Eikonal Regularisation in Physics-Informed Neural Networks for Three-Dimensional Level-Set Advection: Transferability of Two-Dimensional Design Principles

Abstract

Physics-informed neural networks applied to the level-set formulation of interface advection commonly augment the residual and initial-condition losses with an eikonal regulariser, penalising the deviation of ‖∇φ‖ from unity. A previous two-dimensional study identified this weight as the single most influential hyperparameter and observed that its optimum shifts by four orders of magnitude between rigid-body and strongly deforming flows, but left two questions open: whether the design principles established there transfer to three dimensions, where collocation requirements grow substantially, and whether the reported differences survive the run-to-run variability that single-seed experiments cannot assess. This paper answers both. We repeat the weight selection in three dimensions across four benchmarks (a translating sphere, a rotating sphere, a rotating slotted sphere, and a sphere deformed and restored by a reversed vortex), sweeping six weights with three random seeds at the full training budget under a selection rule fixed in advance. The two-dimensional ordering is confirmed: the selected weight tracks how far the exact solution departs from a signed distance function, spanning four decades from 10⁻¹ where the property holds exactly to 10⁻⁵ where the interface is stretched. The specific values, however, transfer only benchmark by benchmark; two of the four carry over from two dimensions unchanged and two do not, so inheritance must be verified rather than assumed. The multi-seed protocol reveals a further effect invisible to single-seed experiments: at small weights the seed-to-seed standard deviation of the error is of the same order as the error itself, and the regulariser reduces it by more than an order of magnitude, so it buys reproducibility as well as accuracy. We benchmark against a fifth-order weighted essentially non-oscillatory (WENO) solver on identical grids and error measures; the classical scheme is the more accurate on all four problems, by two orders of magnitude on smooth rigid advection, with a margin that narrows monotonically with geometric difficulty and is considerably smaller in volume conservation than in the field norm. Finally, we show that the relative L₂ error cannot certify the preservation of thin features, and report a feature-restricted measure that can.