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.

翻译：多尺度偏微分方程在众多应用中出现，且已发展出多种方案以高效求解。均匀化理论是一种强大的方法，它消除了小尺度依赖性，从而得到计算上易于处理的简化方程。在连续介质力学领域，均匀化对于推导包含微观物理特性的本构律至关重要，以制定关键宏观量的平衡律。然而，获得均匀化的本构律通常具有挑战性，因为它们一般不具有解析形式，且可能表现出微观尺度上不存在的现象。为此，数据驱动的本构律学习已被提出适用于此任务。然而，该问题中数据驱动学习方法的一个主要挑战尚未被探索：基础材料中不连续性和角落界面的影响。这些系数的不连续性会影响基础方程解的光滑性。鉴于不连续材料在连续介质力学应用中的普遍性，应对这一场景下的学习挑战至关重要；特别是要发展基础理论以建立数据驱动方法在该科学领域的可靠性。本文通过研究存在此类复杂性时椭圆算子均匀化本构律的可学习性，解决了这一未探索的挑战。文中提出了逼近理论，并通过数值实验验证了该理论在椭圆偏微分方程均匀化中由单元问题定义的解算子上的有效性。