Learning physics-constrained subgrid-scale closures in the small-data regime for stable and accurate LES

A Posteriori Learning for Quasi‐Geostrophic Turbulence Parametrization

“…This approach has been applied to scalar (Frezat et al., 2021; Portwood et al., 2021; Vollant et al., 2017) and momentum (Beck et al., 2019; Gamahara & Hattori, 2017; Xie et al., 2020; Yuan et al., 2020) parametrizations of three‐dimensional turbulence on different configurations. The two‐dimensional case is also well documented in decaying (Guan, Chattopadhyay, et al., 2022; Maulik et al., 2019; Pawar et al., 2020) and double‐gyre (Bolton & Zanna, 2019; Zanna & Bolton, 2020) configurations. We may emphasize that, by construction, the a priori learning strategy shall lead to the best a priori results, which shall translate into a good instantaneous prediction of the SGS term according to metrics $${\mathcal{L}}_{\text{prio}}$$.…”

“…The a posteriori learning strategy states the SGS parametrization problem as the approximation of the true reduced variables according to some a posteriori metrics. This is important since it is possible for a model to perform well a priori while failing a posteriori, the most common factor being numerical instabilities due to the lack of small‐scale energy dissipation (Guan, Chattopadhyay, et al., 2022; Maulik et al., 2019). Let us denote by Φ the flow operator that advances the reduced system in time, that is, $$${{\Phi}}_{\theta }^{{t}_{1}}\left(\bar{\mathbf{y}}\left({t}_{0}\right)\right)=\bar{\mathbf{y}}\left({t}_{0}\right)+\int \nolimits_{{t}_{0}}^{{t}_{1}}g\left(\bar{\mathbf{y}}(t)\right)+\mathcal{M}\left(\bar{\mathbf{y}}(t)\vert \theta \right)\mathrm{d}t.$$$ …”

“…More recently, Guan, Subel, et al. (2022) and Pawar et al. (2022) demonstrated how incorporating physics in the models could lead to stable simulations that requires less data for training and generalizes better.…”

Explaining the physics of transfer learning in data-driven turbulence modeling

“…r is the linear drag coefficient and f ( x , y ) is a time-independent external forcing at wavenumbers m f and n f . This system, with different combinations of f and r , is a fitting prototype for a variety of large-scale geophysical and environmental flows and has been widely used to test novel techniques including data-driven SGS closures [ 40 , 21 , 7 , 41 , 42 , 39 ].…”

“…By changing Re , r , m f , and n f , we have created six distinctly different flows, divided into three cases, each with a base and a target system (Table 1 and Materials and methods). We have shown in previous studies that for various setups of 2D turbulence, CNNs trained on large training sets, or on small training sets with physics-constraints incorporated, produce accurate and stable data-driven closures in a priori (offline) and a posteriori (online) tests [ 21 , 39 ]. These CNN-based closures were found to accurately capture both diffusion and backscattering, and to outperform widely used physics-based SGS closures such as the Smagorinsky, dynamic Smagorinsky, and mixed models in both a priori and a posteriori tests.…”