Projection-based reduced order models (PROMs) have shown promise in representing the behavior of multiscale systems using a small set of generalized (or latent) variables. Despite their success, PROMs can be susceptible to inaccuracies, even instabilities, due to the improper accounting of the interaction between the resolved and unresolved scales of the multiscale system (known as the closure problem). In the current work, we interpret closure as a multifidelity problem and use a multifidelity deep operator network (DeepONet) framework to address it. In addition, to enhance the stability and/or accuracy of the multifidelity-based closure, we employ the recently developed "in-the-loop" training approach from the literature on coupling physics and machine learning models. The resulting approach is tested on shock advection for the one-dimensional viscous Burgers equation and vortex merging for the two-dimensional Navier-Stokes equations. The numerical experiments show significant improvement of the predictive ability of the closure-corrected PROM over the un-corrected one both in the interpolative and the extrapolative regimes.

翻译：基于投影的降阶模型（PROMs）已在利用少量广义（或潜在）变量表征多尺度系统行为方面展现出潜力。尽管取得了成功，但由于未能妥善处理多尺度系统中可解尺度与不可解尺度之间的相互作用（即闭合问题），PROMs仍可能存在不准确性甚至不稳定性。在本文中，我们将闭合问题诠释为一种多保真问题，并采用多保真深度算子网络（DeepONet）框架加以解决。此外，为提升基于多保真闭合的稳定性和/或准确性，我们借鉴了物理与机器学习模型耦合领域的文献最新提出的"套环式"训练方法。所提方法在一维黏性Burgers方程的激波平流问题与二维Navier-Stokes方程的涡旋合并问题中进行了测试。数值实验表明，无论是在插值区域还是外推区域，经闭合校正的PROM模型在预测能力上均显著优于未校正的PROM模型。