Predicting complex nonlinear turbulent dynamical systems is an important and practical topic. However, due to the lack of a complete understanding of nature, the ubiquitous model error may greatly affect the prediction performance. Machine learning algorithms can overcome the model error, but they are often impeded by inadequate and partial observations in predicting nature. In this article, an efficient and dynamically consistent conditional sampling algorithm is developed, which incorporates the conditional path-wise temporal dependence into a two-step forward-backward data assimilation procedure to sample multiple distinct nonlinear time series conditioned on short and partial observations using an imperfect model. The resulting sampled trajectories succeed in reducing the model error and greatly enrich the training data set for machine learning forecasts. For a rich class of nonlinear and non-Gaussian systems, the conditional sampling is carried out by solving a simple stochastic differential equation, which is computationally efficient and accurate. The sampling algorithm is applied to create massive training data of multiscale compressible shallow water flows from highly nonlinear and indirect observations. The resulting machine learning prediction significantly outweighs the imperfect model forecast. The sampling algorithm also facilitates the machine learning forecast of a highly non-Gaussian climate phenomenon using extremely short observations.
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