We parameterize sub-grid scale (SGS) fluxes in sinusoidally forced two-dimensional turbulence on the β-plane at high Reynolds numbers (Re ∼25,000) using simple 2-layer convolutional neural networks (CNN) having only O(1000) parameters, two orders of magnitude smaller than recent studies employing deeper CNNs with 8–10 layers; we obtain stable, accurate, and long-term online or a posteriori solutions at 16× downscaling factors. Our methodology significantly improves training efficiency and speed of online large eddy simulations runs, while offering insights into the physics of closure in such turbulent flows. Our approach benefits from extensive hyperparameter searching in learning rate and weight decay coefficient space, as well as the use of cyclical learning rate annealing, which leads to more robust and accurate online solutions compared to fixed learning rates. Our CNNs use either the coarse velocity or the vorticity and strain fields as inputs, and output the two components of the deviatoric stress tensor, S_d. We minimize a loss between the SGS vorticity flux divergence (computed from the high-resolution solver) and that obtained from the CNN-modeled S_d, without requiring energy or enstrophy preserving constraints. The success of shallow CNNs in accurately parameterizing this class of turbulent flows implies that the SGS stresses have a weak non-local dependence on coarse fields; it also aligns with our physical conception that small-scales are locally controlled by larger scales such as vortices and their strained filaments. Furthermore, 2-layer CNN-parameterizations are more likely to be interpretable.

中文翻译：

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小型局部神经网络的湍流闭合：强制二维和 β 平面流

我们使用仅具有 O(1000) 参数的简单 2 层卷积神经网络 (CNN)，对高雷诺数 (Re ~25,000) β平面上正弦强制二维湍流中的亚网格尺度 (SGS) 通量进行参数化，比最近使用 8-10 层更深的 CNN 的研究小两个数量级；我们在 16 倍缩小因子下获得稳定、准确和长期的在线或后验解。我们的方法显着提高了在线大涡流模拟运行的训练效率和速度，同时提供了对此类湍流中闭合物理学的见解。我们的方法受益于学习率和权重衰减系数空间中广泛的超参数搜索，以及循环学习率退火的使用，与固定学习率相比，这可以带来更稳健和更准确的在线解决方案。我们的 CNN 使用粗略速度或涡度和应变场作为输入，并输出偏应力张量S _{d 的}两个分量。我们最大限度地减少了 SGS 涡度通量散度（由高分辨率求解器计算）与从 CNN 建模的S _d获得的涡度通量散度之间的损失，而不需要能量或熵保持约束。浅层 CNN 在精确参数化此类湍流方面的成功意味着 SGS 应力对粗场具有较弱的非局部依赖性；它也符合我们的物理概念，即小尺度是由较大尺度（例如涡流及其应变细丝）局部控制的。此外，2 层 CNN 参数化更有可能是可解释的。