Learning Homogenization for Elliptic Operators

TL;DR

This paper investigates the ability of data-driven methods to learn homogenized constitutive laws for elliptic PDEs with discontinuous coefficients, providing theoretical insights and validating them through numerical experiments.

Contribution

It develops approximation theory for learning homogenized laws in discontinuous media and validates the theory with numerical experiments on elliptic operators.

Findings

Theoretical bounds on learnability are established.

Numerical experiments confirm the theory's predictions.

Discontinuities significantly impact the smoothness and learnability of solutions.

Abstract

Multiscale partial differential equations (PDEs) arise in various applications, and several schemes have been developed to solve them efficiently. Homogenization theory is a powerful methodology that eliminates the small-scale dependence, resulting in simplified equations that are computationally tractable while accurately predicting the macroscopic response. In the field of continuum mechanics, homogenization is crucial for deriving constitutive laws that incorporate microscale physics in order to formulate balance laws for the macroscopic quantities of interest. However, obtaining homogenized constitutive laws is often challenging as they do not in general have an analytic form and can exhibit phenomena not present on the microscale. In response, data-driven learning of the constitutive law has been proposed as appropriate for this task. However, a major challenge in data-driven learning approaches for this problem has remained unexplored: the impact of discontinuities and corner interfaces in the underlying material. These discontinuities in the coefficients affect the smoothness of the solutions of the underlying equations. Given the prevalence of discontinuous materials in continuum mechanics applications, it is important to address the challenge of learning in this context; in particular, to develop underpinning theory that establishes the reliability of data-driven methods in this scientific domain. The paper addresses this unexplored challenge by investigating the learnability of homogenized constitutive laws for elliptic operators in the presence of such complexities. Approximation theory is presented, and numerical experiments are performed which validate the theory in the context of learning the solution operator defined by the cell problem arising in homogenization for elliptic PDEs.