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

Abstract

Pattern formation is a widely observed phenomenon in diverse fields including materials physics, developmental biology, ecology and atmospheric weather, among many others. The physics underlying the patterns is specific to the mechanisms operating in each field, 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 heat treatments necessary to attain the kinetic rates and thermodynamic driving forces that result in significant evolution of microstructure in materials as well as the processing required for the microscopy that follows translate to notably different characteristics in the data so obtained. 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 bears no spatial correlation to that at another time. This extends to boundary data, when available. 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 uncorrelated, 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

System inference for the spatio-temporal evolution of infectious diseases: Michigan in the time of COVID-19

“…Our approach to inference combines system identification by stepwise regression [12,13] and ODE-constrained optimization using adjoints. We define a loss function that incorporates penalization on θ (leading to ridge regression below):…”

“…There are several possible criteria for eliminating basis terms. Here, we adopt a widely used statistical criterion called the F -test, also used by us previously [12,13]. The significance of the change between the model at iterations j and j − 1 is evaluated by: (20) where p j is the number of bases at iteration j and P = 16 is the total number of operator bases.…”

“…During the COVID-19 Pandemic, the widespread availability of data in the public domain [6][7][8][9][10][11] has served to attract methods of mathematics, computation and data science to analyzing this information, inferring the disease's dynamics and making projections. The present communication is in this spirit, and brings our recent work in large scale computations of partial differential equations (PDEs), system inference and machine learning to this problem [12][13][14][15][16][17].…”

“…To these tasks we have brought the abundance of high-quality, public domain, data on the evolution of the various compartment pertaining to the SIRD model in the US state of Michigan. The temporal resolution by days and spatial resolution by the 85 counties of Michigan has allowed us to apply our methods of Variational System Identification [12,13], PDE-constrained optimization and machine learning [14][15][16][17] to these data.…”

A Bi-fidelity Ensemble Kalman Method for PDE-Constrained Inverse Problems

“…To efficiently solve the variational optimization problem, local gradients of the cost function with respect to the parameters are needed and such derivative information is often obtained via adjoint method [1,2]. Although the adjoint-based variational method has been successfully applied for a variety of PDE-constrained inverse problems [3][4][5][6][7][8], it has two major limitations. First, the development of an adjoint solver is highly code-intrusive, which requires substantial effort to implement, especially for existing large-scale legacy codes of computational mechanics [9].…”