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 -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 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 a factor of five.

中文翻译：

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基于混合快照的对流主导问题的自适应、免训练降阶模型

绝大多数降阶模型 (ROM) 首先从高维模型 (HDM) 训练数据中获得问题的低维表示，然后使用该数据来获得复杂性降低的系统。不幸的是，对流主导的问题通常具有缓慢衰减的柯尔莫哥洛夫方程-width，这使得获得仅从训练数据构建的准确 ROM 变得非常困难。ROM 的准确性可以通过 HDM 解决方案进行丰富来提高；然而，由于对复杂问题进行 H​​DM 评估的计算量很大，因此只能简单地使用它们来获得相关的计算节省。在这项工作中，我们利用这些问题经常表现出的局部空间一致性来导出一种准确、经济高效的方法，该方法可以重复组合 HDM 和 ROM 评估，而无需单独的训练阶段。我们的方法通过完全求解 HDM 或结合部分 HDM 和 ROM 求解来获得给定时间步长的解。动态采样程序识别需要 HDM 解决方案以实现全局精度的区域，并使用 ROM 重建流量提醒。此外，结合了 HDM 和 ROM 的解决方案使用空间滤波来消除可能出现的潜在寄生振荡。我们在无粘可压缩流动问题上测试了所提出的方法，并证明加速速度高达五倍。