  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.
