Minimum distance classification for nonlinear dynamical systems

2026-01-07

arXiv.org

AI Summary

This work addresses the problem of classifying trajectories generated by distinct nonlinear dynamical systems, where each class corresponds to a unique system. The authors propose Dynafit, a novel method that, for the first time, integrates the Koopman operator framework with kernel methods to achieve global linearization of dynamics in a reproducing kernel Hilbert space. By leveraging the kernel trick, Dynafit efficiently computes dynamical distances between trajectories while allowing incorporation of prior knowledge. The approach demonstrates significant performance gains over baseline methods across three diverse tasks: detecting chaos in logistic maps, recognizing handwritten dynamics, and classifying visual dynamic textures. These results validate Dynafit’s effectiveness and generality in multi-class classification of nonlinear dynamical systems.

Technology Category

• Machine Learning: Kernel Methods
• Intelligent Robots: State Estimation
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• nonlinear dynamical systems
• trajectory classification
• dynamical system identification
• minimum distance classification
• chaos detection

Innovation

• Dynafit
• Koopman operator
• kernel method
• nonlinear dynamical systems
• trajectory classification