For many materials, macroscopic mechanical behavior is determined by an intricate microstructure. Understanding the relation between these two scales helps scientists and engineers design better materials. The relation which maps microstructure to bulk mechanical properties can be understood via the well-established theory of homogenization. However inverting the homogenization process, to recover microstructural information from measured macroscopic properties, is fraught with difficulties because of the averaging processes that underlie homogenization. Therefore, scientists and engineers usually need recourse to more invasive, often highly localized, investigations to learn about a microstructure. In this work, we develop a noninvasive methodology by which one can leverage large collections of measured bulk mechanical properties to learn information about the statistics of microstructure at a global level. We call this, distributional inverse homogenization. We study this problem in one and two dimensions, considering both periodic and stochastic homogenization. We demonstrate the methodology in the context of 2D Voronoi constructions and underpin the observed empirical success with theory in 1D. We also show how the natural spatial variability of microstructure can be exploited to gather data that enables distributional inversion. And we concurrently learn a surrogate model, approximating the homogenization map, that accelerates the resulting computations in this setting. The work formulates a new class of inverse problems, bridging ideas from probability and homogenization to facilitate the learning of microstructural material variability from macroscopic measurements.

翻译：对于许多材料而言，其宏观力学行为由复杂的微观结构决定。理解这两个尺度之间的关联有助于科学家和工程师设计更好的材料。将微观结构映射到宏观力学性能的关联可以通过成熟的均匀化理论加以理解。然而，由于均匀化过程中涉及平均化处理，从实测宏观属性反推微观结构信息的逆均匀化过程困难重重。因此，科学家和工程师通常需要借助更具侵入性、往往高度局域化的研究手段来了解微观结构。本文提出一种非侵入性方法，可利用大量实测宏观力学性能的集合，从全局层面学习微观结构的统计信息。我们将其称为"分布逆均匀化"。我们在周期均匀化和随机均匀化两种框架下，分别在一维和二维空间中研究该问题。通过二维Voronoi构型验证该方法的有效性，并在一维空间中用理论支撑所观察到的实证成功。我们还展示了如何利用微观结构固有的空间变异性来收集数据以实现分布反演，同时同步学习一个可加速该场景下相关计算的代理模型（近似均匀化映射）。本研究构建了一类新型逆问题，通过融合概率论与均匀化的思想，促进从宏观测量中学习微观材料变异性。