Learning a generalized multiscale prolongation operator

TL;DR

This paper introduces a deep learning-based approach to efficiently construct multiscale prolongation operators for Darcy flow problems, significantly reducing computation time while maintaining preconditioner effectiveness and demonstrating strong generalization.

Contribution

The study proposes a neural network method to rapidly generate stable multiscale prolongation operators, avoiding repeated spectral problem solutions and enhancing efficiency in high-contrast Darcy flow simulations.

Findings

Deep learning accelerates prolongation operator construction.

The method maintains preconditioner efficiency with reduced computation time.

Neural network generalizes well to unseen permeability fields.

Abstract

In this research, we address Darcy flow problems with random permeability using iterative solvers, enhanced by a two-grid preconditioner based on a generalized multiscale prolongation operator, which has been demonstrated to be stable for high contrast profiles. To circumvent the need for repeatedly solving spectral problems with varying coefficients, we harness deep learning techniques to expedite the construction of the generalized multiscale prolongation operator. Considering linear transformations on multiscale basis have no impact on the performance of the preconditioner, we devise a loss function by the coefficient-based distance between subspaces instead of the plain $l^2$-norm of the difference of the corresponding multiscale bases. We discover that leveraging the inherent symmetry in the local spectral problem can effectively accelerate the neural network training process. In scenarios where training data are limited, we utilize the Karhunen-Lo\`eve expansion to augment the dataset. Extensive numerical experiments with various types of random coefficient models are exhibited, showing that the proposed method can significantly reduce the time required to generate the prolongation operator while maintaining the original efficiency of the two-grid preconditioner. Notably, the neural network demonstrates strong generalization capabilities, as evidenced by its satisfactory performance on unseen random permeability fields.