Solving for high dimensional committor functions using neural network with online approximation to derivatives

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 that involves 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. As the main benefit of this new approach, stochastic gradient descent type algorithms can be applied in the training of the committor function without the need of computing any second-order derivatives. Numerical results are provided to demonstrate the performance of the proposed method.