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Model Selection of PDE for Pattern Forming Systems
Partial differential equations (PDE) have been widely used to reproduce
patterns in nature, and to give an insight on the mechanism underlying
pattern formation. Although enormous number of PDE models have been
proposed, they rely on pre-request knowledge of physical laws and
symmetries, and one has difficulties to develop a model to reproduce a
given desired pattern. Here we show the order parameters extracting
symmetries of a pattern together with Bayesian model selection
successfully estimate parameters in a model as well as the best model to
make the target pattern. We apply our method two-dimensional and
three-dimensional nontrivial patterns, namely quasi-crystal with
twelve-fold symmetry and double gyroid structure reproduced by using a
family of generalised Swift-Hohenberg equations. Our method not only
estimates the parameters to reproduce these patterns, but gives an
insight on the appropriate number of length scales hidden in the patterns.