Model order reduction through the POD-Galerkin method can lead to dramatic gains in terms of computational efficiency in solving physical problems. However, the applicability of the method to non linear high-dimensional dynamical systems such as the Navier-Stokes equations has been shown to be limited, producing inaccurate and sometimes unstable models. This paper proposes a deep learning based closure modeling approach for classical POD-Galerkin reduced order models (ROM). The proposed approach is theoretically grounded, using neural networks to approximate well studied operators. In contrast with most previous works, the present CD-ROM approach is based on an interpretable continuous memory formulation, derived from simple hypotheses on the behavior of partially observed dynamical systems. The final corrected models can hence be simulated using most classical time stepping schemes. The capabilities of the CD-ROM approach are demonstrated on two classical examples from Computational Fluid Dynamics, as well as a parametric case, the Kuramoto-Sivashinsky equation.

翻译：通过POD-Galerkin方法进行模型降阶可以在求解物理问题时显著提升计算效率。然而，该方法在非线性高维动力系统（如Navier-Stokes方程）中的应用已被证明存在局限性，会产生不准确甚至不稳定的模型。本文提出了一种基于深度学习的闭环建模方法，用于经典POD-Galerkin降阶模型（ROM）。该方法具有理论基础，利用神经网络逼近经过充分研究的算子。与以往大多数工作不同，本文提出的CD-ROM方法基于一种可解释的连续记忆公式，该公式源自对部分观测动力系统行为的简单假设。因此，修正后的模型可采用大多数经典的时间步进方案进行模拟。通过计算流体动力学中的两个经典算例以及一个参数化案例——Kuramoto-Sivashinsky方程，验证了CD-ROM方法的能力。