StabOp: A Data-Driven Stabilization Operator for Reduced Order Modeling

TL;DR

The paper tackles the challenge of choosing ROM stabilization filters and their parameters in convection-dominated flows. It introduces StabOp, a data-driven stabilization operator learned via PDE-constrained optimization to replace traditional ROM filters within the Leray ROM, yielding StabOp-L-ROM. Across four representative flows, StabOp-L-ROM substantially surpasses the classical Leray ROM with optimally tuned filters and the baseline G-ROM in predictive accuracy, while offering distinct smoothing behavior. Although the offline training cost is higher, online performance remains competitive, and results indicate a powerful, adaptable framework for QoI-driven ROM stabilization with potential extensions to other stabilizations and FOM-level applications.

Abstract

Spatial filters have played a central role in large eddy simulation and, more recently, in reduced order model (ROM) stabilization for convection-dominated flows. Nevertheless, important open questions remain: in under-resolved regimes, which filter is most suitable for a given stabilization or closure model? Moreover, once a filter is selected, how should its parameters, such as the filter radius, be determined? Addressing these questions is essential for the reliable design and performance of filter-based stabilization strategies. To answer these questions, we propose a novel strategy that differs fundamentally from current filter-based approaches: we replace traditional spatial filters with a data-driven stabilization operator (StabOp) that yields accurate results for a given resolution, quantity of interest, and stabilization strategy. Although the new StabOp can be used for both classical discretizations and ROMs, and for different types of filter-based stabilization or closure, for clarity we focus on ROMs and the Leray stabilization. To build the new StabOp, we postulate its model form as a linear, quadratic, or nonlinear mapping, and then solve a PDE-constrained optimization problem to minimize a given loss function. Using the resulting StabOp in the Leray ROM (L-ROM) yields a new stabilized ROM, StabOp-L-ROM. To assess the StabOp-L-ROM, we compare it with the L-ROM and the standard ROM in numerical simulations of four flows: 2D flow past a cylinder at Re=500, lid-driven cavity at Re=10000, 3D flow past a hemisphere at Re=2200, and minimal channel flow at Re=5000. Our numerical results show that the StabOp-L-ROM can be orders of magnitude more accurate than the classical L-ROM tuned with an optimal filter radius in the predictive regime. Furthermore, while the new StabOp smooths the input flow fields, its smoothing mechanism differs from that of classical spatial filters.

Figures (28)

• Figure 1.1: Effect of the ROM spatial filter in 3D turbulent channel flows tsai2025time. A suitably chosen filter radius $\delta$ yields a physically meaningful solution, while small or large $\delta$ values lead to unphysical or overly smooth results, respectively.
• Figure 6.1: 2D flow past a cylinder at $\rm Re={500}$. The behavior of $\sum^{r}_{i=1} \lambda_i /\sum^{{N}_{\text{train}}}_{i=1} \lambda_i$ as a function of the ROM space dimension, $r$.
• Figure 6.2: 2D flow past a cylinder at $\rm Re=500$. Kinetic energy evolution of the new StabOp-L-ROM, along with the results of the G-ROM, L-ROM with an optimal filter radius, and FOM.
• Figure 6.3: 2D flow past a cylinder at $\rm Re=500$. Mean squared error of kinetic energy with respect to the FOM reference for G-ROM, L-ROM, and new StabOp-L-ROM.
• Figure 6.4: 2D flow past a cylinder at $\rm Re=500$. Energy spectrum for FOM, G-ROM, L-ROM, and StabOp-L-ROMs with ROM dimension $r=4$.

Theorems & Definitions (4)

• Remark 2.1
• Remark 4.1: Is the new StabOp a filter?
• Remark 5.1
• Remark 5.2