  The success of enhanced sampling molecular simulations that accelerate along
collective variables (CVs) is predicated on the availability of variables
coincident with the slow collective motions governing the long-time
conformational dynamics of a system. It is challenging to intuit these slow CVs
for all but the simplest molecular systems, and their data-driven discovery
directly from molecular simulation trajectories has been a central focus of the
molecular simulation community to both unveil the important physical mechanisms
and to drive enhanced sampling. In this work, we introduce state-free
reversible VAMPnets (SRV) as a deep learning architecture that learns nonlinear
CV approximants to the leading slow eigenfunctions of the spectral
decomposition of the transfer operator that evolves equilibrium-scaled
probability distributions through time. Orthogonality of the learned CVs is
naturally imposed within network training without added regularization. The CVs
are inherently explicit and differentiable functions of the input coordinates
making them well-suited to use in enhanced sampling calculations. We
demonstrate the utility of SRVs in capturing parsimonious nonlinear
representations of complex system dynamics in applications to 1D and 2D toy
systems where the true eigenfunctions are exactly calculable and to molecular
dynamics simulations of alanine dipeptide and the WW domain protein.
