Learning Homogenization for Elliptic Operators
Margaret Trautner, California Institute of Technology
Homogenization theory aims to eliminate fast-scale dependence in partial differential equations (PDEs) to obtain homogenized PDEs which produce a good approximate solution of the problem with fast-scales while being more computationally tractable. In continuum mechanics, this methodology is of great practical importance as the constitutive laws derived from physical principles are governed by material behavior at small scales, but the quantities of interest are often relevant on larger scales. These homogenized constitutive laws often do not have a closed analytic form and may have new features not present in the microscale laws. We study the learnability of homogenized constitutive laws in the context of the workhorse of homogenization: the divergence form elliptic PDE. One significant challenge in applications of homogenization in material science arises from the presence of discontinuities and corner interfaces in the underlying material. This leads to roughness in the coefficients and solutions of the associated equations, a phenomenon extensively studied in numerical methods for PDEs. Addressing this challenge in the context of learning remains largely unexplored and is the focus of our work. We develop underlying theory and provide accompanying numerical studies to address learnability in this context.
Abstract Author(s): Margaret Trautner