Image Inversion and Uncertainty Quantification for Constitutive Laws of Pattern Formation

TL;DR

This paper develops a PDE-constrained optimization and Bayesian inference framework to accurately infer and quantify uncertainties of constitutive laws in pattern formation systems from images, even under non-ideal conditions.

Contribution

It introduces a robust inversion algorithm that incorporates physical priors and handles limited data, advancing the understanding of inverse problems in pattern formation.

Findings

The method accurately infers constitutive relations under various imaging conditions.

Uncertainty quantification reveals key factors affecting inference accuracy.

Physical priors improve the robustness of the inverse solutions.

Abstract

The forward problems of pattern formation have been greatly empowered by extensive theoretical studies and simulations, however, the inverse problem is less well understood. It remains unclear how accurately one can use images of pattern formation to learn the functional forms of the nonlinear and nonlocal constitutive relations in the governing equation. We use PDE-constrained optimization to infer the governing dynamics and constitutive relations and use Bayesian inference and linearization to quantify their uncertainties in different systems, operating conditions, and imaging conditions. We discuss the conditions to reduce the uncertainty of the inferred functions and the correlation between them, such as state-dependent free energy and reaction kinetics (or diffusivity). We present the inversion algorithm and illustrate its robustness and uncertainties under limited spatiotemporal resolution, unknown boundary conditions, blurry initial conditions, and other non-ideal situations. Under certain situations, prior physical knowledge can be included to constrain the result. Phase-field, reaction-diffusion, and phase-field-crystal models are used as model systems. The approach developed here can find applications in inferring unknown physical properties of complex pattern-forming systems and in guiding their experimental design.