[论文解读] Minimum distance classification for nonlinear dynamical systems

Dominique Martinez|arXiv (Cornell University)|Jan 7, 2026

Neural Networks and Reservoir Computing被引用 0

一句话总结

Dynafit 通过在核特征空间近似 Koopman 运算符，学习观测轨迹与潜在非线性动力学之间的距离，从而实现对由不同动力系统生成的数据的分类。

ABSTRACT

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.

研究动机与目标

• Motivate classification of trajectory data generated by nonlinear dynamics where each class corresponds to a distinct system.
• Propose a kernel-based method to learn a distance metric between observed trajectories and underlying dynamics.
• Leverage Koopman operator theory to obtain a global linear representation in a (potentially infinite) feature space.
• Enable computation of the distance metric in feature space via the kernel trick without explicit high-dimensional mappings.

提出的方法

• Represent nonlinear dynamics in a high-dimensional feature space via a nonlinear transformation to a (potentially infinite) feature space.
• Use the Koopman operator to obtain a global linear representation: phi(x_{k+1}) = A phi(x_k).
• Model the observed trajectory in feature space as phi(X) and relate it to O_b phi(x0) via a pseudo-inverse to obtain a distance d(X).
• Formulate training by minimizing the sum of feature-space distances across training trajectories to estimate Ob Ob^+.
• Apply the kernel trick to compute distances without explicit feature mappings and allow incorporating prior knowledge of dynamics when available.

Figure 1: Illustration of distance-based classification. A). In traditional classifiers, the distance is a measure of (dis)similarity between a test sample (here the image of Picasso’s signature) and a stored prototype (Pissarro’s signature). B). In Dynafit, the distance is between a trajectory samp

实验结果

研究问题

• RQ1 Can a kernel-based approach learn a distance metric that reflects similarity between observed trajectories and the underlying nonlinear dynamics that generated them?
• RQ2 Does approximating the Koopman operator in feature space enable effective classification of trajectories from different dynamical systems?
• RQ3 How can kernel methods leverage partial prior knowledge of dynamics to improve trajectory-based classification?
• RQ4 Is the proposed Dynafit framework capable of handling various nonlinear dynamical tasks such as chaos detection, handwriting dynamics, and visual dynamic textures?

主要发现

• Dynafit provides a way to compute a distance between observed trajectories and the underlying dynamics in feature space using the kernel trick.
• The method estimates the observability structure Ob Ob^+ and computes distances without requiring explicit (potentially infinite) feature mappings.
• The approach integrates Koopman operator theory with kernel methods to enable global linearization of nonlinear dynamics for classification.
• The framework is demonstrated on applications including chaos detection with the logistic map, handwritten dynamics recognition, and visual dynamic textures.