Errors-in-variables theory for nonlinear weak PDE moments

Develop an errors-in-variables treatment for weak-form estimation of nonlinear dynamics with overlapping space–time PDE moments, and determine whether calibrated coefficient intervals and consistency at fixed nonzero noise can be established for the constrained weak Allen–Cahn identification procedure.

Background

The identification method evaluates nonlinear constitutive features of noisy phase-field observations inside weak space–time moments. Although weak formulations transfer derivatives away from the measured field, nonlinear feature evaluation introduces bias, and overlapping moments create dependence among the resulting equations.

The paper notes that existing weak-identification analyses distinguish noise robustness from unconditional consistency, while an extension of errors-in-variables methods to the present nonlinear PDE-moment setting has not been developed. In particular, the numerical ensembles do not provide calibrated coefficient intervals or establish consistency when the observation noise remains nonzero.