Naik Shibabrat, Krajňák Vladimír, Wiggins Stephen
School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, United Kingdom.
Chaos. 2021 Oct;31(10):103101. doi: 10.1063/5.0062437.
We develop a machine learning framework that can be applied to data sets derived from the trajectories of Hamilton's equations. The goal is to learn the phase space structures that play the governing role for phase space transport relevant to particular applications. Our focus is on learning reactive islands in two degrees-of-freedom Hamiltonian systems. Reactive islands are constructed from the stable and unstable manifolds of unstable periodic orbits and play the role of quantifying transition dynamics. We show that the support vector machines are an appropriate machine learning framework for this purpose as it provides an approach for finding the boundaries between qualitatively distinct dynamical behaviors, which is in the spirit of the phase space transport framework. We show how our method allows us to find reactive islands directly in the sense that we do not have to first compute unstable periodic orbits and their stable and unstable manifolds. We apply our approach to the Hénon-Heiles Hamiltonian system, which is a benchmark system in the dynamical systems community. We discuss different sampling and learning approaches and their advantages and disadvantages.
我们开发了一个机器学习框架，该框架可应用于从哈密顿方程轨迹导出的数据集。目标是学习对与特定应用相关的相空间传输起主导作用的相空间结构。我们的重点是学习二维自由度哈密顿系统中的反应岛。反应岛由不稳定周期轨道的稳定和不稳定流形构成，并起到量化跃迁动力学的作用。我们表明，支持向量机是适用于此目的的机器学习框架，因为它提供了一种找到定性不同的动力学行为之间边界的方法，这符合相空间传输框架的精神。我们展示了我们的方法如何使我们能够直接找到反应岛，即我们不必首先计算不稳定周期轨道及其稳定和不稳定流形。我们将我们的方法应用于亨农 - 海尔斯哈密顿系统，该系统是动力系统领域的一个基准系统。我们讨论了不同的采样和学习方法及其优缺点。
