  In this note we propose a method based on artificial neural network to study
the transition between states governed by stochastic processes. In particular,
we aim for numerical schemes for the committor function, the central object of
transition path theory, which satisfies a high-dimensional Fokker-Planck
equation. By working with the variational formulation of such partial
differential equation and parameterizing the committor function in terms of a
neural network, approximations can be obtained via optimizing the neural
network weights using stochastic algorithms. The numerical examples show that
moderate accuracy can be achieved for high-dimensional problems.
