three different classes of optimized‐EFR (Opt‐EFR) strategies, aiming to optimize one or both parameters. The new Opt‐EFR strategies are tested in the under‐resolved simulation of a turbulent flow past a cylinder at . The Opt‐EFR proved to be more accurate than standard approaches by up to 99, while maintaining a similar computational time. In particular, the key new finding of our analysis is that such accuracy can be obtained only if the optimized objective function includes: (i) aglobalmetric (as the kinetic energy), and (ii)spatial gradients' information. 

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4. Evolve Filter Stabilization Reduced-Order Model for Stochastic Burgers Equation

   https://doi.org/10.3390/fluids3040084  

   Xie, Xuping; Bao, Feng; Webster, Clayton (December 2018, Fluids)

   In this paper, we introduce the evolve-then-filter (EF) regularization method for reduced order modeling of convection-dominated stochastic systems. The standard Galerkin projection reduced order model (G-ROM) yield numerical oscillations in a convection-dominated regime. The evolve-then-filter reduced order model (EF-ROM) aims at the numerical stabilization of the standard G-ROM, which uses explicit ROM spatial filter to regularize various terms in the reduced order model (ROM). Our numerical results are based on a stochastic Burgers equation with linear multiplicative noise. The numerical result shows that the EF-ROM is significantly better than G-ROM. 

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5. Closure Learning for Nonlinear Model Reduction Using Deep Residual Neural Network

   https://doi.org/10.3390/fluids5010039  

   Xie, Xuping; Webster, Clayton; Iliescu, Traian (March 2020, Fluids)

   null (Ed.)

   Developing accurate, efficient, and robust closure models is essential in the construction of reduced order models (ROMs) for realistic nonlinear systems, which generally require drastic ROM mode truncations. We propose a deep residual neural network (ResNet) closure learning framework for ROMs of nonlinear systems. The novel ResNet-ROM framework consists of two steps: (i) In the first step, we use ROM projection to filter the given nonlinear system and construct a spatially filtered ROM. This filtered ROM is low-dimensional, but is not closed. (ii) In the second step, we use ResNet to close the filtered ROM, i.e., to model the interaction between the resolved and unresolved ROM modes. We emphasize that in the new ResNet-ROM framework, data is used only to complement classical physical modeling (i.e., only in the closure modeling component), not to completely replace it. We also note that the new ResNet-ROM is built on general ideas of spatial filtering and deep learning and is independent of (restrictive) phenomenological arguments, e.g., of eddy viscosity type. The numerical experiments for the 1D Burgers equation show that the ResNet-ROM is significantly more accurate than the standard projection ROM. The new ResNet-ROM is also more accurate and significantly more efficient than other modern ROM closure models. 

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