Data-driven Closure Strategies for Parametrized Reduced Order Models via Deep Operator Networks

TL;DR

This work addresses the accuracy and stability challenges of POD-Galerkin reduced order models for parametric turbulent flows by introducing data-driven, nonlinear closure terms learned via neural operators. A parametric extension of closure modeling is achieved with DeepONet to map reduced-state inputs and problem parameters to turbulence closures, and a Multi-Input Operator Network to predict closure corrections that compensate discarded modes. Across three turbulent test cases, the proposed DD-EV-ROM framework consistently improves velocity and especially pressure predictions over the baseline EV-ROM, with the DD-EV-ROM$^{igstar}$ variant offering enhanced stability in time extrapolation and more difficult parameter regimes. The approach demonstrates that physics-based ROMs augmented with learned closures can achieve high-fidelity performance in parametrized settings and paves the way for nonlinear projection strategies to further boost robustness under geometric parametrization.

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 reduced-order approaches are not taken into account. In particular, in this work we focus on a Proper Orthogonal Decomposition (POD)-based formulation and our goal is to build a closure or correction model, aimed to re-introduce the contribution of the discarded modes. The approach has been investigated in previous works, and the goal of this manuscript is to extend the model to a parametric setting making use of machine learning procedures, and, in particular, of deep operator networks. More in detail, we model the closure terms through a deep operator network taking as input the reduced variables and the parameters of the problem. We tested the methods on three test cases with different behaviors: the periodic turbulent flow past a circular cylinder, the unsteady turbulent flow in a channel-driven cavity, and the geometrically-parametrized backstep flow. The performance of the machine learning-enhanced ROM is deeply studied in different modal regimes, and considerably improved the pressure and velocity accuracy with respect to the standard POD-Galerkin approach.

Figures (32)

• Figure 1: Standard version of the DeepONet, as in lu2019deeponet.
• Figure 2: DeepONet architecture used for the eddy viscosity coefficients modeling $\mathcal{G}$.
• Figure 3: MIONet architecture used for the closure modeling $\mathcal{M}$.
• Figure 4: The domain and full order mesh considered for the periodic flow around a circular cylinder.
• Figure 5: Graphical representations of the sets of parameters used in the offline stage for test case ($\bm{a}$), in green, and in the online stage, in red. The green shadow describes the accessible offline/training area.