Multiscale reweighted stochastic embedding (MRSE): Deep learning of collective variables for enhanced sampling

Machine learning methods provide a general framework for automatically finding and representing the essential characteristics of simulation data. This task is particularly crucial in enhanced sampling simulations, where we seek a few generalized degrees of freedom, referred to as collective variables (CVs), to represent and drive the sampling of the free energy landscape. These CVs should separate the different metastable states and correspond to the slow degrees of freedom. For this task, we propose a new method that we call multiscale reweighted stochastic embedding (MRSE). The technique automatically finds CVs by learning a low-dimensional embedding of the high-dimensional feature space to the latent space via a deep neural network. Our work builds upon the popular $t$-distributed stochastic neighbor embedding approach. We introduce several new aspects to stochastic neighbor embedding algorithms that makes MRSE especially suitable for enhanced sampling simulations: (1) a well-tempered selection scheme for the landmark features that gives close to equilibrium representation of the training data; (2) a multiscale representation via Gaussian mixture to model the probabilities of being neighbors in the high-dimensional feature space; and (3) a reweighting procedure to account for the training data being drawn from a biased probability distribution. To test the performance of MRSE, we use it to obtain low-dimensional CVs for three model systems, the Muller-Brown potential, alanine dipeptide, and alanine tetrapeptide, and provide a thorough analysis of the results.