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Earth system models suffer from various structural and parametric errors intheir representation of nonlinear, multi-scale processes, leading touncertainties in their long-term projections. The effects of many of theseerrors (particularly those due to fast physics) can be quantified in short-termsimulations, e.g., as differences between the predicted and observed states(analysis increments). With the increase in the availability of high-qualityobservations and simulations, learning nudging from these increments to correctmodel errors has become an active research area. However, most studies focus onusing neural networks, which while powerful, are hard to interpret, aredata-hungry, and poorly generalize out-of-distribution. Here, we show thecapabilities of Model Error Discovery with Interpretability and DataAssimilation (MEDIDA), a general, data-efficient framework that usessparsity-promoting equation-discovery techniques to learn model errors fromanalysis increments. Using two-layer quasi-geostrophic turbulence as the testcase, MEDIDA is shown to successfully discover various linear and nonlinearstructural/parametric errors when full observations are available. Discoveryfrom spatially sparse observations is found to require highly accurateinterpolation schemes. While NNs have shown success as interpolators in recentstudies, here, they are found inadequate due to their inability to accuratelyrepresent small scales, a phenomenon known as spectral bias. We show that ageneral remedy, adding a random Fourier feature layer to the NN, resolves thisissue enabling MEDIDA to successfully discover model errors from sparseobservations. These promising results suggest that with further development,MEDIDA could be scaled up to models of the Earth system and real observations.

Interpretable structural model error discovery from sparse assimilation  increments using spectral bias-reduced neural networks: A quasi-geostrophic  turbulence test case