  Estimating the likelihood, timing, and nature of events is a major goal of
modeling stochastic dynamical systems. When the event is rare in comparison
with the timescales of simulation and/or measurement needed to resolve the
elemental dynamics, accurate prediction from direct observations becomes
challenging. In such cases a more effective approach is to cast statistics of
interest as solutions to Feynman-Kac equations (partial differential
equations). Here, we develop an approach to solve Feynman-Kac equations by
training neural networks on short-trajectory data. Our approach is based on a
Markov approximation but otherwise avoids assumptions about the underlying
model and dynamics. This makes it applicable to treating complex computational
models and observational data. We illustrate the advantages of our method using
a low-dimensional model that facilitates visualization, and this analysis
motivates an adaptive sampling strategy that allows on-the-fly identification
of and addition of data to regions important for predicting the statistics of
interest. Finally, we demonstrate that we can compute accurate statistics for a
75-dimensional model of sudden stratospheric warming. This system provides a
stringent test bed for our method.
