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（PDE）广泛应用于各类工程与科学领域，目前已发展出多种高效求解方法。均匀化理论作为一种重要方法，通过消除小尺度依赖性，将复杂方程简化为计算可解的数学形式。在连续介质力学中，均匀化对于推导包含微观物理机制的本构关系至关重要，这些本构关系用于建立宏观感兴趣量的平衡方程。然而，获取均匀化后的本构定律通常面临挑战：它们往往缺乏解析表达式，且可能呈现微观尺度不存在的现象。为此，数据驱动的本构定律学习方法被提出作为应对方案。但该方法中存在一个尚未被探索的关键难题：基底材料中不连续性及角点界面的影响。系数中的这些不连续性会影响原始方程解的光滑性。鉴于连续介质力学应用中广泛存在非连续材料，解决此类场景下的学习挑战至关重要——尤其需要发展理论基础，以确立数据驱动方法在该科学领域的可靠性。本文通过研究椭圆算子在存在上述复杂性时均匀化本构定律的可学习性，首次系统应对了这一挑战。文中提出了近似理论，并通过数值实验验证了针对椭圆偏微分方程均匀化中细胞问题定义的解算子的理论有效性。