  Microstructures are critical to the physical properties of materials.
Stochastic microstructures are commonly observed in many kinds of materials and
traditional descriptor-based image analysis of them can be challenging. In this
paper, we introduce a powerful and versatile score-based framework for
analyzing nonstationarity in stochastic materials microstructures. The
framework involves training a parametric supervised learning model to predict a
pixel value using neighboring pixels in images of microstructures~(as known as
micrographs), and this predictive model provides an implicit characterization
of the stochastic nature of the microstructure. The basis for our approach is
the Fisher score vector, defined as the gradient of the log-likelihood with
respect to the parameters of the predictive model, at each micrograph pixel. A
fundamental property of the score vector is that it is zero-mean if the
predictive relationship in the vicinity of that pixel remains unchanged, which
we equate with the local stochastic nature of the microstructure remaining
unchanged. Conversely, if the local stochastic nature changes, then the mean of
the score vector generally differs from zero. Our framework analyzes how the
local mean of the score vector varies across one or more image samples to: (1)
monitor for nonstationarity by indicating whether new samples are statistically
different than reference samples and where they may differ and (2) diagnose
nonstationarity by identifying the distinct types of stochastic microstructures
and labeling accordingly the corresponding regions of the samples. Unlike
feature-based methods, our approach is almost completely general and requires
no prior knowledge of the nature of the nonstationarities. Using a number of
real and simulated micrographs, including polymer composites and multiphase
alloys, we demonstrate the power and versatility of the approach.
