  We present a contribution to the field of system identification of partial
differential equations (PDEs), with emphasis on discerning between competing
mathematical models of pattern-forming physics. The motivation comes from
developmental biology, where pattern formation is central to the development of
any multicellular organism, and from materials physics, where phase transitions
similarly lead to microstructure. In both these fields there is a collection of
nonlinear, parabolic PDEs that, over suitable parameter intervals and regimes
of physics, can resolve the patterns or microstructures with comparable
fidelity. This observation frames the question of which PDE best describes the
data at hand. This question is particularly compelling because identification
of the closest representation to the true PDE, while constrained by the
functional spaces considered relative to the data at hand, immediately delivers
insights to the physics underlying the systems. While building on recent work
that uses stepwise regression, we present advances that leverage the
variational framework and statistical tests. We also address the influences of
variable fidelity and noise in the data.
