A nonlinear optimal smoother and an associated optimal strategy of sampling hidden model trajectories are developed for a rich class of complex nonlinear turbulent dynamical systems with partial and noisy observations. Despite the strong nonlinearity and the significant non-Gaussian characteristics in the underlying systems, both the optimal smoother estimates and the sampled trajectories can be solved via closed analytic formulae. Thus, they are computationally efficient and the methods are applicable to high-dimensional systems. The nonlinear optimal smoother is able to estimate the hidden model states associated with various non-Gaussian phenomena and is particularly skillful in capturing the onset, demise and amplitude of the observed and hidden extreme events. On the other hand, the sampled hidden trajectories succeed in recovering both the dynamical and statistical features of the underlying nonlinear systems, including the fat-tailed non-Gaussian probability density function and the temporal autocorrelation function. In the situations with only a short period of partially observed training time series, the optimal sampling strategy can be used to efficiently create a sufficient number of samples in an unbiased fashion that facilitates an accurate prediction of important non-Gaussian features in both the observed and hidden variables. In addition, the information provided by the sampled trajectories based on imperfect models allows an effective way of quantifying the model error. It also offers a systematic approach to improve approximate models and stochastic parameterizations in highly non-Gaussian systems and thus advances the real-time forecasts.

针对具有部分观测值和嘈杂观测值的一类复杂的复杂非线性湍流动力学系统，开发了一种非线性最优平滑器和相关的最优策略来对隐藏的模型轨迹进行采样。尽管底层系统具有很强的非线性和显着的非高斯特性，但可以通过封闭的解析公式来求解最佳平滑估计和采样轨迹。因此，它们在计算上是有效的，并且这些方法适用于高维系统。非线性最优平滑器能够估计与各种非高斯现象相关的隐藏模型状态，并且特别擅长捕获观察到的和隐藏的极端事件的发作，消亡和振幅。另一方面，采样的隐藏轨迹成功地恢复了底层非线性系统的动力学和统计特征，包括胖尾非高斯概率密度函数和时间自相关函数。在只有短时间部分观测到的训练时间序列的情况下，最佳采样策略可用于无偏见地有效创建足够数量的样本，从而有助于在被观测和非观测两方面都准确预测重要的非高斯特征。隐藏的变量。另外，由基于不完美模型的采样轨迹提供的信息允许量化模型误差的有效方法。