  In this paper, we present large deviation theory that characterizes the
exponential estimate for rare events of stochastic dynamical systems in the
limit of weak noise. We aim to consider next-to-leading-order approximation for
more accurate calculation of mean exit time via computing large deviation
prefactors with the research efforts of machine learning. More specifically, we
design a neural network framework to compute quasipotential, most probable
paths and prefactors based on the orthogonal decomposition of vector field. We
corroborate the higher effectiveness and accuracy of our algorithm with a
practical example. Numerical experiments demonstrate its powerful function in
exploring internal mechanism of rare events triggered by weak random
fluctuations.
