SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1844-1873, August 2024.
Abstract. 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.

中文翻译：

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学习椭圆算子的同质化


《SIAM 数值分析杂志》，第 62 卷，第 4 期，第 1844-1873 页，2024 年 8 月。

抽象的。多尺度偏微分方程（PDE）出现在各种应用中，并且已经开发了几种方案来有效地求解它们。均质化理论是一种强大的方法，可以消除小规模依赖性，从而产生易于计算处理的简化方程，同时准确预测宏观响应。在连续介质力学领域，均质化对于推导包含微观物理的本构定律至关重要，以便为感兴趣的宏观量制定平衡定律。然而，获得均一的本构定律通常具有挑战性，因为它们通常不具有解析形式并且可以表现出微观尺度上不存在的现象。作为回应，有人提出了适合这项任务的数据驱动的宪法学习。然而，针对该问题的数据驱动学习方法的一个主要挑战仍未得到探索：底层材料中不连续性和角界面的影响。系数中的这些不连续性会影响基础方程解的平滑度。鉴于不连续材料在连续介质力学应用中的普遍存在，解决这种背景下的学习挑战非常重要，特别是发展基础理论来建立该科学领域数据驱动方法的可靠性。本文通过研究存在这种复杂性的椭圆算子的均质本构定律的可学习性来解决这一尚未探索的挑战。 提出了近似理论，并进行了数值实验，在学习由椭圆偏微分方程均质化中出现的单元问题定义的解算子的背景下验证了该理论。