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

Z. Wang, X. Huan, K. Garikipati

TL;DR

The variational framework is exploited to make judicious choices of weighting functions and identify PDE operators from the dynamics and a consistency condition arises for parsimonious inference of a minimal set of the spatial operators at steady state.

Abstract

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 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.

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

TL;DR

The variational framework is exploited to make judicious choices of weighting functions and identify PDE operators from the dynamics and a consistency condition arises for parsimonious inference of a minimal set of the spatial operators at steady state.

Abstract

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 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.

Paper Structure

This paper contains 21 sections, 61 equations, 20 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Variational identification of PDE systems from sparse and spatially unrelated data
  3. The Galerkin weak form for system identification
  4. Similarity of snapshots
  5. Two stage variational system identification for dynamics
  6. Candidate basis operators
  7. Weak forms of polynomial approximations of algebraic operators at time $t_n$ with $w^h=1$
  8. Weak form of the Laplacian operator at time $t_n$ with $w^h=C^h$
  9. Weak form of the biharmonic operator $\nabla^4 C_n$ at time $t_n$ with $w^h=C^h$
  10. Identification of basis operators via two stage stepwise regression
  11. Stepwise regression
  12. Variational System Identification with dynamic data
  13. Data preparation
  14. Similarity between snapshots of data in pattern forming physics
  15. Numerical examples
  16. ...and 6 more sections

Figures (20)

  • Figure 1: The sample data shown within the snapshots (yellow square) corresponding to each time instant are only available over subdomains of the full field. They are also spatially unrelated over time.
  • Figure 2: Schematic of the algorithms for stepwise regression and the $F$-test.
  • Figure 3: The three-well non-convex tissue energy density function.
  • Figure 4: Upper plots: Concentration at $t=0$ of two cases representing distinct experimental specimens. The perturbation is re-generated for every simulation. Lower plots: Concentration at $t=20$. The Turing patterns generated from different initial conditions appear statistically similar.
  • Figure 5: The total flux over snapshots of different sizes marked by different colors. The total flux converges to zero with increasing snapshot size. The embedded subplot shows the decreasing standard deviation.
  • ...and 15 more figures