Neural network approach to time-dependent dividing surfaces in classical reaction dynamics

In a dynamical system, the transition between reactants and products is typically mediated by an energy barrier whose properties determine the corresponding pathways and rates. The latter is the flux through a dividing surface (DS) between the two corresponding regions, and it is exact only if it is free of recrossings. For time-independent barriers, the DS can be attached to the top of the corresponding saddle point of the potential energy surface, and in time-dependent systems, the DS is a moving object. The precise determination of these direct reaction rates, e.g., using transition state theory, requires the actual construction of a DS for a given saddle geometry, which is in general a demanding methodical and computational task, especially in high-dimensional systems. In this paper, we demonstrate how such time-dependent, global, and recrossing-free DSs can be constructed using neural networks. In our approach, the neural network uses the bath coordinates and time as input, and it is trained in a way that its output provides the position of the DS along the reaction coordinate. An advantage of this procedure is that, once the neural network is trained, the complete information about the dynamical phase space separation is stored in the network's parameters, and a precise distinction between reactants and products can be made for all possible system configurations, all times, and with little computational effort. We demonstrate this general method for two- and three-dimensional systems and explain its straightforward extension to even more degrees of freedom.

Figure 1

Plot of the LD (1) of the rank-1 saddle defined by the potential (9) for $y=z=v_y=v_z=t=0$. The axes approximate the reaction coordinate $x_{\mathrm{reac}}$ and its associated velocity. The color coding represents the value of the LDs. The stable and unstable manifolds ${\mathcal{W}}_{\text{s,u}}$ are visible as local minima of the LD. The stable manifold ${\mathcal{W}}_{\text{s}}$ is highlighted by the white dash-dotted curve and the unstable manifold ${\mathcal{W}}_{\text{u}}$ by the black dashed one. Their intersection is marked with a black cross ($\times$) and denotes one point of the NHIM of the corresponding saddle. The solid vertical black line is the DS, $\tau$, attached to this point of the NHIM. The choice of the DS is, in general, not unique. We chose the most simple case, viz., a vertical line independent of $v_x$, where we assume that the DS does not cross any of the manifolds in an additional point. Such an intersection exists for each configuration of bath coordinates and for each time. See text for further explanations.

Figure 2

(a) General structure of the neuronal network: time $t$, the bath coordinates ${\mathit{x}}_{\text{bath}}$, and their velocities ${\mathit{v}}_{\text{bath}}$ define the input layer (orange nodes), and the reaction coordinate $x_{\text{reac}}$ is the output layer (cyan node). The gray nodes denote the hidden layers of adjustable depth and number of neurons, and $a(·)$ is the respective activation function. (b) General scheme of the procedure: the time-dependent NHIM is defined by the intersection between the stable and unstable manifolds ${\mathcal{W}}_{\text{s,u}}$ (solid and dashed red lines) for the respective, fixed values of the bath coordinates and for given time $t$. The phase space coordinates of such an intersection point serve as the training set of the neural net.

Figure 3

Extension of the training set in two dimensions. The red points (dark in print) are calculated using the LD minimization procedure according to Eq. (3). The gray points are obtained by using the red points as initial values and propagating the respective particle according to the equations of motion. By this procedure the number of training points for the neural network is easily extended.

Figure 4

Time-dependent NHIM obtained from the NN in dependence of the bath coordinates. Note that $x_{\mathrm{reac}}$ refers to $x$ and $x_{\mathrm{bath}}$ and $v_{\mathrm{bath}}$ to $y$ and its velocity, respectively, in Eq. (9). The color (grayscale in print) of the surface indicates the difference between the NN's output and a spline interpolation of the data on which the NN is trained. Here the smoothing ability of the neural network becomes visible.

Figure 5

Cost functions of the data set for a DS of the system with three degrees of freedom. The cost function represents the difference between the output of the neural network and the actual points in the data set. The training set is a subset of the whole data set on which the neural network is trained, and the verification set is a disjunct second subset on which the neural network is not trained. The cost function of the verification set is a measure of how accurately the neural network interpolates. The training was stopped after ${10}^6$ epochs.

Figure 6

Counted crossings through the DS approximated by a neural network. The corresponding reaction curve can be seen in Fig. 7. The bars corresponding to two and more crossings are recrossings. Taking into account the small number of recrossings, the approximated DS is a recrossing-free DS with only tiny numerical errors.

Figure 7

(a) Behavior of the reactant population over time of a thermal ensemble at $k_{\mathrm{B}}T=4.75$ in units of $E_{\mathrm{b}}/2$. The population is normalized to the amount of particles propagated, which were $250000$ particles. There is also an exponential fit (dashed line) to this reaction curve as in Eq. (13), which yields a reaction rate of $\kappa =3.09232$. (b) Reaction rates determined with fits as in panel (a) for different values of $k_{\mathrm{B}}T$. The fit corresponds to the Arrhenius equation (14).