We present two families of sub-grid scale (SGS) turbulence models developed for large-eddy simulation (LES) purposes. Their development required the formulation of physics-informed robust and efficient Deep Learning (DL) algorithms which, unlike state-of-the-art analytical modeling techniques can produce high-order complex non-linear relations between inputs and outputs. Explicit filtering of data from direct simulations of the canonical channel flow at two friction Reynolds numbers $Re_\tau\approx 395$ and 590 provided accurate data for training and testing. The two sets of models use different network architectures. One of the architectures uses tensor basis neural networks (TBNN) and embeds the simplified analytical model form of the general effective-viscosity hypothesis, thus incorporating the Galilean, rotational and reflectional invariances. The other architecture is that of a relatively simple network, that is able to incorporate the Galilean invariance only. However, this simpler architecture has better feature extraction capacity owing to its ability to establish relations between and extract information from cross-components of the integrity basis tensors and the SGS stresses. Both sets of models are used to predict the SGS stresses for feature datasets generated with different filter widths, and at different Reynolds numbers. It is shown that due to the simpler model's better feature learning capabilities, it outperforms the invariance embedded model in statistical performance metrics. In a priori tests, both sets of models provide similar levels of dissipation and backscatter. Based on the test results, both sets of models should be usable in a posteriori actual LESs.

翻译：我们提出了两类针对大涡模拟（LES）开发的亚格子尺度（SGS）湍流模型。这两类模型的开发需要构建基于物理信息的稳健高效深度学习（DL）算法，与现有最先进的分析建模技术不同，这类算法能在输入与输出之间建立高阶复杂非线性关系。通过显式滤波两个摩擦雷诺数$Re_\tau\approx 395$和590的经典槽道流直接模拟数据，我们获得了用于训练和测试的精确数据集。这两类模型采用不同的网络架构：一类基于张量基神经网络（TBNN），嵌入了广义有效粘性假设的简化分析模型形式，从而纳入了伽利略不变性、旋转不变性和反射不变性；另一类架构则相对简单，仅能纳入伽利略不变性。然而，这一简单架构凭借其从完整基张量交叉分量与SGS应力之间建立关系并提取信息的能力，具备更强的特征提取能力。两类模型均用于预测不同滤波宽度和不同雷诺数下的SGS应力。结果表明，简单模型因具备更强的特征学习能力，在统计性能指标上优于嵌入不变性的模型。在先验测试中，两类模型在耗散和反向散射方面表现相当。基于测试结果，两类模型均可应用于后验实际LES中。