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、精确且可泛化的充分发展湍流降阶模型仍是重大挑战。本文通过建立湍流参数化拉格朗日模型的层级结构来解决该问题，并研究了通过光滑粒子流体动力学（SPH）强制物理结构与依赖神经网络（NN）作为通用函数逼近器两者效果的差异。该模型层级从拉格朗日加速度算子的神经网络参数化出发，逐步融入弱可压缩参数化SPH框架，从而强制满足伽利略不变性、旋转不变性和平移不变性等物理对称性。在此层级中，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，我们实验了不同损失函数，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采用自动微分（AD）和敏感性分析（SA）实现。层级中的每个模型均基于两组与弱可压缩均匀各向同性湍流（HIT）相关的数据集进行训练：（1）使用弱可压缩SPH的验证集；（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结构能提升对不同湍流马赫数和时移的泛化能力，而引入新型参数化光滑核函数则提高了SPH在可解析尺度上的精度。