Building efficient, accurate and generalizable reduced order models of developed turbulence remains a major challenge. This manuscript approaches this problem by developing a hierarchy of parameterized reduced Lagrangian models for turbulent flows, and investigates the effects of enforcing physical structure through Smoothed Particle Hydrodynamics (SPH) versus relying on neural networks (NN)s as universal function approximators. Starting from Neural Network (NN) parameterizations of a Lagrangian acceleration operator, this hierarchy of models gradually incorporates a weakly compressible and parameterized SPH framework, which enforces physical symmetries, such as Galilean, rotational and translational invariances. Within this hierarchy, two new parameterized smoothing kernels are developed in order to increase the flexibility of the learn-able SPH simulators. For each model we experiment with different loss functions which are minimized using gradient based optimization, where efficient computations of gradients are obtained by using Automatic Differentiation (AD) and Sensitivity Analysis (SA). Each model within the hierarchy is trained on two data sets associated with weekly compressible Homogeneous Isotropic Turbulence (HIT): (1) a validation set using weakly compressible SPH; and (2) a high fidelity set from Direct Numerical Simulations (DNS). Numerical evidence shows that encoding more SPH structure improves generalizability to different turbulent Mach numbers and time shifts, and that including the novel parameterized smoothing kernels improves the accuracy of SPH at the resolved scales.

翻译：构建高效、精确且具有泛化能力的充分发展湍流降阶模型仍是一项重大挑战。本文通过发展一系列参数化湍流拉格朗日降阶模型来应对这一问题，并探究了通过光滑粒子流体动力学（SPH）施加物理结构相较于依赖神经网络（NN）作为通用函数逼近器的效果差异。从神经网络参数化的拉格朗日加速度算子出发，该模型层级逐步融入弱可压缩参数化SPH框架，从而强制施加伽利略不变性、旋转不变性及平移不变性等物理对称性。在该层级框架下，我们开发了两种新型参数化光滑核函数以增强可学习SPH模拟器的灵活性。针对每个模型，我们尝试了不同损失函数，并通过基于梯度的优化方法进行最小化，其中梯度的高效计算借助自动微分（AD）和灵敏度分析（SA）实现。层级中的每个模型均基于与弱可压缩均匀各向同性湍流（HIT）相关的两个数据集进行训练：（1）采用弱可压缩SPH的验证数据集；（2）来自直接数值模拟（DNS）的高保真数据集。数值结果表明，编码更多SPH结构可提升对不同湍流马赫数和时间偏移的泛化能力，且引入新型参数化光滑核函数能提高SPH在可解析尺度上的精度。