A deep learning based reduced order modeling for stochastic underground flow problems

Abstract

In this paper, we propose a deep learning based reduced order modeling method for stochastic underground flow problems in highly heterogeneous media. We aim to utilize supervised learning to build a reduced surrogate mapping from the stochastic parameter space that characterizes the possible highly heterogeneous media to the solution space of a stochastic flow problem. Offline preparation includes collecting a set of full-order solutions to form a global snapshot space and extracting dominating POD modes using a particular well-designed spectral problem to represent full-order solutions. Due to the small dimension of reduced-order solutions, the complexity of neural network is significantly lightened, which effectively relieves the burden of training. We adopt the generalized multiscale finite element method (GMsFEM), where a set of local multiscale basis functions that can capture the heterogeneity of the media and source information are constructed to efficiently generate globally defined snapshot space. In an online stage, one can rapidly obtain the reduced-order pressure and then corresponding full-order solution is recovered using a linear transformation. Due to the decoupling of offline and online procedures, the proposed reduced order method serves as a non-intrusive tool that can achieve fast simulations independent of input parameters. Rigorous theoretical analyses are provided and extensive numerical experiments for linear and nonlinear stochastic flows are provided to verify the superior performance of the proposed method.