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This paper proposes a new method based on neural networks for computing thehigh-dimensional committor functions that satisfy Fokker-Planck equations.Instead of working with partial differential equations, the new method workswith an integral formulation based on the semigroup of the differentialoperator. The variational form of the new formulation is then solved byparameterizing the committor function as a neural network. There are two majorbenefits of this new approach. First, stochastic gradient descent typealgorithms can be applied in the training of the committor function without theneed of computing any mixed second-order derivatives. Moreover, unlike theprevious 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 proposedmethod.

A semigroup method for high dimensional committor functions based on neural network