Topology-aware neural operators for free boundary problems with certified uncertainty

Free boundary problems govern phenomena whose spatial domain is unknown in advance and evolve in time, from solidification to tumour growth. Neural operators have recently been extended to this setting by transporting every evolving domain, through a learned diffeomorphism, onto one fixed reference domain. A diffeomorphism cannot change topology, so the resulting model class excludes coalescence, splitting and hole closure, and the method returns an interface with no measure of confidence. We represent the geometry instead as a level set of a continuous phase field on a fixed background domain, so that topological transitions are ordinary events and the existing operator approximation theory applies once the diffuse problem is well posed. We prove a level-set stability estimate that converts a uniform field error below an explicit threshold into a Hausdorff error of the interface when the level set is regular, a measure bound on the reconstructed domain that holds through transitions, a posteriori residual bounds, a conditional consistency estimate with the sharp interface before the first singular time, and a finite-sample marginal coverage guarantee for a confidence tube around the discretized interface. A phase-field Stefan system is analysed formally; no operator is trained for it. The experiments concern the two-dimensional Allen-Cahn flow at one interface width and a short time horizon. Of 2200 instances used for evaluation, 46.1% change topology, mostly by the disappearance of small drops; coalescence occurs in 12.5% and is an effect of the finite interface width. An ensemble of ten Fourier neural operators has a mean relative \(L²\) error of 0.325% against the numerical reference and predicts the number of components correctly in 99.5% of the instances. A U-Net is equally accurate, physics-informed loss terms do not improve the accuracy, and the residual bounds are uninformative. The coverage guarantee is marginal and does not certify topology: on data generated after all settings were fixed, at the nominal level 0.90 and on average over calibration sets, the coverage is 0.892 overall and 0.805 for the topology-changing instances that retain an interface.