A posteriori reduced-order models (ROM), e.g. based on proper orthogonal decomposition (POD), are essential to affordably tackle realistic parametric problems. They rely on a trustful training set, that is a family of full-order solutions (snapshots) representative of all possible outcomes of the parametric problem. Having such a rich collection of snapshots is not, in many cases, computationally viable. A strategy for data augmentation, designed for parametric laminar incompressible flows, is proposed to enrich poorly populated training sets. The goal is to include in the new, artificial snapshots emerging features, not present in the original basis, that do enhance the quality of the reduced basis (RB) constructed using POD dimensionality reduction. The methodologies devised are based on exploiting basic physical principles, such as mass and momentum conservation, to construct physically-relevant, artificial snapshots at a fraction of the cost of additional full-order solutions. Interestingly, the numerical results show that the ideas exploiting only mass conservation (i.e., incompressibility) are not producing significant added value with respect to the standard linear combinations of snapshots. Conversely, accounting for the linearized momentum balance via the Oseen equation does improve the quality of the resulting approximation and therefore is an effective data augmentation strategy in the framework of viscous incompressible laminar flows. Numerical experiments of parametric flow problems, in two and three dimensions, at low and moderate values of the Reynolds number are presented to showcase the superior performance of the data-enriched POD-RB with respect to the standard ROM in terms of both accuracy and efficiency.

翻译：后验降阶模型（ROM），例如基于本征正交分解（POD）的模型，对于经济高效地处理实际参数化问题至关重要。这类模型依赖于可信的训练集，即一组能代表参数化问题所有可能结果的全阶解（快照）。然而在许多情况下，获取如此丰富的快照集合在计算上是不可行的。本文提出一种专为参数化层流不可压缩流动设计的数据增强策略，用于扩充样本不足的训练集。其目标是在新生成的人工快照中引入原始基中未出现的新特征，从而提升通过POD降维构建的降阶基（RB）的质量。所设计的方法基于基本物理原理（如质量和动量守恒）来构建物理意义明确的人工快照，其计算成本仅为额外全阶解的很小一部分。值得注意的是，数值结果表明：仅利用质量守恒（即不可压缩性）的思路相较于标准的快照线性组合并未产生显著的附加价值。相反，通过Oseen方程考虑线性化动量平衡的做法确实改善了最终近似解的质量，因此在粘性不可压缩层流框架中是一种有效的数据增强策略。本文展示了二维和三维参数化流动问题在低雷诺数和中等雷诺数下的数值实验，证明数据增强的POD-RB在精度和效率方面均优于标准ROM。