{
  "abstract": "We extend to the Willmore flow the phase-field neural operator approach proposed by Bretin, Denis, Masnou, and Terii (2022) for mean curvature motion in two and three dimensions. The resulting lightweight, single-time-step architecture is motivated by a Bence-Merriman-Osher-type expansion of the Allen-Cahn semigroup, and is trained using trajectories generated by a hybrid minimizing movement scheme. In our experiments with non-oriented interfaces, unconstrained convolution kernels produce anisotropic dynamics, whereas radial kernels enhance the evolution process. We also construct three-dimensional kernels from trained two-dimensional kernels via their radial Fourier profiles, which provides an effective initialization for the non-oriented 3D model. Finally, we show applications to curve and surface reconstruction from unoriented point clouds, including reconstruction using the transferred model without the need for additional 3D training.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03534",
  "author_total": 4,
  "authors": [
    {
      "name": "Elie Bretin"
    },
    {
      "name": "Roland Denis"
    },
    {
      "name": "Tokuhiro Eto"
    },
    {
      "name": "Simon Masnou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3d6t5m5xqvvdr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03534",
  "fulltext_url": null,
  "id": "3d6t5m5xqvvdr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03534",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.03534",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3d6t5m5xqvvdr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03534",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03534"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Learning the Willmore flow with a lightweight neural operator",
  "title_text": "Learning the Willmore flow with a lightweight neural operator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/3d6t5m5xqvvdr",
  "venue": "arXiv",
  "versions": []
}