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, 356:44–74, 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.

模式形成是在包括材料物理学，发育生物学和生态学等许多领域的广泛观察到的现象。模式背后的物理学是机制特有的，由偏微分方程（PDE）编码。为了发现隐藏的物理学，我们先前提出了一种变通的方法，以面对变化保真度高的噪声数据来识别此类PDE系统（《应用力学与工程计算机方法》，第356页）：44-74，2019）。在这里，我们扩展了变分系统识别方法，以解决材料物理学中微结构图像数据所带来的挑战。在时间间隔和空间域的组合上，PDE正式地被表示为初始值和边值问题，其演化是固定的或可以被跟踪的。然而，用于在给定材料系统中发展微观结构的绝大多数显微镜技术在不同时刻在彼此之间没有关系的区域上提供了图案演变的显微照片。时间分辨率很少能捕捉到主导早期动态的最快时间尺度，并且噪声比比皆是。此外，很可能从不同的物理样本中获得用于在材料系统中发生相同现象的数据。在空间无关，稀疏和多源数据的背景下，我们利用变分框架来明智地选择加权函数，并从动力学中识别PDE运算符。对于稳态条件下空间运算符的最小集合的简约推断，出现了一个一致性条件。它辅以确认测试，为接受推断的操作员提供了清晰的条件。整个框架在合成数据上得到了证明，这些数据反映了实验材料显微镜图像的特征。它辅以确认测试，为接受推断的操作员提供了清晰的条件。整个框架在合成数据上得到了证明，这些数据反映了实验材料显微镜图像的特征。它辅以确认测试，为接受推断的操作员提供了清晰的条件。整个框架在合成数据上得到了证明，这些数据反映了实验材料显微镜图像的特征。