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[Submitted on 18 Sep 2026] Title:A Data-Free Physics-Informed Neural Operator for Level-Set Interface Advection View a PDF of the paper titled A Data-Free Physics-Informed Neural Operator for Level-Set Interface Advection, by Muhammad Akbar Khan View PDF HTML (experimental) Abstract:Operators for interfacial problems are trained on reference solutions produced by the solver they are intended to replace. This work develops a data-free physics-informed neural operator for level-set interface advection, in which the interface is the equation's unknown and the operator maps an initial interface to the full spatiotemporal trajectory under a prescribed flow. Training uses only the transport residual and a geometric constraint; no reference solution enters the objective at any point. A spacetime Fourier backbone emits the entire trajectory in one pass, and the initial condition is imposed by construction rather than by penalty, which removes the competition between the anchoring term and the residual that otherwise arises when no solution data are available. Supervised and hybrid operators are trained under an identical architecture, family, budget and test set, and are reported throughout as baselines that quantify what refusing labels costs. On a reversed single vortex the data-free operator reaches 1.614 +/- 0.067% relative L2 error on 100 held-out initial interfaces against 0.369 +/- 0.035% for the supervised baseline, a factor of 4.4; on solid-body rotation the corresponding figures are 3.804 +/- 1.075% and 2.576 +/- 0.159%, a factor of 1.5. The two benchmarks rank the arms differently, and the difference is attributable to the eikonal constraint: the exact solution violates |grad phi| = 1 over 0.3% of the domain under rotation and 86.9% under the vortex. Where the constraint is valid the physics-trained operator conserves enclosed area 2.7 times better than the supervised baseline despite a larger field error, and a hybrid arm using eight reference solutions outperforms a supervised arm using sixteen. Comments: 27 pages, 8 figures, 6 tables Subjects: Machine Learning (cs.LG); Fluid Dynamics (physics.flu-dyn) MSC classes: 65M99, 68T07, 35Q35 ACM classes: G.1.8; I.2.6 Cite as: arXiv:2609.38195 [cs.LG] (or arXiv:2609.38195v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2609.38195 arXiv-issued DOI via DataCite Submission history From: Muhammad Akbar Khan [view email] [v1] Fri, 18 Sep 2026 07:09:13 UTC (1,330 KB) Full-text links: Access Paper: View a PDF of the paper titled A Data-Free Physics-Informed Neural Operator for Level-Set Interface Advection, by Muhammad Akbar Khan View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG new | recent | 2026-09 Change to browse by: cs physics physics.flu-dyn References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) IArxiv recommender toggle IArxiv Recommender (What is IArxiv?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)