Minimising Willmore Energy via Neural Flow
This work addresses the open problem of finding minimal Willmore surfaces in computational geometry, with incremental contributions in applying neural networks to a known bottleneck.
The paper tackles the problem of minimizing Willmore energy for surfaces in 3D space by introducing a neural flow method, achieving results that reproduce known solutions like the round sphere for genus 0 and Clifford torus for genus 1, and offering a novel approach for genus 2 surfaces.
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.