Inferring constitutive forces from noisy Allen-Cahn movies with moment equations

Abstract

Inferring an Allen-Cahn constitutive force from images requires more than an accurate optimizer: nonlinear observation noise alters the mean of force features, shared noise couples instruments to the response, and a temporal balance borrowed from another solver can remain biased even on clean data. We develop observation-aware, noise-adjusted moment equations that address these effects before inversion. The surface-tension-calibrated constitutive transformer (SCCT) couples the resulting moments to a positive Bernstein inverse and can adapt its prior and precision from images and moments. A conditional stability bound separates moment residuals from prior error. Synthetic tests show that correcting the estimating equation matters more than increasing regularizer flexibility; the benefit of learned adaptation is smaller and depends on the training comparison. Forces inferred from planar movies predict unseen planar and three-dimensional evolutions. A ring-cage breakup shows why diffuse-interface accuracy distinguishes forces even when their topological predictions agree, while external surface tension sets the energy scale separately.