A perspective on regression and Bayesian approaches for system identification of pattern formation dynamics

We present two approaches to system identification, i.e. the identification of partial differentialequations (PDEs) from measurement data. The first is a regression-based variational systemidentification procedure that is advantageous in not requiring repeated forward model solves andhas good scalability to large number of differential operators. However it has strict data typerequirements needing the ability to directly represent the operators through the available data.The second is a Bayesian inference framework highly valuable for providing uncertaintyquantification, and flexible for accommodating sparse and noisy data that may also be indirectquantities of interest. However, it also requires repeated forward solutions of the PDE modelswhich is expensive and hinders scalability. We provide illustrations of results on a model problemfor pattern formation dynamics, and discuss merits of the presented methods.

Mechanochemical spinodal decomposition: a phenomenological theory of phase transformations in multi-component, crystalline solids

We present a phenomenological treatment of diffusion-driven martensitic phase transformations in multi-component crystalline solids that arise from non-convex free energies in mechanical and chemical variables.The treatment describes diffusional phase transformations that are accompanied by symmetry-breaking structural changes of the crystal unit cell and reveals the importance of a mechanochemical spinodal,defined as the region in strain–composition space,where the free-energy density function is non-convex.The approach is relevant to phase transformations wherein the structural order parameters can be expressed as linear combinations of strains relative to a high-symmetry reference crystal.The governing equations describing mechanochemical spinodal decomposition are variationally derived from a free-energy density function that accounts for interfacial energy via gradients of the rapidly varying strain and composition fields.A robust computational framework for treating the coupled,higher-order diffusion and nonlinear strain gradient elasticity problems is presented.Because the local strains in an inhomogeneous,transforming microstructure can be finite,the elasticity problem must account for geometric nonlinearity.An evaluation of available experimental phase diagrams and first-principles free energies suggests that mechanochemical spinodal decomposition should occur in metal hydrides such as ZrH_(2−2c).The rich physics that ensues is explored in several numerical examples in two and three dimensions,and the relevance of the mechanism is discussed in the context of important electrode materials for Li-ion batteries and high-temperature ceramics.