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、准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。本文通过发展一组参数化降阶拉格朗日模型来逼近湍流问题，并研究通过光滑粒子流体动力学（SPH）强制施加物理结构相对于依赖神经网络（NN）作为通用函数逼近器的影响。从拉格朗日加速度算子的神经网络参数化出发，该模型层次逐步引入弱可压缩参数化SPH框架，强制施加伽利略不变性、旋转不变性和平移不变性等物理对称性。在此层次中，我们开发了两种新的参数化光滑核函数，以增强可学习SPH模拟器的灵活性。针对每个模型，我们尝试了不同的损失函数，并通过基于梯度的优化方法最小化目标，利用自动微分（AD）和灵敏度分析（SA）实现梯度的高效计算。层次中的每个模型均在两个与弱可压缩均匀各向同性湍流（HIT）相关的数据集上训练：（1）基于弱可压缩SPH的验证数据集；（2）基于直接数值模拟（DNS）的高保真数据集。数值实验表明，编码更多SPH结构可提高模型对不同湍流马赫数及时间偏移的泛化能力，而引入新型参数化光滑核函数则提升了SPH在解析尺度上的精度。