A semigroup method for high dimensional committor functions based on neural network
This method offers a more efficient and robust way to compute committor functions for researchers and practitioners working with high-dimensional stochastic systems, particularly in fields like chemistry and physics.
This paper introduces a neural network-based method for calculating high-dimensional committor functions that adhere to Fokker-Planck equations. The method leverages an integral formulation derived from the semigroup of the differential operator, rather than directly solving partial differential equations. This approach allows for the use of stochastic gradient descent without computing mixed second-order derivatives and automatically incorporates boundary conditions.
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.