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We present a contribution to the field of system identification of partialdifferential equations (PDEs), with emphasis on discerning between competingmathematical models of pattern-forming physics. The motivation comes fromdevelopmental biology, where pattern formation is central to the development ofany multicellular organism, and from materials physics, where phase transitionssimilarly lead to microstructure. In both these fields there is a collection ofnonlinear, parabolic PDEs that, over suitable parameter intervals and regimesof physics, can resolve the patterns or microstructures with comparablefidelity. This observation frames the question of which PDE best describes thedata at hand. This question is particularly compelling because identificationof the closest representation to the true PDE, while constrained by thefunctional spaces considered relative to the data at hand, immediately deliversinsights to the physics underlying the systems. While building on recent workthat uses stepwise regression, we present advances that leverage thevariational framework and statistical tests. We also address the influences ofvariable fidelity and noise in the data.

Variational system identification of the partial differential equations governing pattern-forming physics: Inference under varying fidelity and noise