Constrained weak identification of Allen--Cahn free energies with surface-tension calibration

Abstract: Phase-field trajectories governed by the Allen--Cahn equation determine the effective force $G=F&#39;/\eps<sup>2$, but not the free-energy potential $F$ and interfacial scale $\eps$ separately. We introduce surface-tension-calibrated Allen--Cahn identification (STAC), a two-step method that resolves this ambiguity using an independently supplied surface tension. First, space--time weak moments produce a linear inverse problem without differentiating the measured field. A Bernstein representation with positive coefficients enforces two stable pure phases and a single central energy maximum. Second, the surface tension fixes the remaining scale algebraically. We prove scale equivalence of the forward model, uniqueness of the calibrated pair within this class, and finite perturbation bounds that remain valid when the potential vanishes at the pure phases. Experiments with resolved synthetic phase fields separate the effects of the structural constraint, ridge regularization, and constitutive approximation error. Across 80 shared noisy records, unconstrained degree-five regression produces six potentials with an incorrect well structure. Positivity combined with ridge regularization eliminates these violations and reduces the mean force error from 8.70\% to 5.66\%. Lower-degree models are nevertheless more accurate in some tests, and an admissible law outside the positive Bernstein family shows how the constraint can introduce approximation bias. STAC thus recovers a structurally admissible, physically scaled potential while making explicit which information comes from the observed dynamics and which comes from the independent surface-tension datum.