  We show that unsupervised machine learning can be used to learn physical and
chemical transformation pathways from the observational microscopic data, as
demonstrated for atomically resolved images in Scanning Transmission Electron
Microscopy (STEM) and ferroelectric domain structures in Piezoresponse Force
Microscopy (PFM). To enable this analysis in STEM, we assumed the existence of
atoms, a discreteness of atomic classes, and the presence of an explicit
relationship between the observed STEM contrast and the presence of atomic
units. In PFM, we assumed the uniquely-defined relationship between the
measured signal and polarization distribution. With only these postulates, we
developed a machine learning method leveraging a rotationally-invariant
variational autoencoder (rVAE) that can identify the existing structural units
observed within a material. The approach encodes the information contained in
image sequences using a small number of latent variables, allowing the
exploration of chemical and physical transformation pathways via the latent
space of the system. The results suggest that the high-veracity imaging data
can be used to derive fundamental physical and chemical mechanisms involved, by
providing encodings of the observed structures that act as bottom-up
equivalents of structural order parameters. The approach also demonstrates the
potential of variational (i.e., Bayesian) methods for physical sciences and
will stimulate the development of new ways to encode physical constraints in
the encoder-decoder architectures, and generative physical laws, topological
invariances, and causal relationships in the latent space of VAEs.
