Bifurcation Preservation as a Physics Diagnostic for Neural Phase-Field Surrogates

A common approach for evaluating neural surrogates of phase-field equations is aggregate field error against a reference solver, a measure that can overlook bifurcations: abrupt shifts between qualitatively distinct outcomes, e.g., whether a phase-field droplet dissolves or persists. We propose evaluating neural phase-field surrogates in terms of their capacity for bifurcation preservation. We demonstrate the diagnostic on the Cahn-Hilliard (CH) critical droplet boundary using a droplet-aware Fourier Neural Operator, which reaches a moderate held-out rollout error, with relative $L^2 = 0.153$, however, it shifts the threshold-stable $m = 8$ dissolve/persist boundary by $16.2\%$. This persists for different random seeds, training budgets, exact zero-mode conservation, a U-Net baseline, and an entire 2D $(m, R_0)$ transition curve including out-of-training regimes. We then compared Allen-Cahn (AC), conservative AC, and CH models using the same epoch-checkpoint protocol. AC preserved the bifurcation once training had converged but fails when undertrained into the CH rollout-error band, and adding mass conservation to AC reduces preservation from from $13/24$ checkpoints to $4/24$. Conservation is implicated as a contributing CH-specific feature, with operator order, free-energy curvature, and branch stability as remaining candidates.