Minimising Willmore Energy via Neural Flow

TL;DR

The paper introduces a neural flow method to minimize Willmore energy of surfaces, successfully reproducing known minimal surfaces and exploring new solutions for genus 2 surfaces.

Contribution

It presents a neural flow approach using PINNs to minimize Willmore energy, including novel results for genus 2 surfaces.

Findings

Reproduces round sphere and Clifford torus as minimizers for genus 0 and 1.

Provides a new method to search for minimal Willmore surfaces in genus 2.

Demonstrates neural architectures effectively model surface embeddings for energy minimization.

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.