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

TL;DR

This paper introduces a deep learning-based reduced order model for stochastic underground flow in heterogeneous media, enabling fast online simulations by combining POD modes, spectral analysis, and GMsFEM.

Contribution

It presents a novel integration of deep learning, POD, spectral methods, and GMsFEM for efficient stochastic flow modeling in complex media.

Findings

The method achieves significant reduction in neural network complexity.

Numerical experiments verify high accuracy for linear and nonlinear stochastic flows.

The approach outperforms traditional methods in computational efficiency.

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 model from the stochastic parameter space that characterizes the possible highly heterogeneous media to the solution space of a stochastic flow problem to have fast online simulations. Dominant POD modes obtained from a well-designed spectral problem in a global snapshot space are used to represent the solution of the flow problem. Due to the small dimension of the solution, the complexity of the neural network is significantly reduced. We adopt the generalized multiscale finite element method (GMsFEM), in which 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. 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.