In turbulence modeling, and more particularly in the Large-Eddy Simulation (LES) framework, we are concerned with finding closure models that represent the effect of the unresolved subgrid scales on the resolved scales. Recent approaches gravitate towards machine learning techniques to construct such models. However, the stability of machine-learned closure models and their abidance by physical structure (e.g. symmetries, conservation laws) are still open problems. To tackle both issues, we take the `discretize first, filter next' approach, in which we apply a spatial averaging filter to existing energy-conserving (fine-grid) discretizations. The main novelty is that we extend the system of equations describing the filtered solution with a set of equations that describe the evolution of (a compressed version of) the energy of the subgrid scales. Having an estimate of this energy, we can use the concept of energy conservation and derive stability. The compressed variables are determined via a data-driven technique in such a way that the energy of the subgrid scales is matched. For the extended system, the closure model should be energy-conserving, and a new skew-symmetric convolutional neural network architecture is proposed that has this property. Stability is thus guaranteed, independent of the actual weights and biases of the network. Importantly, our framework allows energy exchange between resolved scales and compressed subgrid scales and thus enables backscatter. To model dissipative systems (e.g. viscous flows), the framework is extended with a diffusive component. The introduced neural network architecture is constructed such that it also satisfies momentum conservation. We apply the new methodology to both the viscous Burgers' equation and the Korteweg-De Vries equation in 1D and show superior stability properties when compared to a vanilla convolutional neural network.

翻译：在湍流建模中，特别是在大涡模拟（LES）框架下，我们关注寻找能够表征未解析亚格子尺度对解析尺度影响的闭合模型。近期研究倾向于采用机器学习技术构建此类模型。然而，机器学习闭合模型的稳定性及其对物理结构（如对称性、守恒律）的遵循仍属开放问题。为解决这两个问题，我们采用"先离散、后滤波"方法，即对现有能量守恒（粗网格）离散格式施加空间平均滤波。核心创新在于：我们通过一组描述亚格子尺度能量（压缩版本）演化的方程来扩展滤波解的系统方程。通过估算该能量值，可运用能量守恒概念推导稳定性。采用数据驱动技术确定压缩变量，使得亚格子尺度能量得以匹配。对于扩展系统，闭合模型需满足能量守恒，我们为此提出具有该性质的新型斜对称卷积神经网络架构。该架构可确保稳定性不受网络实际权重和偏置的影响。重要的是，本框架允许解析尺度与压缩亚格子尺度间的能量交换，从而能够实现反向散射。对于耗散系统（如粘性流），我们通过添加扩散分量扩展该框架。所构建的神经网络架构同时满足动量守恒。我们将新方法应用于一维粘性Burgers方程和Korteweg-De Vries方程，结果表明其稳定性显著优于传统卷积神经网络。