(103b) Supervised Learning and the Finite-Temperature String Method for Computing Committor Functions and Reaction Rates

The discovery of transition pathways governing kinetic processes at the microscopic level is of fundamental interest in chemistry. Common to these problems is the presence of high-energy barriers, which make the pathway discovery difficult due to the need to sample rare events. The committor function, the probability that a trajectory starting from some initial configuration enters the product state before the reactant state, encodes the complete mechanistic details of these pathways, including the reaction rates and transition-state ensembles. Computing the committor function by traditional means is prohibitively expensive, requiring hundreds of trajectories per configuration to yield sufficient estimates along with the need for rare configurations along the transition pathway. Recent approaches have utilized the framework of transition path theory, in which the committor is a solution of a high-dimensional partial differential equation, to create data-driven approaches to estimate the committor function. Common to these approaches is the use of importance sampling, in which high probability regions of the transition pathway are targeted, and machine learning, in which an artificial neural network approximates the committor function. However, these prior approaches require either knowledge of suitable collective variables to perform metadynamics [1], or adaptively training the neural network via umbrella sampling [2], which can be demonstrated to not homogeneously sample the entire transition pathway without extensive fine-tuning.

Our work [3] extends the previous approach of [2] by utilizing supervised learning, in which sample-mean estimates of the committor function obtained via short simulation trajectories are used to fit the neural network, and the finite-temperature string method, a path-finding algorithm that enables homogeneous sampling across the transition pathway. We demonstrate these modifications on a two-dimensional Muller-Brown system as well as a model dimerization problem from molecular simulation, showing that they yield accurate estimates of the committor function and reaction rates. Additionally, an error analysis for our algorithm is developed from which the reaction rates can be accurately estimated via a small number of samples.

[1] Q. Li, B. Lin, and W. Ren. “Computing committor functions for the study of rare events using deep learning.” The Journal of Chemical Physics 151, 054112 (2019).

[2] G. M. Rotskoff, A. R. Mitchell, and E. Vanden-Eijnden. “Active importance sampling for variational objectives dominated by rare events: Consequences for optimization and generalization.” arXiv preprint arXiv:2008.06334v2 (2021).

[3] M. R. Hasyim, C. H. Batton, and K. K. Mandadapu. “Supervised Learning and the Finite-Temperature String Method for Computing Committor Functions and Reaction Rates.” arXiv preprint arXiv:2107.13522 (2021).