Computing committors in collective variables via Mahalanobis diffusion maps

The study of rare events in molecular and atomic systems such as conformal changes and cluster rearrangements has been one of the most important research themes in chemical physics. Key challenges are associated with long waiting times rendering molecular simulations inefficient, high dimensionality impeding the use of PDE-based approaches, and the complexity or breadth of transition processes limiting the predictive power of asymptotic methods. Diffusion maps are promising algorithms to avoid or mitigate all these issues. We adapt the diffusion map with Mahalanobis kernel proposed by Singer and Coifman (2008) for the SDE describing molecular dynamics in collective variables in which the diffusion matrix is position-dependent and, unlike the case considered by Singer and Coifman, is not associated with a diffeomorphism. We offer an elementary proof showing that one can approximate the generator for this SDE discretized to a point cloud via the Mahalanobis diffusion map. We use it to calculate the committor functions in collective variables for two benchmark systems: alanine dipeptide, and Lennard-Jones-7 in 2D. For validating our committor results, we compare our committor functions to the finite-difference solution or by conducting a "committor analysis" as used by molecular dynamics practitioners. We contrast the outputs of the Mahalanobis diffusion map with those of the standard diffusion map with isotropic kernel and show that the former gives significantly more accurate estimates for the committors than the latter.
Comment: Restructured introduction, additional Theorem 3.1 and Appendix A, B