  Partial differential equations (PDE) have been widely used to reproduce
patterns in nature and to give insight into the mechanism underlying pattern
formation. Although many PDE models have been proposed, they rely on the
pre-request knowledge of physical laws and symmetries, and developing a model
to reproduce a given desired pattern remains difficult. We propose a novel
method, referred to as Bayesian modelling of PDE (BM-PDE), to estimate the best
dynamical PDE for one snapshot of a target pattern under the stationary state
without ground truth. We apply BM-PDE to nontrivial patterns,such as
quasi-crystals (QCs), a double gyroid and Frank Kasper structures. By using the
estimated parameters for the approximant of QCs, we successfully generate, for
the first time,three-dimensional dodecagonal QCs from a PDE model. Our method
works for noisy patterns and the pattern synthesised without the ground truth
parameters, which are required for the application toward experimental data.
