Minimising Willmore Energy via Neural Flow

2026-04-05

AI Summary

This study addresses the open problem of minimizing Willmore energy for closed surfaces of high genus in three-dimensional Euclidean space. We introduce, for the first time, a neural flow approach to this domain by modeling the embedding map from a two-dimensional topological domain into three-dimensional space using neural networks. A physics-informed neural network (PINN)-based loss function is designed to drive the evolution of the Willmore flow. Our method successfully reproduces the known minimizers—the round sphere for genus 0 and the Clifford torus for genus 1—and, for genus 2, yields a novel candidate surface with lower Willmore energy than previously reported configurations. This work thus establishes an effective new pathway for exploring Willmore minimizers of higher genus.

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.

Problem

• Willmore energy
• minimal surfaces
• neural flow
• genus-2 surfaces
• mean curvature

Innovation

• Neural Willmore Flow
• Willmore Energy
• Physics-Informed Neural Networks
• Minimal Surfaces
• Geometric Deep Learning