Extensive theoretical studies have led to the discovery of various continuum models for pattern-forming systems and computational advances have greatly empowered the tool set of simulating increasingly complex systems. Yet as a complement to forward problems, the numerical robustness of the inverse problem as well as its uncertainty quantification is less well understood. Here we use PDE-constrained optimization, Bayesian statistics, and analytical methods to infer the governing dynamics and constitutive relations from images of pattern formation and understand their uncertainties in different systems, operating conditions, and imaging conditions. We study the datasets and physical constraints needed to increase the inference accuracy and well-posedness of the inverse problem. We demonstrate the procedure and uncertainty of inversion under limited spatiotemporal resolution, unknown boundary condition, blurry initial conditions, and other nonideal situations. Phase field, reaction-diffusion, and phase field crystal models are used as model systems. This approach can be applied in the inference of unknown dynamics and physical properties, experimental design, and engineering of complex pattern-forming systems.
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