We address the problem of classifying trajectory data generated by some nonlinear dynamics, where each class corresponds to a distinct dynamical system. We propose Dynafit, a kernel-based method for learning a distance metric between training trajectories and the underlying dynamics. New observations are assigned to the class with the most similar dynamics according to the learned metric. The learning algorithm approximates the Koopman operator which globally linearizes the dynamics in a (potentially infinite) feature space associated with a kernel function. The distance metric is computed in feature space independently of its dimensionality by using the kernel trick common in machine learning. We also show that the kernel function can be tailored to incorporate partial knowledge of the dynamics when available. Dynafit is applicable to various classification tasks involving nonlinear dynamical systems and sensors. We illustrate its effectiveness on three examples: chaos detection with the logistic map, recognition of handwritten dynamics and of visual dynamic textures.

翻译：本文研究由非线性动力学生成的轨迹数据分类问题，其中每个类别对应一个不同的动力系统。我们提出Dynafit——一种基于核函数的方法，用于学习训练轨迹与底层动力学之间的距离度量。根据习得的度量标准，新观测数据将被分配至具有最相似动力学的类别。该学习算法通过逼近Koopman算子，在核函数关联的（可能无限维）特征空间中实现动力学的全局线性化。距离度量在特征空间中计算，并借助机器学习中常用的核技巧使其维度无关性。我们还证明，当具备部分动力学先验知识时，核函数可进行针对性设计以融合这些信息。Dynafit适用于涉及非线性动力系统及传感器的多种分类任务。我们通过三个示例验证其有效性：基于逻辑映射的混沌检测、手写动力学识别以及视觉动态纹理识别。