G. Simpson

D. Aristoff, M. Johnson, G. Simpson et al.

In the study of stochastic systems, the committor function describes the probability that a system starting from an initial configuration $x$ will reach a set $B$ before a set $A$. This paper introduces an efficient and interpretable algorithm for approximating the committor, called the "fast committor machine" (FCM). The FCM uses simulated trajectory data to build a kernel-based model of the committor. The kernel function is constructed to emphasize low-dimensional subspaces that optimally describe the $A$ to $B$ transitions. The coefficients in the kernel model are determined using randomized linear algebra, leading to a runtime that scales linearly in the number of data points. In numerical experiments involving a triple-well potential and alanine dipeptide, the FCM yields higher accuracy and trains more quickly than a neural network with the same number of parameters. The FCM is also more interpretable than the neural net.

D. Aristoff, S. Chill, G. Simpson

Adaptive Kinetic Monte Carlo combines the simplicity of Kinetic Monte Carlo (KMC) with a Molecular Dynamics (MD) based saddle point search algorithm in order to simulate metastable systems. Key to making Adaptive KMC effective is a stopping criterion for the saddle point search. In this work, we examine a recent criterion, due to S. Chill and G. Henkelman, that is based on the fraction of total reaction rate found instead of the fraction of observed saddles. The criterion uses the Eyring-Kramers law to estimate the reaction rate at the MD search temperature. We also consider a related criterion that remains valid when the Eyring-Kramers law is not. We examine the mathematical properties of both estimators and prove their mean square errors are well behaved, vanishing as the simulation continues to run.