  Starting with sets of disorganized observations of spatially varying and
temporally evolving systems, obtained at different (also disorganized) sets of
parameters, we demonstrate the data-driven derivation of parameter dependent,
evolutionary partial differential equation (PDE) models capable of generating
the data. This tensor type of data is reminiscent of shuffled
(multi-dimensional) puzzle tiles. The independent variables for the evolution
equations (their "space" and "time") as well as their effective parameters are
all "emergent", i.e., determined in a data-driven way from our disorganized
observations of behavior in them. We use a diffusion map based "questionnaire"
approach to build a parametrization of our emergent space/time/parameter space
for the data. This approach iteratively processes the data by successively
observing them on the "space", the "time", and the "parameter" axes of a
tensor. Once the data are organized, we use machine learning (here, neural
networks) to approximate the operators governing the evolution equations in
this emergent space. Our illustrative example is based on a previously
developed vertex-plus-signaling model of Drosophila embryonic development. This
allows us to discuss features of the process like symmetry breaking,
translational invariance, and autonomousness of the emergent PDE model, as well
as its interpretability.
