In turbulence modeling, we are concerned with finding closure models that represent the effect of the 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 this approach we apply a spatial averaging filter to existing fine-grid discretizations. The main novelty is that we introduce an additional set of equations which dynamically model the energy of the subgrid scales. Having an estimate of the energy of the subgrid scales, we can use the concept of energy conservation to derive stability. The subgrid energy containing variables are determined via a data-driven technique. The closure model is used to model the interaction between the filtered quantities and the subgrid energy. Therefore the total energy should be conserved. Abiding by this conservation law yields guaranteed stability of the system. In this work, we propose a novel skew-symmetric convolutional neural network architecture that satisfies this law. The result is that stability is guaranteed, independent of the weights and biases of the network. Importantly, as our framework allows for energy exchange between resolved and subgrid scales it can model 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. The novel architecture displays superior stability properties when compared to a vanilla convolutional neural network.

翻译：在湍流建模中，我们关注的是寻找能够表征亚格子尺度对可解尺度影响的闭合模型。近年来的研究趋势倾向于利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、守恒定律）的遵从性仍是悬而未决的问题。为同时解决这两个问题，我们采用“先离散后滤波”的范式：对现有细网格离散格式施加空间平均滤波器。主要创新在于引入一组额外方程来动态建模亚格子尺度的能量。通过获取亚格子能量的估计值，可借助能量守恒概念推导系统稳定性。亚格子能量相关变量通过数据驱动方法确定，闭合模型用于描述滤波量域亚格子能量之间的相互作用，因而需确保总能量守恒。遵循该守恒律可保证系统具有确定稳定性。本研究提出一种满足该守恒律的新型斜对称卷积神经网络架构，其稳定性与网络权重和偏置无关。特别地，本框架允许可解尺度与亚格子尺度间的能量交换，从而能够模拟反向散射。为模拟耗散系统（如黏性流），我们在框架中引入扩散组分。所构建的神经网络架构同时满足动量守恒定律。我们将新方法应用于一维黏性Burgers方程和Korteweg-De Vries方程。相较于传统卷积神经网络，该新型架构展现出更优异的稳定性特征。