标题： 嵌入不变性机器学习亚网格尺度应力模型用于中尺度飓风边界层流模拟 I：模型开发和$\textit{a priori}$研究 显示英文标题

标题： Invariance-embedded Machine Learning Sub-grid-scale Stress Models for Meso-scale Hurricane Boundary Layer Flow Simulation I: Model Development and $\textit{a priori}$ Studies

摘要： 本研究开发了一种嵌入不变性特征的机器学习亚网格尺度（SGS）应力模型，该模型允许湍流动能（TKE）向更大尺度的飓风边界层流动的大涡模拟（LES）中出现反向耗散。 新的机器学习SGS模型由两部分组成：一个分类模型用于区分具有强能量级联或能量反向耗散的区域与具有温和TKE传输的区域；以及一个回归模型，用于计算具有强TKE传输区域的SGS应力。 为了简化模型在计算流体力学（CFD）求解器中的实现，采用带有符号系数 $C_s$ 的Smagorinsky模型作为机器学习模型的载体，其中正值表示能量级联，负值表示能量反向耗散。 为了提高模型的鲁棒性和通用性，湍流流动的物理不变性和几何不变性特征被嵌入到分类和回归模型的输入中，且符号Smagorinsky模型系数被用作回归模型的输出。 不同的机器学习方法和输入设置已被用于测试分类模型的性能。 嵌入物理和几何不变性特征的分类模型的F1分数（衡量模型精度和召回率的平衡指标），比未考虑几何不变性的模型提高了约 $17\%$。 基于集成神经网络的回归模型在预测符号Smagorinsky模型系数方面表现出色，在 $\textit{a priori}$ 次测试中优于动态Smagorinsky模型。 显示英文摘要

摘要： This study develops invariance-embedded machine learning sub-grid-scale (SGS) stress models admitting turbulence kinetic energy (TKE) backscatter towards more accurate large eddy simulation (LES) of meso-scale turbulent hurricane boundary layer flows. The new machine learning SGS model consists of two parts: a classification model used to distinguish regions with either strong energy cascade or energy backscatter from those with mild TKE transfer and a regression model used to calculate SGS stresses in regions with strong TKE transfer. To ease model implementation in computational fluid dynamics (CFD) solvers, the Smagorinsky model with a signed coefficient $C_s$, where a positive value indicates energy cascade while a negative one indicates energy backscatter, is employed as the carrier of the machine learning model. To improve its robustness and generality, both physical invariance and geometric invariance features of turbulent flows are embedded into the model input for classification and regression, and the signed Smagorinsky model coefficient is used as the output of the regression model. Different machine-learning methods and input setups have been used to test the classification model's performance. The F1-scores, which measure balanced precision and recall of a model, of the classification models with physical and geometric invariance embedded can be improved by about $17\%$ over those without considering geometric invariance. Regression models based on ensemble neural networks have demonstrated superior performance in predicting the signed Smagorinsky model coefficient, exceeding that of the dynamic Smagorinsky model in $\textit{a priori}$ tests.