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. 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 to establish 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 for the solution operator
defined by the cell-problem arising in homogenization for elliptic PDEs.

多尺度偏微分方程中的组分理论及椭圆 PDE 的合成定律的数据驱动学习方法。