Using Nearest-Neighbor Distributions to Quantify Machine Learning of Materials’ Microstructures

Machine learning strategies for the semantic segmentation of materials' micrographs, such as U-Net, have been employed in recent years to enable the automated identification of grain-boundary networks in polycrystals. For example, most recently, this architecture has allowed researchers to address the long-standing problem of automated image segmentation of thin-film microstructures in bright-field TEM micrographs. Such approaches are typically based on the minimization of a binary cross-entropy loss function that compares constructed images to a ground truth at the pixel level over many epochs. In this work, we quantify the rate at which the underlying microstructural features embodied in the grain-boundary network, as described stereologically, are also learned in this process. In particular, we assess the rate of microstructural learning in terms of the moments of the <i>k</i>-th nearest-neighbor pixel distributions and associated metrics, including a microstructural cross-entropy, that embody the spatial correlations among the pixels through a hierarchy of <i>n</i>-point correlation functions. From the moments of these distributions, we obtain so-called learning functions that highlight the rate at which the important topological features of a grain-boundary network appear. It is found that the salient features of network structure emerge after relatively few epochs, suggesting that grain size, network topology, etc., are learned early (as measured in epochs) during the segmentation process.