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PreprintCC BY-NC-ND 4.0arXiv

Learning the Willmore flow with a lightweight neural operator

  • Elie Bretin
  • Roland Denis
  • Tokuhiro Eto
  • Simon Masnou
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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.

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Cite

BibTeXRIS
BibTeX
@misc{bretin2026learning,
  title = {{Learning the Willmore flow with a lightweight neural operator}},
  author = {Elie Bretin and Roland Denis and Tokuhiro Eto and Simon Masnou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.03534},
  eprint = {2610.03534},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.03534},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Learning the Willmore flow with a lightweight neural operator
AU  - Elie Bretin
AU  - Roland Denis
AU  - Tokuhiro Eto
AU  - Simon Masnou
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.03534
UR  - https://arxiv.org/abs/2610.03534
AB  - 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.
ER  -