Most global ocean circulation models used in Earth System Modeling operate at non-eddy-resolving or eddy-permitting resolutions due to computational constraints. As a result, these models often fail to properly resolve small-scale processes, which are important for ocean circulations and related tracer redistribution. The unresolved effects are necessarily approximated using eddy parameterization. While machine learning, particularly convolutional neural networks (CNNs), has emerged as a powerful tool for learning subgrid eddy parameterizations from high-resolution simulations, purely data-driven approaches often produce physically inconsistent results (e.g., violating energy conservation) and struggle to generalize beyond their training regimes, especially in chaotic, nonlinear systems...[ Read more ]

Most global ocean circulation models used in Earth System Modeling operate at non-eddy-resolving or eddy-permitting resolutions due to computational constraints. As a result, these models often fail to properly resolve small-scale processes, which are important for ocean circulations and related tracer redistribution. The unresolved effects are necessarily approximated using eddy parameterization. While machine learning, particularly convolutional neural networks (CNNs), has emerged as a powerful tool for learning subgrid eddy parameterizations from high-resolution simulations, purely data-driven approaches often produce physically inconsistent results (e.g., violating energy conservation) and struggle to generalize beyond their training regimes, especially in chaotic, nonlinear systems like ocean turbulence.

To address these challenges, we propose integrating domain knowledge into CNN training through two approaches. First, we choose training datasets based on a priori knowledge. In an idealized Quasi-Geostrophic (QG) model, training CNNs with only physically relevant information significantly enhances their robustness while maintaining strong performance. Second, we embed CNNs into a differentiable low-resolution QG model to create a hybrid system. This approach, referred to as online learning, allows CNNs to train through direct interaction with the dynamical model, enabling adaptive corrections during simulations. We demonstrate how to implement online learning and testing in a fully differentiable two-layer QG model. Compared to a purely data-driven (offline learning) approach, online learning shows superior performance in both a priori and a posteriori testing. The online hybrid model exhibits improved stability and generalization by leveraging both the provided training data and real-time feedback from the dynamic model. Our results highlight the importance of coupling data-driven methods with physical constraints to improve the fidelity and reliability of subgrid eddy parameterizations in ocean modeling.