材料中微观结构演化控制偏微分方程的变分系统辨识：在稀疏且空间无关数据上的推断
Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data
AI总结：
本文扩展变分系统辨识方法，针对材料微观结构显微图像中空间无关、稀疏且多源的噪声数据，通过选择加权函数从动力学中辨识偏微分方程算子，并提出一致性条件与确认测试，在合成数据上验证了框架有效性。
摘要：
图样形成是在材料物理、发育生物学和生态学等众多领域中广泛观察到的现象。这些图样背后的物理机制因具体机制而异，并由偏微分方程（PDEs）编码。为了发现隐藏的物理规律，我们此前提出了一种变分方法，用于在具有不同保真度的噪声数据面前识别此类 PDEs 系统（Computer Methods in Applied Mechanics and Engineering, 353:201-216, 2019）。在此，我们将变分系统辨识方法进行扩展，以应对材料物理中微观结构图像数据所带来的挑战。PDEs 被正式表述为在时间区间和空间域组合上的初值和边值问题，其演化要么是固定的，要么是可以被追踪的。然而，绝大多数用于观测给定材料系统中演化微观结构的显微技术，所提供的图样演化显微图像在不同时刻的空间域之间毫无关联。时间分辨率极少能捕捉到主导早期动力学的最快时间尺度，且噪声无处不在。此外，同一材料系统中同一现象演化的数据，很可能来自不同的物理样本。面对这种空间无关、稀疏且多源数据的背景，我们利用变分框架，明智地选择加权函数，并从动力学中辨识 PDE 算子。由此产生的一致性条件，用于在稳态下对最小空间算子集合进行简约推断。作为补充，一个确认测试为接受所推断的算子提供了严格的条件。整个框架在反映实验材料显微图像特征的合成数据上得到了验证。
英文摘要：
Pattern formation is a widely observed phenomenon in diverse fields including materials physics, developmental biology and ecology, among many others. The physics underlying the patterns is specific to the mechanisms, and is encoded by partial differential equations (PDEs). With the aim of discovering hidden physics, we have previously presented a variational approach to identifying such systems of PDEs in the face of noisy data at varying fidelities (Computer Methods in Applied Mechanics and Engineering, 353:201-216, 2019). Here, we extend our variational system identification methods to address the challenges presented by image data on microstructures in materials physics. PDEs are formally posed as initial and boundary value problems over combinations of time intervals and spatial domains whose evolution is either fixed or can be tracked. However, the vast majority of microscopy techniques for evolving microstructure in a given material system deliver micrographs of pattern evolution over domains that bear no relation with each other at different time instants. The temporal resolution can rarely capture the fastest time scales that dominate the early dynamics, and noise abounds. Furthermore, data for evolution of the same phenomenon in a material system may well be obtained from different physical specimens. Against this backdrop of spatially unrelated, sparse and multi-source data, we exploit the variational framework to make judicious choices of weighting functions and identify PDE operators from the dynamics. A consistency condition arises for parsimonious inference of a minimal set of the spatial operators at steady state. It is complemented by a confirmation test that provides a sharp condition for acceptance of the inferred operators. The entire framework is demonstrated on synthetic data that reflect the characteristics of the experimental material microscopy images.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
