Parametric Intrusive Reduced Order Models enhanced with Machine Learning Correction Terms

TL;DR

This work addresses the instability and limited accuracy of standard POD-Galerkin reduced-order models for turbulent, advection-dominated flows by introducing parametric, machine-learning–driven corrections. It integrates two closure strategies into a PPE-ROM framework: physics-based eddy-viscosity closures learned via neural networks and purely data-driven correction terms learned from exact closures, with a hybrid approach combining both. The authors demonstrate significant accuracy and stability gains on two turbulence test cases (periodic cylinder flow and channel-driven cavity) across online Reynolds-number extrapolation, with LSTM-based corrections often providing strongest performance. The methods offer a practical pathway to reliable, real-time ROMs for parametrized turbulent flows, though limitations of linear POD motivate future work toward nonlinear latent representations and adaptive, post-hoc learning.

Abstract

In this paper, we propose an equation-based parametric Reduced Order Model (ROM), whose accuracy is improved with data-driven terms added into the reduced equations. These additions have the aim of reintroducing contributions that in standard ROMs are not taken into account. In particular, in this work we consider two types of contributions: the turbulence modeling, added through a reduced-order approximation of the eddy viscosity field, and the correction model, aimed to re-introduce the contribution of the discarded modes. Both approaches have been investigated in previous works and the goal of this paper is to extend the model to a parametric setting making use of ad-hoc machine learning procedures. More in detail, we investigate different neural networks' architectures, from simple dense feed-forward to Long-Short Term Memory neural networks, in order to find the most suitable model for the re-introduced contributions. We tested the methods on two test cases with different behaviors: the periodic turbulent flow past a circular cylinder and the unsteady turbulent flow in a channel-driven cavity. In both cases, the parameter considered is the Reynolds number and the machine learning-enhanced ROM considerably improved the pressure and velocity accuracy with respect to the standard ROM.

Figures (21)

• Figure 2: Graphical representations of the sets of parameters used in the offline stage ($\text{T}_{\text{offline}}$ and $\mathcal{N}_{\text{train}}$), in green, and in the online stage ($\text{T}_{\text{online}}$ and $\mathcal{N}_{\text{test}}$), in red. The green shadow describes the accessible offline/training area. The parameters are also specified in Table \ref{['tab:table-params']}.
• Figure 3: The domain and full order mesh considered for the periodic flow around a circular cylinder.
• Figure 4: Cumulative eigenvalues and eigenvalues decay for the test case a.
• Figure 5: Eddy viscosity coefficients for $\nu=8.33e-5m \per s$ (in $\mathcal{N}_{\text{train}}$): neural networks predictions' average and corresponding confidence interval, and POD projected exact coefficients.
• Figure 6: Correction terms' coefficients for $\nu=8.33e-5m \per s$ (in $\mathcal{N}_{\text{train}}$): neural networks predictions' average and corresponding confidence interval, and POD projected exact coefficients.