Inverse Modeling and System Inference from Data

In this chapter we turn to questions of inference, specifically, the identification of partial differential equations (PDEs), with an emphasis on discerning between competing mathematical models of pattern-forming physics. There are many reasons to consider such questions, which form a class of inverse modeling problems. Some of the motivation comes from developmental biology: pattern formation is central to the development of any multicellular organism, and from materials physics where phase transitions form microstructure. These phenomena are modeled by nonlinear, parabolic PDEs of different forms that, over certain parameter ranges, can resolve the patterns or microstructures with comparable fidelity. This naturally leads us to ask which PDE best describes the data at hand—particularly compelling question because identification of the best representation in PDE form, while considering suitable functional spaces for the data, provides insights to the physics underlying the systems.