Operators for interfacial problems are trained on reference solutions produced by the solver they are intended to replace.
This work develops a data-free physics-informed neural operator for level-set interface advection, in which the interface is the equation's unknown and the operator maps an initial interface to the full spatiotemporal trajectory under a prescribed flow.
Training uses only the transport residual and a geometric constraint; no reference solution enters the objective at any point.
A spacetime Fourier backbone emits the entire trajectory in one pass, and the initial condition is imposed by construction rather than by penalty, which removes the competition between the anchoring term and the residual that otherwise arises when no solution data are available.
Supervised and hybrid operators are trained under an identical architecture, family, budget and test set, and are reported throughout as baselines that quantify what refusing labels costs.
On a reversed single vortex the data-free operator reaches 1.614 +/- 0.067% relative L2 error on 100 held-out initial interfaces against 0.369 +/- 0.035% for the supervised baseline, a factor of 4.4;
on solid-body rotation the corresponding figures are 3.804 +/- 1.075% and 2.576 +/- 0.159%, a factor of 1.5.
The two benchmarks rank the arms differently, and the difference is attributable to the eikonal constraint: the exact solution violates |grad phi| = 1 over 0.3% of the domain under rotation and 86.9% under the vortex.
Where the constraint is valid the physics-trained operator conserves enclosed area 2.7 times better than the supervised baseline despite a larger field error, and a hybrid arm using eight reference solutions outperforms a supervised arm using sixteen.