Data-Driven Committor Approximation for Anisotropic Diffusions in Collective Variables

Molecular simulation commonly deals with systems that reside in stable states over very large timescales and transition quickly between on extremely small scales. These transitions are crucial to molecular simulations but difficult to characterize due to the timescale gap. Transition Path Theory (TPT) is a powerful mathematical framework describing the transitions, but involves solving the committor equation, an elliptic boundary value problem on the high dimensional state space. To circumvent high dimensionality, practical use of TPT involves: (a) collective variables (CVs) and (b) assuming that the transition mechanism is restricted to thin channels: the "transition tube" assumption . A common issue with the use of CVs is anisotropy from local metric distortion. Hence, we propose a data-driven method to correct for anisotropy and solve the committor equation without the transition tube assumption in the space of CVs. We provide illustrative examples that our approach approximates the committor and more generally is invariant to the choice of CVs up to diffeomorphism.