[2604.04321] Minimising Willmore Energy via Neural Flow

Mathematics > Differential Geometry arXiv:2604.04321 (math) [Submitted on 6 Apr 2026] Title:Minimising Willmore Energy via Neural Flow Authors:Edward Hirst, Henrique N. Sá Earp, Tomás S. R. Silva View a PDF of the paper titled Minimising Willmore Energy via Neural Flow, by Edward Hirst and 2 other authors View PDF HTML (experimental) Abstract:The neural Willmore flow of a closed oriented $2$-surface in $\mathbb{R}^3$ is introduced as a natural evolution process to minimise the Willmore energy, which is the squared $L^2$-norm of mean curvature. Neural architectures are used to model maps from topological $2d$ domains to $3d$ Euclidean space, where the learning process minimises a PINN-style loss for the Willmore energy as a functional on the embedding. Training reproduces the expected round sphere for genus $0$ surfaces, and the Clifford torus for genus $1$ surfaces, respectively. Furthermore, the experiment in the genus $2$ case provides a novel approach to search for minimal Willmore surfaces in this open problem. Comments: Subjects: Differential Geometry (math.DG); Machine Learning (cs.LG) Cite as: arXiv:2604.04321 [math.DG]   (or arXiv:2604.04321v1 [math.DG] for this version)   https://doi.org/10.48550/arXiv.2604.04321 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Edward Hirst [view email] [v1] Mon, 6 Apr 2026 00:02:56 UTC (12,847 KB) Full-text links: Access Paper: View a PDF of the paper titled Minimising Willmore Ene...