  The development of enhanced sampling methods has greatly extended the scope
of atomistic simulations, allowing long-time phenomena to be studied with
accessible computational resources. Many such methods rely on the
identification of an appropriate set of collective variables. These are meant
to describe the system's modes that most slowly approach equilibrium. Once
identified, the equilibration of these modes is accelerated by the enhanced
sampling method of choice. An attractive way of determining the collective
variables is to relate them to the eigenfunctions and eigenvalues of the
transfer operator. Unfortunately, this requires knowing the long-term dynamics
of the system beforehand, which is generally not available. However, we have
recently shown that it is indeed possible to determine efficient collective
variables starting from biased simulations. In this paper, we bring the power
of machine learning and the efficiency of the recently developed on-the-fly
probability enhanced sampling method to bear on this approach. The result is a
powerful and robust algorithm that, given an initial enhanced sampling
simulation performed with trial collective variables or generalized ensembles,
extracts transfer operator eigenfunctions using a neural network ansatz and
then accelerates them to promote sampling of rare events. To illustrate the
generality of this approach we apply it to several systems, ranging from the
conformational transition of a small molecule to the folding of a mini-protein
and the study of materials crystallization.
