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, e.g., 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.

翻译：地球系统模型在表征非线性、多尺度过程时存在多种结构与参数误差，导致其长期预测存在不确定性。许多此类误差（尤其是快速物理过程相关误差）的影响可通过短期模拟量化，例如预测状态与观测状态之间的差异（即分析增量）。随着高质量观测与模拟数据的增加，从这些增量中学习 nudging 以修正模型误差已成为一个活跃的研究领域。然而，多数研究聚焦于神经网络，尽管这些方法功能强大，却难以解释、数据需求量大且分布外泛化能力差。本文展示了具备可解释性与数据同化的模型误差发现框架（MEDIDA）的能力——这是一种通用、数据高效的方法，利用稀疏化促进的方程发现技术从分析增量中学习模型误差。以双层准地转湍流为试验案例，MEDIDA 被证明能在全观测条件下成功发现多种线性和非线性结构/参数误差。研究发现，从空间稀疏观测中发现误差需要极高精度的插值方案。尽管神经网络在近期研究中已被证明作为插值器的有效性，但由于其难以精确表征小尺度现象（即谱偏差问题），在此应用中表现不足。我们证明了一种通用解决方案——在神经网络中添加随机傅里叶特征层——可解决该问题，使 MEDIDA 能够从稀疏观测中成功发现模型误差。这些令人鼓舞的结果表明，通过进一步开发，MEDIDA 可扩展至地球系统模型与实际观测。