A highly efficient Physics-Constrained Explainable Artificial Intelligence for Forecasting Subsurface Mesoscale Eddy Trajectories

Mesoscale eddies play a central role in oceanic energy transport, material exchange, and climate variability. Compared with surface eddies, subsurface eddies are more difficult to detect and predict because of sparse direct observations and more complex vertical dynamical mechanisms. Despite their significance, operational forecasting tools for subsurface eddy trajectories remain underdeveloped, and many existing data-driven models lack physical consistency, highlighting the need for approaches that jointly enforce dynamical constraints and predictive skill. To address this challenge, we develop a physics-constrained neural network framework, the subsurface eddy trajectory prediction network (SETPNET), which uses vector-geometry methods to reconstruct 3-D eddy structures from the GLORYS12V1 reanalysis dataset. SETPNET explicitly leverages zonal and meridional velocity components, eddy radius, and vorticity, variables that capture depth-dependent dynamics and rotational structure, to forecast the trajectories of subsurface mesoscale eddies. As mesoscale eddies evolve under a quasi-geostrophic balance, we incorporate simplified horizontal momentum equations, including pressure-gradient and Coriolis forces, into the model loss function to ensure dynamical consistency in the predicted trajectories. Relative to long short-term memory (LSTM) baselines, SETPNET reduces Week-3 root mean square error (RMSE) and median angular deviations by approximately 30% and 20%, respectively, demonstrating its ability to mitigate error accumulation overextended forecast horizons. Ablation studies further reveal that incorporating the pressure gradient force (PGF) term into the loss function contributes to a 6%–7% reduction in Week-3 RMSE. Furthermore, a SHapley Additive exPlanations (SHAP) analysis confirms the dominant roles of velocity components and eddy structural parameters, providing physically interpretable insights into the model behavior. Overall, SETPNET offers a physically con...