A systematic study of physics-informed neural networks for the level-set interface advection

Muhammad Akbar Khan, Fahim Raees

Abstract We present a systematic ablation study of physics-informed neural networks (PINNs) for level-set advection across four benchmarks: 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 w_eik = 1.0 is optimal (mean L² error E_L2 = 2.09×10⁻⁴). For RO, cosine annealing (CosineAnnealing, minimum learning rate η_min = 10⁻⁵) outperforms StepLR, establishing that scheduler choice is benchmark-specific and cannot be transferred. For RV, reducing w_eik from 1.0 to 10⁻⁴ yields an 82× improvement (E_L2 = 1.51×10⁻³, mean relative L² error E^rel_L2 = 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 w_eik = 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^rel_L2 = 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 (w_eik = 10⁻⁴) distorts the signed-distance field and underperforms the tanh baseline, whereas higher bandwidth (σ = 5) is compatible with moderate regularization (w_eik = 10⁻³) and yields the global best result. This bandwidth-dependent constraint has not previously been identified in the PINN literature.