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 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 using 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）在利用少量广义（或潜在）变量描述多尺度系统行为方面展现出潜力。然而，由于未能正确处理多尺度系统中已分辨尺度与未分辨尺度之间的相互作用（即封闭问题），PROMs 可能产生不准确甚至不稳定的结果。本研究将封闭问题视为多保真问题，并采用多保真深度算子网络（DeepONet）框架加以解决。此外，为增强基于多保真封闭的稳定性与准确性，我们借鉴了近期关于物理与机器学习模型耦合文献中提出的"in-the-loop"训练方法。所提方法在一维粘性Burgers方程的激波平流问题及二维Navier-Stokes方程的涡旋合并问题中进行了测试。数值实验表明，无论是在插值还是外推场景下，经过封闭校正的PROMs的预测能力均较未校正版本有显著提升。