  The predictability of a certain effect or phenomenon is often equated with
the knowledge of relevant physical laws, typically understood as a functional
or numerically derived relationship between the observations and known states
of the system. Correspondingly, observations inconsistent with prior knowledge
can be used to derive new knowledge on the nature of the system or indicate the
presence of yet unknown mechanisms. Here we explore the applicability of
Gaussian Processes (GP) to establish predictability and uncertainty of local
behaviors from multimodal observations, providing an alternative to this
classical paradigm. Using atomic-resolution Scanning Transmission Electron
Microscopy (STEM) of multiferroic Sm-doped BiFeO3 across a broad composition
range, we directly visualize the atomic structure and structural, physical, and
chemical order parameter fields for the material. GP regression is used to
establish the predictability of the local polarization field from different
groups of parameters, including the adjacent polarization values and several
combinations of physical and chemical descriptors, including lattice
parameters, column intensities, etc. We observe that certain elements of
microstructure including charged and uncharged domain walls and interfaces with
the substrate are best predicted with specific combinations of descriptors, and
this predictability and their associated uncertainties are consistent across
the composition series. The associated generative physical mechanisms are
discussed. We argue that predictability and uncertainty in observational data
offers a new pathway to probe the physics of condensed matter systems from
multimodal local observations.
