The paper presents a framework for online learning of the Koopman operator using streaming data. Many complex systems for which data-driven modeling and control are sought provide streaming sensor data, the abundance of which can present computational challenges but cannot be ignored. Streaming data can intermittently sample dynamically different regimes or rare events which could be critical to model and control. Using ideas from subspace identification, we present a method where the Grassmannian distance between the subspace of an extended observability matrix and the streaming segment of data is used to assess the `novelty' of the data. If this distance is above a threshold, it is added to an archive and the Koopman operator is updated if not it is discarded. Therefore, our method identifies data from segments of trajectories of a dynamical system that are from different dynamical regimes, prioritizes minimizing the amount of data needed in updating the Koopman model and furthermore reduces the number of basis functions by learning them adaptively. Therefore, by dynamically adjusting the amount of data used and learning basis functions, our method optimizes the model's accuracy and the system order.

翻译：本文提出了一种利用流数据在线学习Koopman算子的框架。许多需要数据驱动建模与控制的复杂系统会产生流式传感器数据，这些海量数据可能带来计算挑战但不可忽视。流数据可能间歇性地采样动态特性不同的状态区间或罕见事件，而这些对建模与控制至关重要。借鉴子空间辨识的思想，我们提出一种方法：通过扩展可观性子空间与数据流片段在Grassmann流形上的距离来评估数据的"新颖性"。若该距离超过阈值，则将该数据存入档案库并更新Koopman算子；否则予以丢弃。因此，我们的方法能够识别来自动态系统不同状态区间的轨迹片段数据，优先考虑最小化更新Koopman模型所需的数据量，并通过自适应学习基函数进一步减少基函数数量。通过动态调整所用数据量与学习基函数，本方法实现了模型精度与系统阶次的最优化。