A posteriori reduced-order models, e.g. proper orthogonal decomposition, 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-order solution. The methodologies devised are based on exploiting basic physical principles, such as mass and momentum conservation, to devise 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.

翻译：后验降阶模型（如本征正交分解）对于经济地处理实际参数化问题至关重要。这类模型依赖于可靠的训练集，即一组能够代表参数化问题所有可能结果的全局解（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并不可行。本文提出了一种专为参数化层流不可压缩流动设计的数据增强策略，旨在丰富样本匮乏的训练集。其目标是在新生成的人工快照中包含原始基中未出现的新特征，从而提升降阶解的精度。所提出的方法基于质量守恒和动量守恒等基本物理原理，以远低于额外全局解的计算成本生成具有物理意义的人工快照。有趣的是，数值结果表明，仅利用质量守恒（即不可压缩性）的策略相较于标准快照线性组合并未产生显著增益。相反，通过奥辛方程引入线性化动量平衡后，所得近似解的质量得到提升，因此对于粘性不可压缩层流问题而言，这是一种有效的数据增强策略。