Minimising Willmore Energy via Neural Flow

The neural Willmore flow of a closed oriented 2-surface in ℝ3 is introduced as a natural evolution process to minimise the Willmore energy, which is the squared L2-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.