  Rapid emergence of the multimodal imaging in scanning probe, electron, and
optical microscopies have brought forth the challenge of understanding the
information contained in these complex data sets, targeting both the intrinsic
correlations between different channels and further exploring the underpinning
causal physical mechanisms. Here, we develop such analysis framework for the
Piezoresponse Force Microscopy. We argue that under certain conditions, we can
bootstrap experimental observations with the prior knowledge of materials
structure to get information on certain non-observed properties, and
demonstrate linear causal analysis for PFM observables. We further demonstrate
that this approach can be extended to complex descriptors using the deep kernel
learning (DKL) model. In this DKL analysis, we use the prior information on
domain structure within the image to predict the physical properties. This
analysis demonstrates the correlative relationships between morphology,
piezoresponse, elastic property, etc. at nanoscale. The prediction of
morphology and other physical parameters illustrates a mutual interaction
between surface condition and physical properties in ferroelectric materials.
This analysis is universal and can be extended to explore the correlative
relationships of other multi-channel datasets.
