Real-time data analysis requires the ability to accurately and adaptively address nonlinear dynamics in a nonstationary data stream while preserving computational efficiency. However, nonlinear dynamics are so complex that capturing dynamically changing nonlinear patterns and utilizing them for downstream tasks under strict time constraints is nontrivial. To bridge the gap between nonlinear complexity and computational tractability, this study applies Koopman operator theory, which states that nonlinear dynamics can be represented as linear transitions in an infinite-dimensional space. Building upon finite-dimensional approximations of this operator, we present AdaKoop, an efficient streaming algorithm for modeling nonlinear dynamics over nonstationary data streams. Our approach utilizes a probabilistic framework grounded in Koopman operator theory, treating both raw observations and reproducing kernel Hilbert space (RKHS) features as emissions from latent vectors. This dual-view formulation allows nonlinear dynamics to be expressed as a tractable linear system. Therefore, AdaKoop enables the efficient and stable modeling of nonlinear dynamics in a streaming fashion, avoiding the prohibitive computational costs of iterative nonlinear optimization. Furthermore, to address nonstationarity in data streams, AdaKoop adaptively detects the switching of patterns via statistical hypothesis testing for abrupt pattern shifts and incrementally updates model parameters to handle continuous changes. Extensive experiments on a total of 71 practical benchmark datasets across various domains demonstrate that AdaKoop outperforms state-of-the-art methods in terms of real-time forecasting accuracy and computational efficiency.

翻译：实时数据分析要求能够准确且自适应地处理非平稳数据流中的非线性动力学问题，同时保持计算效率。然而，非线性动力系统极其复杂，在严格时间约束下捕捉动态变化的非线性模式并用于下游任务极具挑战性。为弥合非线性复杂度与计算可行性之间的鸿沟，本研究应用库普曼算子理论——该理论指出非线性动力系统可在无限维空间中表示为线性变换。基于该算子的有限维近似，我们提出AdaKoop，一种面向非平稳数据流非线性动力学建模的高效流式算法。该方法构建于库普曼算子理论的概率框架之上，将原始观测值与再生核希尔伯特空间特征均视为隐含向量的发射观测。这种双视角表达允许非线性动力系统转化为可解线性系统。因此，AdaKoop能够以流式方式实现非线性动力系统的高效稳定建模，避免了迭代非线性优化带来的高昂计算成本。此外，为应对数据流的非平稳性，AdaKoop通过统计假设检验自适应检测模式突变，并增量更新模型参数以处理连续变化。在涵盖多个领域的71个实际基准数据集上的大量实验表明，AdaKoop在实时预测精度与计算效率方面均优于现有最优方法。