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.

翻译：多尺度偏微分方程（PDEs）在各类应用中广泛出现，已有多种高效求解方案被提出。均匀化理论是一种强有力的方法，它消除小尺度依赖性，得到简化方程，这些方程在计算上易于处理，同时能准确预测宏观响应。在连续介质力学领域，均匀化对于推导将微观物理纳入的本构律至关重要，从而为感兴趣的宏观量制定平衡律。然而，获取均匀化本构律通常具有挑战性，因为它们一般不具有解析形式，并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该任务。但该方法面临的一个主要挑战尚未被探索：底层材料中不连续性和角界面的影响。这些系数的间断性会影响底层方程解的平滑性。鉴于连续介质力学应用中不连续材料的普遍存在，解决该背景下的学习挑战具有重要意义；特别是要建立支撑理论，确保数据驱动方法在该科学领域的可靠性。本文通过探究椭圆算子在存在此类复杂性时均匀化本构律的可学习性，来解决这一未探索的挑战。文中提出了逼近理论，并通过数值实验，在椭圆偏微分方程均匀化中所定义的胞元问题求解算子学习的背景下验证了该理论。