math.DGApr 6, 2026
SaveMinimising 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.
Figures & tables
Figure 1.1 : The final trained embedded surfaces for each considered genus , 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 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 .
Figure 4.3 : Training losses for the genus 1 setup. Willmore loss reaches the expected theoretical minimum for the Clifford torus from the 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 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 , using point colours matching the fundamental domain input, with values of the Willmore energy .
Figure 4.5 : Training losses for the genus 2 setup. Willmore loss has a more erratic decline but drops to a stable region. Regularity loss stays low throughout affirming a smooth consistent surface. Since this setup used a disconnected fundamental domain a gluing loss also was used which was a weighted sum of components, this also dropped substantially over the training as the gluing junction was smoothened out; at epoch 50 when the component was introduced there is a spike in both gluing and Willmore losses, and the same at epoch 150 when component was introduced as the surface restabilises. Total loss is not shown here since component weight scheduling makes it non-monotonic and less informative.
Figure 4.6 : Visualisation of the embedding evolution over training for the genus 2 setup. The fundamentaxl domain is shown in (a) where the coordinates denote the architecture input (after conversion to Fourier modes), there’re 2 domains for each tori used in gluing. Then (b)-(i) are embedding outputs of the model at various epochs through training, using point colours matching the fundamental domain input. values through training are given for the Willmore energy , we note a caveat that before gluing is enforced at epoch 150 the measure is unphysical.
Appendix figures & tables2 assets
Supplementary material from the paper’s appendix.
Appendix
Figure B.1 : Training losses for the twisted genus 1 setup. Willmore loss reaches the expected theoretical minimum for the Clifford torus from the twisted start point; regularity loss stays at exactly 0 throughout (below the ReLU-gated thresholds) affirming a smooth consistent surface.
Figure B.2 : Visualisation of the embedding evolution over training for the genus 1 setup. The fundamental domain is shown in (a) where the 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 , using point colours matching the fundamental domain input, with values of the Willmore energy .