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.

翻译：本文提出一种基于方程的参数化降阶模型（ROM），其精度通过向降阶方程中添加数据驱动项得到提升。这些添加项旨在重新引入标准ROM中未考虑的贡献项。具体而言，本研究考虑两类贡献：通过涡粘性场的降阶近似引入的湍流建模，以及旨在重新引入被舍弃模态贡献的修正模型。两种方法在先前研究中已有探讨，本文目标在于利用专用机器学习流程将该模型扩展至参数化场景。更具体地，我们研究了从简单密集前馈网络到长短期记忆神经网络等不同神经网络架构，以寻找最适合重新引入贡献项的模型。我们在两个具有不同特性的测试案例中验证了该方法：圆柱绕流的周期性湍流和驱动腔槽内的非定常湍流。两种案例均以雷诺数为参数，相较于标准ROM，经机器学习增强的ROM在压力和速度精度上均取得显著提升。