Data-driven prediction and physics-agnostic machine-learning methods have attracted increased interest in recent years achieving forecast horizons going well beyond those to be expected for chaotic dynamical systems. In a separate strand of research data-assimilation has been successfully used to optimally combine forecast models and their inherent uncertainty with incoming noisy observations. The key idea in our work here is to achieve increased forecast capabilities by judiciously combining machine-learning algorithms and data assimilation. We combine the physics-agnostic data-driven approach of random feature maps as a forecast model within an ensemble Kalman filter data assimilation procedure. The machine-learning model is learned sequentially by incorporating incoming noisy observations. We show that the obtained forecast model has remarkably good forecast skill while being computationally cheap once trained. Going beyond the task of forecasting, we show that our method can be used to generate reliable ensembles for probabilistic forecasting as well as to learn effective model closure in multi-scale systems.

数据驱动的预测和与物理无关的机器学习方法近年来引起了越来越多的兴趣，其实现的预测范围远远超出了混沌动力学系统的预期范围。在单独的研究链中，数据同化已成功用于将预测模型及其固有的不确定性与传入的噪声观测值进行最佳组合。我们这里工作的关键思想是通过明智地将机器学习算法和数据同化相结合来提高预测能力。我们在集合卡尔曼滤波数据同化程序中结合了随机特征图的物理不可知数据驱动方法作为预测模型。机器学习模型是通过合并传入的噪声观测值而顺序学习的。我们表明，所获得的预测模型具有出色的预测技能，而一旦训练，其计算量就很便宜。超出了预测的任务，我们证明了我们的方法可用于为概率预测生成可靠的集合，以及在多尺度系统中学习有效的模型闭合。