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

• URL: http://arxiv.org/abs/2608.08322v1
• Date: Sat, 08 Aug 2026 20:20:13 GMT
• Title: Eikonal Regularisation in Physics-Informed Neural Networks for Three-Dimensional Level-Set Advection: Transferability of Two-Dimensional Design Principles
• Abstract summary: We implement an eikonal regulariser to penalise the deviation of $|nabla|$ from unity.<n>We benchmark against a fifth-order WENO solver on identical grids and error measures.<n>We show that the relative $L$ error cannot certify the preservation of thin features.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\|\nablaφ\|$ from unity. A previous two-dimensional study identified this weight as the dominant hyperparameter and found its optimum shifts by four orders of magnitude between rigid-body and deforming flows, but left open whether these principles transfer to three dimensions and whether single-seed results survive run-to-run variability. We answer both by repeating the weight selection across four 3D benchmarks (translating sphere, rotating sphere, slotted sphere, reversed vortex), sweeping six weights with three seeds at full training budget under a pre-registered selection rule. The ordering transfers: the selected weight tracks how far the exact solution departs from the signed-distance property, spanning four decades from $10^{-1}$ where it holds exactly to $10^{-5}$ where the interface is stretched. Values transfer only benchmark by benchmark; two of four carry over unchanged and two do not, so inheritance must be verified. The multi-seed protocol reveals that at small weights the seed-to-seed standard deviation equals the error itself, and the regulariser reduces it by more than an order of magnitude, buying reproducibility as well as accuracy. We benchmark against a fifth-order WENO solver on identical grids and error measures; the classical scheme is more accurate on all four problems, by two orders of magnitude on smooth rigid advection, with a margin that narrows with geometric difficulty and is smaller in volume conservation than in the field norm. Finally, we show that the relative $L_2$ error cannot certify the preservation of thin features, and report a feature-restricted measure that can.

Related papers

• A variational physics-informed graph neural network for heterogeneous solid mechanics [0.0]
  This work presents a variational, label-free physics-informed graph neural network (PI-GNN) in which the conforming is carried by the discretization rather than by the trial field.
  arXiv  Detail & Related papers  (2026-09-10T02:02:32Z)
• Feature Priming in Online Linear Regression: Sparse-Regret Lower Bounds and a Tight Univariate Rate [47.36630149538994]
  In high-dimensional online prediction, the best predictor may depend on only a few features, so regret should scale with sparsity rather than the ambient dimension.<n>Warmuth and Amid asked at COLT 2023 whether any of three such rules admits a competitive online regret guarantee.<n>Our analysis identifies a common obstruction: cheap nuisance causes the refit to underweight the truly predictive coordinate.
  arXiv  Detail & Related papers  (2026-08-18T09:32:03Z)
• Adaptive operator-generated subspaces for effective many-body Hamiltonians [0.0]
  We present the Adaptive Clifford-Algebra Subspacesolver (A-CASE)<n>Overlap, Hamiltonian, observable, and response matrices are reconstructed from one shared Pauli-expectation bank.<n>A grouped iteration propagates finite-shot variability through thresholding, root matching, spectral weights, susceptibility, and broadening.
  arXiv  Detail & Related papers  (2026-08-01T09:51:25Z)
• Dynamical and Optimization Trade-offs of Levi--Civita Coordinates for Learned Close-Encounter Dynamics [0.0]
  We show that a Levi--Civita Hamiltonian splitting holds the maximum relative energy error near $2.1times10-5$.<n>Regularized models produce finite rollouts in $40$ runs versus $0/40$ for Cartesian.<n>Four neural residual objectives fail to approach the analytic result.<n>This is a controlled falsification-plus-trade-off study, not a solution to learned close-encounter dynamics.
  arXiv  Detail & Related papers  (2026-07-22T14:56:15Z)
• From Direction to Magnitude: How Multimodal Instruction-Tuning Reorganizes the Geometric Encoding of Identity-Specifying Prompts in Transformer Hidden States [0.0]
  We investigate whether identity-specifying system prompts produce statistically distinguishable geometric fingerprints in the hidden-state trajectories of four open-weight transformer language models.<n>Three prompt conditions are compared via five metrics, principally the 1-Wasserstein distance between edge-wise of Ollivier-Ricci curvature on kNN trajectory graphs.
  arXiv  Detail & Related papers  (2026-07-10T17:22:36Z)
• Level-Crossing Density as a Mesh-Free High-Frequency Auxiliary Loss for Implicit Neural Representations [0.0]
  We derive smooth Monte-Carlo estimators for a continuous neural field using no grid, no complex, no persistence.<n>The estimator is accurate to 1-3% against exact topology in 2D and 3D, and costs about 3 ms per iteration.<n>In 3D neural-SDF fitting, however, a failure mode we believe general to any sampled soft topology objective appears.
  arXiv  Detail & Related papers  (2026-07-07T04:23:23Z)
• Normalized Architectures are Natively 4-Bit [49.13186675123547]
  nGPT, an architecture that constrains weights and hidden representations to the unit hypersphere, is inherently more robust to low-precision arithmetic.<n>We validate this approach on both a 1.2B dense model and hybrid (Mamba-Transformer) MoE models of up to 3B/30B parameters.
  arXiv  Detail & Related papers  (2026-05-07T11:54:07Z)
• Robust inverse material design with physical guarantees using the Voigt-Reuss Net [0.0]
  We propose a spectrally normalized surrogate for forward and inverse mechanical homogenization with hard physical guarantees.<n>In 3D linear elasticity on an open dataset of biphasic microstructures, a fully connected Voigt-Reuss net trained on $>!7.5times 105$ FFT-based labels with 236 isotropy-in descriptors.<n>Overall, the Voigt-Reuss net unifies accurate, physically admissible forward prediction with large-batch, constraint-consistent inverse design.
  arXiv  Detail & Related papers  (2025-11-14T15:17:37Z)
• Quantized Visual Geometry Grounded Transformer [67.15451442018258]
  This paper proposes the first Quantization framework for VGGTs, namely QuantVGGT.<n>We introduce Dual-Smoothed Fine-Grained Quantization, which integrates pre-global Hadamard rotation and post-local channel smoothing.<n>We also design Noise-Filtered Diverse Sampling, which filters outliers via deep-layer statistics.
  arXiv  Detail & Related papers  (2025-09-25T15:17:11Z)
• CWF: Consolidating Weak Features in High-quality Mesh Simplification [50.634070540791555]
  We propose a smooth functional that simultaneously considers all of these requirements. The functional comprises a normal anisotropy term and a Centroidal Voronoi Tessellation (CVT) energy term.
  arXiv  Detail & Related papers  (2024-04-24T05:37:17Z)
• Quantum control landscape for generation of $H$ and $T$ gates in an open qubit with both coherent and environmental drive [57.70351255180495]
  An important problem in quantum computation is generation of single-qubit quantum gates such as Hadamard ($H$) and $pi/8$ ($T$) Here we consider the problem of optimal generation of $H$ and $T$ gates using coherent control and the environment as a resource acting on the qubit via incoherent control.
  arXiv  Detail & Related papers  (2023-09-05T09:05:27Z)
• Transformers as Support Vector Machines [54.642793677472724]
  We establish a formal equivalence between the optimization geometry of self-attention and a hard-margin SVM problem. We characterize the implicit bias of 1-layer transformers optimized with gradient descent. We believe these findings inspire the interpretation of transformers as a hierarchy of SVMs that separates and selects optimal tokens.
  arXiv  Detail & Related papers  (2023-08-31T17:57:50Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.