Most global ocean circulation models used in Earth System Modeling operate at resolutions too coarse to resolve small-scale processes due to computational constraints. These unresolved processes significantly influence ocean circulation and tracer redistribution. Current models address this by approximating their effects through eddy parameterizations. While machine learning, particularly convolutional neural networks (CNNs), has emerged as a powerful tool for eddy parameterizations, purely data-driven approaches often produce physically inconsistent results (e.g., violating energy conservation) and struggle to generalize beyond their training regimes.
To overcome these limitations, we propose integrating domain knowledge into CNN training through two approaches. First, we choose training datasets based on priori knowledge. Training CNNs with only physically relevant information significantly enhances their robustness while maintaining strong performance. Second, we embed CNNs into a differentiable model to create a hybrid system, which is referred to as online learning. Compared to a purely data-driven (offline learning) approach, 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 eddy parameterizations in ocean modeling.