Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data

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 discoveri...

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Bibliographic Details
Published in:Computer methods in applied mechanics and engineering Vol. 377; p. 113706
Main Authors: Wang, Z., Huan, X., Garikipati, K.
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 15.04.2021
Elsevier BV
Subjects:
Acceptance
Boundary value problems
Developmental biology
Domains
Evolution
Identification methods
Incomplete data
Inference
Inverse problems
Mathematical analysis
Microscopy
Microstructure
Operators (mathematics)
Partial differential equations
Pattern formation
Photomicrographs
Physics
System identification
Temporal resolution
Weighting functions
Pattern formation
Incomplete data
Inverse problems
System identification
ISSN:0045-7825, 1879-2138
Online Access:Get full text
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Summary: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, 356:44–74, 2019). Here, we extend our variational system identification 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 over domains that bear no relation with each other at different time instants. The temporal resolution can rarely capture the fastest time scales that dominate the early dynamics, and noise abounds. Furthermore, data for evolution of the same phenomenon in a material system may well be obtained from different physical specimens. Against this backdrop of spatially unrelated, sparse and multi-source data, we exploit the variational framework to make judicious choices of weighting functions and identify PDE operators from the dynamics. A consistency condition arises for parsimonious inference of a minimal set of the spatial operators at steady state. It is complemented by a confirmation test that provides a sharp condition for acceptance of the inferred operators. The entire framework is demonstrated on synthetic data that reflect the characteristics of the experimental material microscopy images. •Our study focuses on inferring the equations that govern pattern formation.•We use a variational approach to system identification.•Our approach works with spatially unrelated, sparse and multi-source data.•We introduce a Confirmation Test of Consistency with Operator Suppression.•Synthetic data are used reflecting the characteristics of experimental images.
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ISSN:0045-7825
1879-2138
DOI:10.1016/j.cma.2021.113706