Neural network approach for the dynamics on the normally hyperbolic invariant manifold of periodically driven systems

Martin Tschöpe, Matthias Feldmaier, Jörg Main, and Rigoberto Hernandez

Phys. Rev. E 101, 022219 – Published 24 February 2020

Abstract

Chemical reactions in multidimensional systems are often described by a rank-1 saddle, whose stable and unstable manifolds intersect in the normally hyperbolic invariant manifold (NHIM). Trajectories started on the NHIM in principle never leave this manifold when propagated forward or backward in time. However, the numerical investigation of the dynamics on the NHIM is difficult because of the instability of the motion. We apply a neural network to describe time-dependent NHIMs and use this network to stabilize the motion on the NHIM for a periodically driven model system with two degrees of freedom. The method allows us to analyze the dynamics on the NHIM via Poincaré surfaces of section (PSOS) and to determine the transition-state (TS) trajectory as a periodic orbit with the same periodicity as the driving saddle, viz. a fixed point of the PSOS surrounded by near-integrable tori. Based on transition state theory and a Floquet analysis of a periodic TS trajectory we compute the rate constant of the reaction with significantly reduced numerical effort compared to the propagation of a large trajectory ensemble.

Received 16 March 2019
Accepted 29 January 2020

DOI:https://doi.org/10.1103/PhysRevE.101.022219

Physics Subject Headings (PhySH)

Atomic & molecular collisions Chemical reactions

Artificial neural networks Dynamical systems Stochastic dynamical systems

Phase space methods

Atomic, Molecular & Optical

Authors & Affiliations

Martin Tschöpe ¹, Matthias Feldmaier ¹, Jörg Main ¹, and Rigoberto Hernandez ^2,*

¹Institut für Theoretische Physik 1, Universität Stuttgart, 70550 Stuttgart, Germany
²Department of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218, USA

^*Corresponding author: r.hernandez@jhu.edu

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Vol. 101, Iss. 2 — February 2020

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Images

Figure 1
Basic construction of a feed-forward neural network (a) and an individual neuron (b). Here we use such networks for the multidimensional regression task of interpolating the $(2 d - 2)$ -dimensional NHIM of a $(2 d)$ -dimensional, time-dependent system. For any given time $t$ , the reaction coordinates $(x, v_{x})$ are continuously obtained using the network with the $(2 d - 2)$ phase-space coordinates $(y, v_{y})$ and the time $t$ as input. The weights $w$ and the biases $b$ are obtained using the Adam optimizer (see text).
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Figure 2
PSOS of the system (12) with amplitude $\hat{x} = 0.4$ using a stroboscopic map with barrier phase $t_{B} = 0$ . Trajectories with various initial conditions have been stabilized on the NHIM via a NN and propagated for 100 periods of the driving potential. The fixed point at the center of the regular toruslike structures marks the TS trajectory.
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Figure 3
(a) Centroid search for the TS trajectory of system (12) with $\hat{x} = 0.4$ . Stroboscopic view of trajectories for 10 periods determined using stabilization generated with a NN. Three selected initial points of trajectories are marked as 1, 2, and 3 and correspond to the stroboscoped points marked as blue red dots, green stars, and blue crosses, respectively. The centroid of the $n th$ trajectory corresponds to the initial point of the $(n + 1) st$ trajectory, e.g., point 2 is the centroid of the red points of trajectory 1. The algorithm converges to the TS trajectory, located, for the given barrier phase $t_{B} = 0$ , at $y \approx 0.00, v_{y} \approx - 0.72$ . (b) Friction search for the TS trajectory. The algorithm starts with large friction (red line) for the TS trajectory starting point. The trajectory forms a spiral, starting in the upper right corner and converging to a point that is closer to a periodic TS trajectory than the initial point. During the propagation of the particle the friction was reduced twice (green and blue line). The initial point of the green curve corresponds to the final point of the red curve. The orange dot marks the point to which the friction search finally converges by successively reducing the friction. Details of both algorithms are given in the text.
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Figure 4
(a) Projection of the TS trajectory, determined by the centroid search. Trajectories are plotted over 2.5 periods for both, the trajectory determination with and without the stabilization on the NHIM. (b) Distance $Δ x$ between the $x$ coordinates of the two trajectories calculated with and without the stabilization over two periods. Within one period the separation is two orders of magnitudes smaller than the actual motion in $x$ direction.
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