Support vector machines for learning reactive islands

TL;DR

The paper tackles identifying phase-space structures that govern transport in two-degree-of-freedom Hamiltonian systems by learning reactive islands directly from trajectory data. It introduces a trajectory-based SVM framework with three learning modes (fixed data, active learning, and trajectory-geometry features via the Lagrangian descriptor) to classify points on a 2D section as leading to imminent escape or not, without explicitly computing invariant manifolds. Key contributions include near-perfect boundary learning across energies on the Henon–Heiles system, robustness to parameter changes, and data-efficient strategies suitable for expensive trajectory integration. This approach offers a scalable, generalizable method to map phase-space bottlenecks and could extend to higher-dimensional reaction-dynamics models and system-bath settings.

Abstract

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 support vector machines (SVM) is 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.

Figures (9)

• Figure 1: Four sample trajectories initialized on the section $y = 0, p_y > 0$ with $p_x = 0.516,0.07,0.526,0.08$ in (a-d), respectively, projected on the configuration space. The total energy $E = 0.17$ is slightly above the energy of the index one saddles and escape times $T_E$ are shown on each plot.
• Figure 2: (a) Cylindrical (or tube) manifolds, stable in green and unstable in red, of the hyperbolic periodic orbits associated with the top index-one saddle in the Hénon-Heiles Hamiltonian. The energy of the hyperbolic periodic orbit and the invariant manifolds are at the total energy, $E = 0.17$ and mediate the trajectories that escape via top saddle as shown in Fig. \ref{['fig:pes_proj_HH']}(b,d). (b) Stable manifolds projected on the configuration space reveal the geometry of imminent escape from the potential well via the three bottlenecks. Only the segment of the stable manifolds from the hyperbolic periodic orbits to the intersection with the Poincaré section (shown as a black line) is shown for the energy, $E = 0.19$.
• Figure 3: An illustration of a decision boundary (black) between two classes of data (blue and orange) calculated by SVC with radial basis function kernel. The distance between the boundary and the closest points $P_i$ of every class, in this case the support vectors, is highlighted in green.
• Figure 4: Colormap showing accuracy for different combination of radial basis function parameters, $(C,\gamma)$. The accuracy is obtained using a 5-fold cross validation over the grid of values for $C$ and $\gamma$. The pair of value which gives maximum accuracy is chosen for training the support vector classifier.
• Figure 5: Fixed training data set. Reactive islands identified by the support vector classifier trained using fixed size data set shown as dashed curves. The overlayed continuous curve is obtained using direct computation of tube manifolds at energy $E = 0.17, 0.18, 0.19, 0.20$ in first, second, third, fourth column, respectively. The magenta curve denotes the intersection of the energy surface with the two dimensional section. The cyan dots denote the support vectors used by the classifier in learning the reactive islands as decision surfaces. Two sections with $(x,p_x)$ coordinates are shown in top and bottom rows: (a-d) $y_c = 0$ (e-h) $y_c = -0.25$ with $p_y > 0$.