A systematic study of physics-informed neural networks for the level-set interface advection
TL;DR
A key finding is an RFF–eikonal joint design constraint: at low bandwidth, weak eikonal regularization distorts the signed-distance field and underperforms the tanh baseline, whereas higher bandwidth is compatible with moderate regularization and yields the global best result.
Abstract
We present a systematic ablation study of physics-informed neural networks (PINNs) for level-set advection across four benchmarks of increasing complexity: linear translation (TR), solid-body rotation (RO), reversed vortex deformation (RV), and the Zalesak rotating slotted disc (ZD), covering 69 experiments. For TR, a step learning rate scheduler (StepLR) with eikonal weight weik=1.0 is optimal (mean L2 error E―L2=2.09×10−4). For RO, cosine annealing (CosineAnnealing, minimum learning rate ηmin=10−5) outperforms StepLR, establishing that scheduler choice is benchmark-specific and cannot be transferred. For RV, reducing weik from 1.0 to 10−4 yields an 82× improvement ( E―L2=1.51×10−3, mean relative L2 error E―L2rel=0.43%, final time T=2); extending to T=8 with causal weighting and residual-based adaptive distribution and refinement (RAD + RAR) achieves 0.63% with a standard tanh network, outperforming the PirateNet state-of-the-art (Sota) of Mullins et al (2025) ( 0.85%). For ZD, four studies are conducted: the eikonal weight study (S1), the progressive sampling study (S2), the architecture study (S3), and the adaptive sampling study (S4). Random Fourier feature (RFF) encoding (bandwidth σ=2) with weik=10−2 and causal weighting achieves E―L2=5.74×10−4 in the architecture study (S3), a 10× reduction over the tanh baseline; adding RAD + RAR with RFF σ=5 and M=32 causal chunks achieves the best overall ZD result: E―L2=4.64×10−4 ( E―L2rel=0.13%) in 15.2 min in the adaptive sampling study (S4), outperforming the published Sota. A key finding is an RFF–eikonal joint design constraint: at low bandwidth ( σ=2), weak eikonal regularization ( weik=10−4) distorts the signed-distance field and underperforms the tanh baseline, whereas higher bandwidth ( σ=5) is compatible with moderate regularization ( weik=10−3) and yields the global best result. To the best of our knowledge, this bandwidth-dependent constraint has not previously been identified in the PINN literature.