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Minimising Willmore Energy via Neural Flow

Edward Hirst, Henrique N. Sá Earp, Tomás S. R. Silva

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.

Minimising Willmore Energy via Neural Flow

Abstract

The neural Willmore flow of a closed oriented -surface in is introduced as a natural evolution process to minimise the Willmore energy, which is the squared -norm of mean curvature. Neural architectures are used to model maps from topological domains to 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 surfaces, and the Clifford torus for genus surfaces, respectively. Furthermore, the experiment in the genus case provides a novel approach to search for minimal Willmore surfaces in this open problem.

Paper Structure

This paper contains 31 sections, 33 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Willmore energy
  4. Known minimisers by genus
  5. Genus $0$.
  6. Genus $1$.
  7. Genus $g \geq 2$.
  8. PINNs
  9. PINNs for geometric variational problems
  10. The neural architecture
  11. Sampling
  12. Genus 0.
  13. Genus 1.
  14. Genus 2.
  15. Loss functions
  16. ...and 16 more sections

Figures (9)

  • Figure 1.1: The final trained embedded surfaces for each considered genus $\{0, 1, 2\}$, from the neural Willmore flow learning.
  • Figure 4.1: Training losses for the genus 0 setup. Willmore loss reaches the expected theoretical minimum for the round sphere; regularity loss stays exceptionally low throughout (far lower order of magnitude) affirming a smooth consistent surface.
  • Figure 4.2: Visualisation of the embedding evolution over training for the genus 0 setup. The fundamental domain is shown in (a) where the $(u,v)$ coordinates denote the architecture input (after conversion to spherical harmonics), then (b)-(f) are embedding outputs of the model at various epochs through training, using point colours matching the fundamental domain input, with values of the Willmore energy $\widehat{\mathcal{W}}$.
  • Figure 4.3: Training losses for the genus 1 setup. Willmore loss reaches the expected theoretical minimum for the Clifford torus from the $\tau = 0.1i$ start point; regularity loss stays at exactly 0 throughout (below the ReLU-gated thresholds) affirming a smooth consistent surface.
  • Figure 4.4: Visualisation of the embedding evolution over training for the genus 1 setup. The fundamental domain is shown in (a) where the $(u,v)$ coordinates denote the architecture input (after conversion to Fourier modes), then (b)-(f) are embedding outputs of the model at various epochs through training from $\tau = 0.1i$, using point colours matching the fundamental domain input, with values of the Willmore energy $\widehat{\mathcal{W}}$.
  • ...and 4 more figures