SDF-Aware Weighting: Adaptive Eikonal Regularisation for Three-Dimensional Level-Set Physics-Informed Neural Networks
Adaptive loss-balancing schemes for physics-informed neural networks rest on a premise that every residual should be driven to zero. For level-set advection with an eikonal regulariser that premise fails: the eikonal term penalises deviation of $\lVert\nabla\phi\rVert$ from unity, a property transport preserves only under rigid motion; where the exact solution departs from a signed-distance function the eikonal residual of the correct answer is nonzero, and driving it to zero moves the network away from that answer. We show that standard gradient-norm balancing fails in exactly this way, its weight remaining near its initial value throughout training on benchmarks where the property is violated, and we introduce SDF-Aware Weighting (SAW), which combines a residual-quantile gate with a gradient-norm ratio so that points exhibiting legitimate departure are excluded before the surviving term is scaled. Across four three-dimensional benchmarks SAW selects an eikonal weight within an order of magnitude of the value located by an eighteen-run manual sweep, spanning four decades from $10^{-1}$ to $10^{-5}$ with a single fixed configuration. On the slotted sphere, where the initial field is non-differentiable at reentrant edges, SAW attains a lower error than any weight in that sweep. Two smooth rigid benchmarks serve as controls: SAW is worse there, as expected when its premise does not hold. An ablation with the gate disabled shows the slot is nearly entirely filled while the relative $L_2$ error reads $1.06\%$, indistinguishable from a field that never represented the slot. We give a feature-restricted measure that separates the two cases.