  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 reaction rates, eg 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.
