A hybrid minimizing movement and neural network approach to Willmore flow
Numerical Analysis
2026-02-12 v2 Numerical Analysis
Abstract
We present a hybrid method combining a minimizing movement scheme with neural operators for the simulation of phase field-based Willmore flow. The minimizing movement component is based on a standard optimization problem on a regular grid whereas the functional to be minimized involves a neural approximation of mean curvature flow proposed by Bretin et al. Numerical experiments confirm stability for large time step sizes, consistency and significantly reduced computational cost compared to a traditional finite element method. Moreover, applications demonstrate its effectiveness in surface fairing and reconstructing of damaged shapes. Thus, the approach offers a robust and efficient tool for geometry processing.
Cite
@article{arxiv.2502.14656,
title = {A hybrid minimizing movement and neural network approach to Willmore flow},
author = {Martin Rumpf and Josua Sassen and Christoph Smoch},
journal= {arXiv preprint arXiv:2502.14656},
year = {2026}
}