Testing the Limits of Surface QG Theory With Deep Learning

Abstract Surface Quasi‐Geostrophic (SQG) theory provides a closed‐form mapping from surface buoyancy to interior stream function under the assumption that interior potential vorticity (PV) vanishes . In realistic flows, however, the interior PV is typically nonzero. Here we use machine learning to determine whether surface buoyancy can be used to infer additional information about interior flow when the interior PV is non‐zero and SQG theory is not expected to work well. We perform controlled numerical experiments with a multilayer QG model spanning three regimes: a canonical SQG case and two multilayer quasi‐geostrophic cases with linear and exponential zonal velocity shear profiles, correspondingly. We then use these data to train a Vision Transformer (ViT) to predict the streamfunction at all levels from the surface buoyancy in each regime, and we compare it with the predictions of SQG theory. In the SQG‐consistent regime, the ViT recovers the SQG analytical predictions to within normalized error. When interior PV is present, the SQG calculation develops systematic depth‐dependent amplitude biases and pattern degradation, as expected; the ViT shows better skill in predicting the interior flow and reduces these errors by approximately 50%–, depending on depth and regime. These results demonstrate that learned operators can accurately predict interior structure from surface data alone, even when classical assumptions break down. This suggests a potential for an improved theory in the regime of non‐zero interior PV.