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网格尺度对可解尺度影响的闭合模型。近期研究方法倾向于利用机器学习技术构建此类模型。然而，机器学习获得的闭合模型稳定性及其对物理结构（如对称性、守恒律）的遵循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"的方法。该方法通过对现有细网格离散化方案施加空间平均滤波，其主要创新在于引入一组额外方程来动态建模亚网格尺度的能量。通过估算亚网格能量，我们可利用能量守恒概念推导稳定性。包含亚网格能量的变量通过数据驱动技术确定，闭合模型用于建模滤波量与亚网格能量之间的相互作用。由于总能量必须守恒，遵循该守恒律可保证系统的稳定性。本文提出一种满足该守恒律的新型斜对称卷积神经网络架构，其优势在于不论网络权重与偏置如何，稳定性均可得到保证。值得注意的是，由于该框架允许可解尺度与亚网格尺度之间的能量交换，可有效模拟背向散射。针对耗散系统（如粘性流动），该框架通过引入扩散分量进行扩展。所提出的神经网络架构还经过专门设计以满足动量守恒。我们将新方法分别应用于一维粘性Burgers方程和Korteweg-De Vries方程。相较于传统卷积神经网络，新型架构展现出更优异的稳定性特征。