Physics‐Informed Deep Operator Learning Based on Reduced‐Order Modeling for Retrieving the Ocean Interior Density From the Surface

Abstract Exploring methods to reconstruct the ocean interior from surface data is a crucial focus in the study of ocean processes and phenomena due to the shortage of subsurface and deep‐sea data. Nonetheless, the existing methods predominantly concentrate on either data‐driven or dynamical methodologies, with limited exploration of integrating the strengths of both approaches. To combine the advantages of these two methods for reconstructing the subsurface density field from surface data, a novel dynamics‐constrained deep operator learning network based on reduced‐order model is proposed. Encoding the mean‐squared residuals of the reduced‐order equation along with the mean‐squared error between the network outputs and targets into the loss function effectively merges the dynamical and data constraints during the training process. This integration makes the network outputs and inputs approximately satisfy a specific form of the equation, allowing for interpretability, and once the network is well‐trained, rapid reconstruction evaluation can be performed. The reduced‐order equation is established by the Galerkin projection of quasi‐geostrophic equation onto the low‐dimensional subspace identified via reduced‐basis, which explains the vertical variation of ocean density. The developed model can tackle the challenge of directly measuring subsurface potential vorticity and predicting subsurface density. Evaluation is conducted using simulation data from the Max‐Planck‐Institute ocean model, indicating that it can offer precise estimations, outperforms the purely data‐driven algorithm presented in the paper and the interior plus surface quasi‐geostrophic method, and enables model sharing across different regions.