Construction of elliptic stochastic partial differential equations solver in groundwater flow with convolutional neural networks

Abstract Elliptic stochastic partial differential equations (SPDEs) play an indispensable role in mathematics, engineering and other fields, and its solution methods emerge in endlessly with the progress of science and technology. In this paper, we make use of the convolutional neural networks (CNNs), which are widely used in machine learning, to construct a solver for SPDEs. The SPDEs with Neumann boundary conditions are considered, and two CNNs are employed. One is used to deal with the essential equation, and the other satisfies the boundary conditions. With the help of the length factor, the integrated neural network model can predict the solution of the equations accurately. We show an example of groundwater flow to evaluate the model proposed with Gaussian random field (GRF). The experimental results show that the proposed neural network solver can approximate the traditional numerical algorithm, and has high computational efficiency.