The mean exit time escaping basin of attraction in the presence of white
noise is of practical importance in various scientific fields. In this work, we
propose a strategy to control mean exit time of general stochastic dynamical
systems to achieve a desired value based on the quasipotential concept and
machine learning. Specifically, we develop a neural network architecture to
compute the global quasipotential function. Then we design a systematic
iterated numerical algorithm to calculate the controller for a given mean exit
time. Moreover, we identify the most probable path between metastable
attractors with help of the effective Hamilton-Jacobi scheme and the trained
neural network. Numerical experiments demonstrate that our control strategy is
effective and sufficiently accurate
