  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.
