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1. Bridging Large Eddy Simulation and Reduced-Order Modeling of Convection-Dominated Flows through Spatial Filtering: Review and Perspectives

   https://doi.org/10.3390/fluids9080178  

   Quaini, Annalisa; San, Omer; Veneziani, Alessandro; Iliescu, Traian (August 2024, Fluids)

   Reduced-order models (ROMs) have achieved a lot of success in reducing the computational cost of traditional numerical methods across many disciplines. In fluid dynamics, ROMs have been successful in providing efficient and relatively accurate solutions for the numerical simulation of laminar flows. For convection-dominated (e.g., turbulent) flows, however, standard ROMs generally yield inaccurate results, usually affected by spurious oscillations. Thus, ROMs are usually equipped with numerical stabilization or closure models in order to account for the effect of the discarded modes. The literature on ROM closures and stabilizations is large and growing fast. In this paper, instead of reviewing all the ROM closures and stabilizations, we took a more modest step and focused on one particular type of ROM closure and stabilization that is inspired by large eddy simulation (LES), a classical strategy in computational fluid dynamics (CFD). These ROMs, which we call LES-ROMs, are extremely easy to implement, very efficient, and accurate. Indeed, LES-ROMs are modular and generally require minimal modifications to standard (“legacy”) ROM formulations. Furthermore, the computational overhead of these modifications is minimal. Finally, carefully tuned LES-ROMs can accurately capture the average physical quantities of interest in challenging convection-dominated flows in science and engineering applications. LES-ROMs are constructed by leveraging spatial filtering, which is the same principle used to build classical LES models. This ensures a modeling consistency between LES-ROMs and the approaches that generated the data used to train them. It also “bridges” two distinct research fields (LES and ROMs) that have been disconnected until now. This paper is a review of LES-ROMs, with a particular focus on the LES concepts and models that enable the construction of LES-inspired ROMs and the bridging of LES and reduced-order modeling. This paper starts with a description of a versatile LES strategy called evolve–filter–relax (EFR) that has been successfully used as a full-order method for both incompressible and compressible convection-dominated flows. We present evidence of this success. We then show how the EFR strategy, and spatial filtering in general, can be leveraged to construct LES-ROMs (e.g., EFR-ROM). Several applications of LES-ROMs to the numerical simulation of incompressible and compressible convection-dominated flows are presented. Finally, we draw conclusions and outline several research directions and open questions in LES-ROM development. While we do not claim this review to be comprehensive, we certainly hope it serves as a brief and friendly introduction to this exciting research area, which we believe has a lot of potential in the practical numerical simulation of convection-dominated flows in science, engineering, and medicine. 

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2. Pressure Stabilization Strategies for a LES Filtering Reduced Order Model

   https://doi.org/10.3390/fluids6090302  

   Girfoglio, Michele; Quaini, Annalisa; Rozza, Gianluigi (September 2021, Fluids)

   We present a stabilized POD–Galerkin reduced order method (ROM) for a Leray model. For the implementation of the model, we combine a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. In both steps of the EF algorithm, velocity and pressure fields are approximated using different POD basis and coefficients. To achieve pressure stabilization, we consider and compare two strategies: the pressure Poisson equation and the supremizer enrichment of the velocity space. We show that the evolve and filtered velocity spaces have to be enriched with the supremizer solutions related to both evolve and filter pressure fields in order to obtain stable and accurate solutions with the supremizer enrichment method. We test our ROM approach on a 2D unsteady flow past a cylinder at Reynolds number 0≤Re≤100. We find that both stabilization strategies produce comparable errors in the reconstruction of the lift and drag coefficients, with the pressure Poisson equation method being more computationally efficient. 

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3. 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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4. 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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5. Data‐driven variational multiscale reduced order modeling of vaginal tissue inflation

   https://doi.org/10.1002/cnm.3660  

   Snyder, William; McGuire, Jeffrey A.; Mou, Changhong; Dillard, David A.; Iliescu, Traian; De Vita, Raffaella (November 2022, International Journal for Numerical Methods in Biomedical Engineering)

   Abstract The vagina undergoes large finite deformations and has complex geometry and microstructure, resulting in material and geometric nonlinearities, complicated boundary conditions, and nonhomogeneities within finite element (FE) simulations. These nonlinearities pose a significant challenge for numerical solvers, increasing the computational time by several orders of magnitude. Simplifying assumptions can reduce the computational time significantly, but this usually comes at the expense of simulation accuracy. This study proposed the use of reduced order modeling (ROM) techniques to capture experimentally measured displacement fields of rat vaginal tissue during inflation testing in order to attain both the accuracy of higher‐fidelity models and the speed of simpler simulations. The proper orthogonal decomposition (POD) method was used to extract the significant information from FE simulations generated by varying the luminal pressure and the parameters that introduce the anisotropy in the selected constitutive model. A new data‐driven (DD) variational multiscale (VMS) ROM framework was extended to obtain the displacement fields of rat vaginal tissue under pressure. For comparison purposes, we also investigated the classical Galerkin ROM (G‐ROM). In our numerical study, both the G‐ROM and the DD‐VMS‐ROM decreased the FE computational cost by orders of magnitude without a significant decrease in numerical accuracy. Furthermore, the DD‐VMS‐ROM improved the G‐ROM accuracy at a modest computational overhead. Our numerical investigation showed that ROM has the potential to provide efficient and accurate computational tools to describe vaginal deformations, with the ultimate goal of improving maternal health. 

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