  This paper proposes a new method based on neural networks for computing the
high-dimensional committor functions that satisfy Fokker-Planck equations.
Instead of working with partial differential equations, the new method works
with an integral formulation based on the semigroup of the differential
operator. The variational form of the new formulation is then solved by
parameterizing the committor function as a neural network. There are two major
benefits of this new approach. First, stochastic gradient descent type
algorithms can be applied in the training of the committor function without the
need of computing any mixed second-order derivatives. Moreover, unlike the
previous methods that enforce the boundary conditions through penalty terms,
the new method takes into account the boundary conditions automatically.
Numerical results are provided to demonstrate the performance of the proposed
method.
