Identification of the partial differential equations governing microstructure evolution in materials: Inference over incomplete, sparse and spatially non-overlapping data
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Pattern formation is a widely observed phenomenon in diverse fields including materials physics, developmental biology and ecology among many others. The physics underlying the patterns is specific to the mechanisms, and is encoded by partial differential equations (PDEs). With the aim of discovering hidden physics, we have previously presented a variational approach to identifying such systems of PDEs in the face of noisy data at varying fidelities (Computer Methods in Applied Mechanics and Engineering, 353:201-216, 2019). Here, we extend our methods to address the challenges presented by image data on microstructures in materials physics. PDEs are formally posed as initial and boundary value problems over combinations of time intervals and spatial domains whose evolution is either fixed or can be tracked. However, the vast majority of microscopy techniques for evolving microstructure in a given material system deliver micrographs of pattern evolution wherein the domain at one instant does not spatially overlap with that at another time. The temporal resolution can rarely capture the fastest time scales that dominate the early dynamics, and noise abounds. Finally data for evolution of the same phenomenon in a material system may well be obtained from different physical samples. Against this backdrop of spatially non-overlapping, sparse and multi-source data, we exploit the variational framework to make judicious choices of moments of fields and identify PDE operators from the dynamics. This step is preceded by an imposition of consistency to parsimoniously infer a minimal set of the spatial operators at steady state. The framework is demonstrated on synthetic data that reflects the characteristics of the experimental material microscopy images.
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