Understanding the role of predictive time delay and biased propagator in RAVE

In this work, we revisit our recent iterative machine learning (ML)-molecular dynamics (MD) technique "Reweighted autoencoded variational Bayes for enhanced sampling" [J. M. L. Ribeiro et al., J. Chem. Phys. 149, 072301 (2018) and Y. Wang, J. M. L. Ribeiro, and P. Tiwary, Nat. Commun. 10, 3573 (2019)] and analyze and formalize some of its approximations. These include (a) the choice of a predictive time-delay, or how far into the future should the ML try to predict the state of a given system output from MD, and (b) that for short time-delays, how much of an error is made in approximating the biased propagator for the dynamics as the unbiased propagator. We demonstrate through a master equation framework as to why the exact choice of time-delay is irrelevant as long as a small non-zero value is adopted. We also derive a correction to reweight the biased propagator, and somewhat to our dissatisfaction but also to our reassurance, we find that it barely makes a difference to the intuitive picture we had previously derived and used.