用神经网络学习浅水方程亚网格通量，提升长期模拟能量守恒性。
Parametrization of subgrid scales in long-term simulations of the shallow-water equations using machine learning and convex limiting
- 基于前馈神经网络和局部平均变量构建四点模板的局域参数化方法
- 长期湍流模拟中显著改善能量平衡，且能准确复现具体解
- 训练外动态下仍可靠，可与通量限制结合抑制激波附近振荡
本文提出一种浅水方程中亚网格过程的参数化方法。通过定义粗粒度变量和局部空间平均，利用前馈神经网络学习亚网格通量。该方法实现基于四点计算模板的局域参数化，相比全局耦合参数化具有多重优势。数值实验表明，该方法在长期湍流模拟中显著改善能量平衡，并能精确再现个别解。长期模拟指流体流动持续时间足够长以达到统计稳态的数值研究。神经网络参数化可简便地与通量限制结合，降低激波附近的振荡。更重要的是，该方法在训练数据未涵盖的动力学状态中仍能提供可靠参数化。
原文摘要 · Abstract (English)
We present a method for parametrizing sub-grid processes in the Shallow Water equations. We define coarse variables and local spatial averages and use a feed-forward neural network to learn sub-grid fluxes. Our method results in a local parametrization that uses a four-point computational stencil, which has several advantages over globally coupled parametrizations. We demonstrate numerically that our method improves energy balance in long-term turbulent simulations and also accurately reproduces individual solutions. The long-term simulations refer to numerical studies where a fluid flow is simulated over a duration long enough to reach a statistical steady state. The neural network parametrization can be easily combined with flux limiting to reduce oscillations near shocks. More importantly, our method provides reliable parametrizations, even in dynamical regimes that are not included in the training data.
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