Accurate numerical simulation of fluid flows often requires solving the governing equations using full-order models (FOMs) with millions or even billions of degrees of freedom. Although such high-fidelity simulations can deliver highly accurate results, they remain computationally demanding. Reduced-order models (ROMs) have emerged as an efficient alternative, enabling the analysis of complex flow phenomena with significantly reduced computational cost.
A key advantage of modern ROM techniques is their offline-online decomposition: the most computationally demanding tasks are performed during an offline stage. In contrast, the online stage consists of low-dimensional and inexpensive computations for evaluating new flow states. Owing to this efficiency, ROMs are particularly attractive for applications requiring repeated simulations, such as design optimization, uncertainty quantification, optimal control, inverse problems, and data assimilation.
Without loss of generality, consider a nonlinear dynamical system
with the corresponding weak formulation
During the offline stage, the FOM is solved for selected parameter values to generate a reduced basis. Projecting the governing equations onto this low-dimensional subspace leads to the compact Galerkin reduced-order model (G-ROM) dynamical system
The resulting online simulations typically require several orders of magnitude fewer computational cost than their full-order counterparts. Despite their efficiency, standard Galerkin ROM often exhibit poor performance in convection-dominated regimes, such as turbulent or transitional flows. These systems require a large number of modes to accurately represent the dynamics, while practical ROMs retain only a small subset to preserve efficiency. This truncation often leads to spurious oscillations and a significant loss of accuracy.
To address these limitations, two main strategies have been developed: numerical stabilization techniques and ROM closure modeling. Stabilization approaches include projection-based stabilization, subspace rotation, variational multiscale (VMS) methods, SUPG stabilization, filtering techniques, and the enforcement of physical constraints.
An alternative and complementary approach is the ROM closure modeling, in which additional terms are introduced to capture the effect of discarded modes on the resolved dynamics:
where the second term on the RHS represents the ROM closure (or correction) term, explicitly accounting for the influence of unresolved modes on the reduced system.