Interpretable Structural Model Error Discovery From Sparse Assimilation Increments Using Spectral Bias‐Reduced Neural Networks: A Quasi‐Geostrophic Turbulence Test Case.

Earth system models suffer from various structural and parametric errors in their representation of nonlinear, multi‐scale processes, leading to uncertainties in their long‐term projections. The effects of many of these errors (particularly those due to fast physics) can be quantified in short‐term simulations, for example, as differences between the predicted and observed states (analysis increments). With the increase in the availability of high‐quality observations and simulations, learning nudging from these increments to correct model errors has become an active research area. However, most studies focus on using neural networks, which while powerful, are hard to interpret, are data‐hungry, and poorly generalize out‐of‐distribution. Here, we show the capabilities of Model Error Discovery with Interpretability and Data Assimilation (MEDIDA), a general, data‐efficient framework that uses sparsity‐promoting equation‐discovery techniques to learn model errors from analysis increments. Using two‐layer quasi‐geostrophic turbulence as the test case, MEDIDA is shown to successfully discover various linear and nonlinear structural/parametric errors when full observations are available. Discovery from spatially sparse observations is found to require highly accurate interpolation schemes. While NNs have shown success as interpolators in recent studies, here, they are found inadequate due to their inability to accurately represent small scales, a phenomenon known as spectral bias. We show that a general remedy, adding a random Fourier feature layer to the NN, resolves this issue enabling MEDIDA to successfully discover model errors from sparse observations. These promising results suggest that with further development, MEDIDA could be scaled up to models of the Earth system and real observations. Plain Language Summary: Numerical models are used to predict the Earth system, for example, from daily weather to the next‐century climate. These models have been developed and validated against observations over decades, however, they still have shortcomings (errors) in their representations of many complex processes, particularly those that are nonlinear and span many scales in time and space. The rapid improvements in the quality and quantity of observational data from the Earth system and advances in machine learning (ML) algorithms provide an opportunity to try to reduce these errors. However, the challenge is that many ML methods require a lot of training data, and it is also often difficult to explain how they are reducing the error. Here, we show the capabilities of a framework called MEDIDA (Model Error Discovery with Interpretability and Data Assimilation), which uses a class of ML methods that provide closed‐form (thus interpretable) equations for what they are learning from the differences between observations and model predictions. We show the success of MEDIDA when applied to a model of atmospheric turbulent circulation. Even when observations are only sparsely (not at every location) available, we show that MEDIDA works accurately once we leverage more recent advances in ML. Key Points: Model error discovery with interpretability and data assimilation is scaled up to geostrophic turbulence and sparse observationsNaive use of neural nets (NNs) as interpolator does not capture small scales due to spectral bias, failing discoveries of closed‐form errorsReducing this bias using random Fourier features enables NNs to represent the full range of scales, leading to successful error discoveries [ABSTRACT FROM AUTHOR]