Learning nonlinear turbulent dynamics from partial observations is an important and challenging topic. In this article, an efficient learning algorithm based on the expectation-maximization approach is developed for a rich class of complex nonlinear turbulent dynamics using short training data. Despite the significant nonlinear and non-Gaussian features in these models, the analytically solvable conditional statistics allows the development of an exact and accurate nonlinear optimal smoother for recovering the hidden variables, which facilitates an efficient learning of these fully nonlinear models with extreme events. Then three additional ingredients are incorporated into the basic algorithm for improving the learning process. First, the physics constraint that requires the conservation of energy in the quadratic nonlinear terms is taken into account. It plays an important role in preventing the finite-time blowup of the solution and various pathological behavior of the recovered model. Second, a judicious block decomposition is applied to many large-dimensional nonlinear systems. It greatly accelerates the calculation of high-dimensional conditional covariance matrix and provides an extremely cheap parallel computation for learning the model parameters. Third, sparse identification of the complex turbulent models is combined with the learning algorithm that leads to parsimonious models. Numerical tests show the skill of the algorithm in learning the nonlinear dynamics and non-Gaussian statistics with extreme events in both perfect model and model error scenarios. It is also shown that in the presence of noise and partial observations, the model is not uniquely identified. Different nonlinear models all perfectly capture the key non-Gaussian features and obtain the same ensemble forecast skill of the observed variables as the perfect model, but they may have distinct model responses to external perturbations.

从局部观测中学习非线性湍流动力学是一个重要且具有挑战性的主题。在本文中，使用短训练数据为一类丰富的复杂非线性湍流动力学开发了一种基于期望最大化方法的有效学习算法。尽管这些模型具有显着的非线性和非高斯特性，但可解析的条件统计量允许开发出精确且准确的非线性最优平滑器，以恢复隐藏变量，这有助于有效学习这些带有极端事件的完全非线性模型。然后，将三个附加成分合并到基本算法中，以改善学习过程。第一，考虑了需要以二次非线性项守恒能量的物理约束。它在防止溶液的有限时间膨胀和恢复模型的各种病理行为中起着重要作用。其次，明智的块分解应用于许多大型非线性系统。它极大地加速了高维条件协方差矩阵的计算，并为学习模型参数提供了非常便宜的并行计算。第三，将复杂湍流模型的稀疏识别与导致简约模型的学习算法相结合。数值测试表明，该算法具有在完美模型和模型误差情况下学习具有极端事件的非线性动力学和非高斯统计量的能力。还表明，在存在噪声和局部观测的情况下，无法唯一识别模型。不同的非线性模型都可以完美地捕获关键的非高斯特征，并获得与理想模型相同的观测变量的整体预测技能，但是它们对外部扰动可能具有不同的模型响应。