Journal Article

Turbulence Closure With Small, Local Neural Networks: Forced Two‐Dimensional and β‐Plane Flows

McWilliams, James C., Chekroun, Mickaël D., Srinivasan, Kaushik

2024

Overview

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, Sd. We minimize a loss between the SGS vorticity flux divergence (computed from the high‐resolution solver) and that obtained from the CNN‐modeled Sd, 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. Plain Language Summary In this study, we demonstrate that simple, shallow neural networks can be used to effectively model complex turbulent flows in the atmosphere and oceans. By using these simpler NNs, we can improve the efficiency of our simulations and better understand the underlying physics of turbulent flows. We also explore different training techniques to make these models more accurate and robust. Our findings suggest that the stress in these turbulent flows has only a weak spatial dependence on larger‐scale features, which has important implications for our understanding of how turbulence behaves. Overall, our work can help improve climate and weather models by providing a more efficient and interpretable way to simulate turbulence. Key Points Shallow convolutional neural networks (CNNs) accurately parameterize high Reynolds number forced2D turbulence, with efficient training and high online large eddy simulations accuracy Extensive hyperparameter searching and cyclical learning rates yield robust and accurate online solutions for CNN‐based turbulence models The success of shallow CNNs implies nearly‐local dependence of SGS stresses providing insights into turbulent flow interactions

• Publisher: John Wiley & Sons, Inc,American Geophysical Union (AGU)
• Subject: Algebra / closure / Dimensional analysis / Distance learning / Dynamical systems / Enstrophy / geostrophic turbulence / Large eddy simulations / machine learning / Neural networks / Physics / Simulation / Turbulence / turbulence parameterization / two‐dimensional turbulence / Vortices / Vorticity