Jonathan Weare (Courant Institute)

Estimating long-time rare-event statistics from short trajectories

Long-time rare-event statistics can be estimated from ensembles of trajectories that are much shorter than the event timescale. Committors, mean first passage times, and related rare event statistics all solve Bellman equations defined in terms of stopped trajectories. Markov state models are Galerkin approximations of these equations. On finite-state spaces we can establish relative error bounds for these quantities that are nearly independent of a condition number or related quantity (return-time, spectral gap, etc), partially explaining the empirical success of these methods on rare event problems (where condition numbers are often extremely large). I'll also describe recent approaches to non-linear parametrization of these equations as well as their failure, in the case of state-of-the-art AI weather emulators, when training data does not sufficiently cover dynamically important regions of state space and show that combining parametrized approximation of rare event statistics with rare-event simulation is one promising route to generating the necessary training data.