2020/12/12 by Haoya Li, Yuehaw Khoo, Li, Haoya +5 · 1 citation
Decision Sciences · Mathematics · Physics and Astronomy · #FOS: Computer and information sciences #FOS: Mathematics #Fractional Differential Equations Solutions #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.2012.06727
openalex publication_date 2020/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.