SDF-Aware Weighting: Adaptive Eikonal Regularisation for Three-Dimensional Level-Set Physics-Informed Neural Networks
Abstract
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∇ϕrVert 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 L2 error reads 1.06%, indistinguishable from a field that never represented the slot. We give a feature-restricted measure that separates the two cases.
Related
- A gradient-enhanced physics-informed neural network with adaptive loss weighting for high-dimensional non-linear sine-Gordon problems
- Physics-informed neural networks by Gradient-Guided Gaussian Adaptive Sampling (3GAS-PINNs)
- Adaptive Physics-Informed Neural Networks for the Blasius Boundary-Layer Problem
- Physics-Informed Classical and Quantum Neural Networks for One-Dimensional Schrodinger Eigenvalue Problems
- Analytic results for one-loop integrals in dimensional regularisation
BibTeX
@misc{khan2026sdfaware,
title = {{SDF-Aware Weighting: Adaptive Eikonal Regularisation for Three-Dimensional Level-Set Physics-Informed Neural Networks}},
author = {Muhammad Akbar Khan},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.08322},
eprint = {2608.08322},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.08322},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - SDF-Aware Weighting: Adaptive Eikonal Regularisation for Three-Dimensional Level-Set Physics-Informed Neural Networks
AU - Muhammad Akbar Khan
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.08322
UR - https://arxiv.org/abs/2608.08322
AB - 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∇ϕrVert 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.
ER -