A deep neural network method for solving partial differential equations with complex boundary in groundwater seepage

Abstract

High dimensional stochastic partial differential equations (SPDEs) attract a lot of attention because of their application in uncertainty quantification (UQ) of ore deposits, petroleum geology and other fields. The groundwater flow in heterogeneous media affects all geological processes, including diagenesis, mineralization and oil accumulation. A point of the deep neural network (DNN) method in machine learning is that it can effectively output the solution of the SPDEs with the complex boundaries such as Newman boundary in a direct way. We propose a novel methodology to construct approximate solutions for SPDEs in groundwater flow by combining two deep convolutional residual networks, which can complete the training without interference in the help of an adaptive functional factor. One of the two network models deals with the inner observed points and the other one is required to satisfy the boundary conditions. This model has great potential for solving more complex boundary problems. Another contribution of this work is that the loss function to train the proposed DNN is obtained by deducing the energy functional based on the variational principles, which integrates the physical meaning into the proposed DNN. In addition, we theoretically prove that the minimization loss function problem has a unique solution, and it always conforms to the weak solution form of the problem no matter how the uncertain parameters change. An unconstrained optimization problem, rather than a constrained problem, can be solved without adding any penalty terms. The practical application shows that our model has low computational cost and strong efficiency for solving high-dimensional SPDEs.