Data-driven closure strategies for parametrized reduced-order models via deep operator networks

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.