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

Z. Wang, Xun Huan+1 more|arXiv (Cornell University)|Jan 11, 2020

Model Reduction and Neural Networks48 references4 citations

TL;DR

This paper introduces a variational system identification (VSI) framework to discover partial differential equations (PDEs) governing microstructure evolution in materials from sparse, spatially unrelated, and noisy microscopy data. By leveraging a variational formulation with adaptive weighting functions and a consistency-based confirmation test, the method enables robust inference of minimal, parsimonious PDE operators even with limited dynamic data, achieving accurate identification of diffusion and reaction terms up to scaling factors 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.

Motivation & Objective

• Address the challenge of identifying governing PDEs for microstructure evolution when data are sparse, spatially disconnected, and temporally undersampled.
• Overcome limitations of traditional PDE system identification under incomplete and noisy image data from materials microscopy.
• Develop a variational framework that enables consistent inference of minimal sets of spatial differential and algebraic operators from dynamic and steady-state data.
• Introduce a confirmation test for operator suppression that provides a mathematically rigorous validation of inferred PDEs without requiring additional data or prior knowledge.
• Enable robust inference of kinetic coefficients (e.g., diffusivities) by combining dynamic data with high-resolution steady-state data.

Proposed method

• Formulate the PDE system identification problem as a variational optimization over a library of candidate spatial and temporal operators, using a weak form to handle noisy and sparse data.
• Employ adaptive weighting functions in the variational loss to emphasize regions of high data fidelity and reduce sensitivity to noise and spatial misalignment.
• Apply a two-stage variational system identification (VSI) process: first identify operators at steady state using consistency conditions, then refine dynamic operators using time-resolved snapshots.
• Implement a Confirmation Test of Consistency with Operator Suppression to validate inferred operators by testing their robustness under systematic suppression of candidate terms.
• Use a parsimony-promoting regularization strategy to favor minimal operator sets, reducing overfitting and improving generalizability.
• Integrate forward trajectories from inferred models into the loss function to improve stability and accuracy, particularly for extreme parameter regimes.

Experimental results

Research questions

• RQ1 Can a variational system identification framework reliably infer the underlying PDEs from sparse, spatially unrelated, and noisy microstructure image data?
• RQ2 How can consistency and robustness of identified PDE operators be verified without relying on additional data or prior knowledge?
• RQ3 To what extent can steady-state data be used to pre-identify the minimal set of governing operators before dynamic data refinement?
• RQ4 How does the proposed method handle the challenge of non-uniform temporal sampling and missing fast dynamics in microstructure evolution?
• RQ5 What role does operator suppression testing play in distinguishing true physical mechanisms from spurious or redundant terms in the identified PDEs?

Key findings

• The method successfully identifies all governing PDE operators—both differential and algebraic—up to a scaling factor using only steady-state data, demonstrating the feasibility of zeroth-order inference.
• The Confirmation Test of Consistency with Operator Suppression provides a mathematically rigorous, data-driven validation of inferred operators, reducing reliance on cross-validation or external benchmarks.
• Combining dynamic data with high-resolution steady-state data significantly improves the identification of small kinetic parameters, such as diffusivities, by providing a stable reference point.
• The two-stage VSI approach enables accurate inference of the full dynamic system even when temporal resolution is insufficient to capture early transient dynamics.
• The framework remains robust to noise and spatial misalignment, as demonstrated on synthetic data mimicking real microscopy conditions.
• The method can distinguish first-order parabolic PDEs from second-order hyperbolic systems through the structure of the loss and operator consistency checks, suggesting potential for broader applicability beyond parabolic systems.