Neural numerical homogenization based on Deep Ritz corrections

TL;DR

This paper introduces a neural homogenization technique that combines the localized orthogonal decomposition method with Deep Ritz corrections to efficiently approximate solutions to elliptic PDEs with complex coefficients, especially under uncertainty.

Contribution

It proposes a novel neural-based correction method for the LOD approach, enabling better handling of temporal variations and uncertainties in coefficients.

Findings

Effective in approximating solutions to elliptic PDEs with oscillatory coefficients

Handles non-periodic and non-smooth coefficients well

Demonstrates promising results on parabolic model problems

Abstract

Numerical homogenization methods aim at providing appropriate coarse-scale approximations of solutions to (elliptic) partial differential equations that involve highly oscillatory coefficients. The localized orthogonal decomposition (LOD) method is an effective way of dealing with such coefficients, especially if they are non-periodic and non-smooth. It modifies classical finite element basis functions by suitable fine-scale corrections. In this paper, we make use of the structure of the LOD method, but we propose to calculate the corrections based on a Deep Ritz approach involving a parametrization of the coefficients to tackle temporal variations or uncertainties. Numerical examples for a parabolic model problem are presented to assess the performance of the approach.