The vast majority of reduced-order models (ROMs) first obtain a low dimensional representation of the problem from high-dimensional model (HDM) training data which is afterwards used to obtain a system of reduced complexity. Unfortunately, convection-dominated problems generally have a slowly decaying Kolmogorov n-width, which makes obtaining an accurate ROM built solely from training data very challenging. The accuracy of a ROM can be improved through enrichment with HDM solutions; however, due to the large computational expense of HDM evaluations for complex problems, they can only be used parsimoniously to obtain relevant computational savings. In this work, we exploit the local spatial and temporal coherence often exhibited by these problems to derive an accurate, cost-efficient approach that repeatedly combines HDM and ROM evaluations without a separate training phase. Our approach obtains solutions at a given time step by either fully solving the HDM or by combining partial HDM and ROM solves. A dynamic sampling procedure identifies regions that require the HDM solution for global accuracy and the reminder of the flow is reconstructed using the ROM. Moreover, solutions combining both HDM and ROM solves use spatial filtering to eliminate potential spurious oscillations that may develop. We test the proposed method on inviscid compressible flow problems and demonstrate speedups up to an order of magnitude.

翻译：绝大多数降阶模型首先从高维模型训练数据中获取问题的低维表示，随后利用该表示构建降阶复杂系统。然而，对流主导问题通常具有缓慢衰减的Kolmogorov n-宽度，这使得仅依靠训练数据构建精确的降阶模型极具挑战性。通过高维模型解进行丰富性增强可提升降阶模型精度；但由于复杂问题中高维模型评估的计算开销巨大，只能谨慎使用此类方法以实现显著计算节省。本研究利用此类问题常展现的局部时空相干性，提出一种精确且经济高效的方法，无需单独训练阶段即可反复结合高维模型与降阶模型评估。该方法通过完全求解高维模型或结合部分高维模型与降阶模型求解，获取特定时间步的解。动态采样过程识别需要高维模型解以保证全局精度的区域，其余流场通过降阶模型重构。此外，结合高维模型与降阶模型求解的方案采用空间滤波消除可能产生的伪振荡。我们在无粘可压缩流动问题上测试了所提方法，并展示了高达一个数量级的加速效果。