Willmore Energy Minimization with Neural Networks

Learn optimal surface embeddings φ:(u,v)→(x,y,z) that minimize the Willmore energy functional ∫∫ H² dA using PyTorch.

Quick Start

# Setup (see environment/README.md for details)
conda create -n willmore python=3.10
conda activate willmore
pip install -r environment/requirements.txt

# Train (defaults to configs/config_genus2.yaml)
python run.py

# Train with a specific config
python run.py --config configs/config_genus1.yaml

# Visualize training evolution
python visualisation/visualise.py

# Visualize analytical reference surfaces
python visualisation/visualise_analytic.py --genus 1

# Visualize supervised pretraining results
python visualisation/visualise_supervised.py --genus 1

Core Concept

The network learns a surface embedding φ:(u,v)→(x,y,z), minimizing:

$$W = \int\int H^2 \sqrt{EG - F^2} , du , dv$$

where H is mean curvature computed via automatic differentiation of the embedding.

Usage

# Resume from a checkpoint
python run.py --config configs/config_genus1.yaml --resume checkpoints/run_1/latest_model.pt

# Visualize with both surface and loss plots
python visualisation/visualise.py --mode both

# Visualize analytical surfaces (genus 0, 1, or 2)
python visualisation/visualise_analytic.py --genus 1

# Supervised pretraining visualization
python visualisation/visualise_supervised.py --genus 1 --points 20000

Config files are in configs/. The --config flag accepts any path; it defaults to configs/config_genus2.yaml.

Files

• model.py - EmbeddingNetwork: φ(u,v)→(x,y,z) with Fourier features for periodicity
• losses.py - Willmore energy ∫∫H²dA computed via autodiff of fundamental forms
• sampling.py - Parameter space sampling for torus, sphere, double torus, etc.
• run.py - Training loop with checkpointing
• utils.py - Visualization utilities and run management
• visualisation/ - Visualization scripts:
  • visualise.py - Training evolution visualization
  • visualise_analytic.py - Analytical reference surface visualization
  • visualise_supervised.py - Supervised pretraining visualization

Theory

Willmore Energy: W = ∫∫ H² √(EG-F²) du dv

where H = (EN-2FM+GL)/(2(EG-F²)) is mean curvature, computed from:

• First fundamental form: E = ⟨φ_u,φ_u⟩, F = ⟨φ_u,φ_v⟩, G = ⟨φ_v,φ_v⟩
• Second fundamental form: L = ⟨φ_uu,n⟩, M = ⟨φ_uv,n⟩, N = ⟨φ_vv,n⟩
• Unit normal: n = (φ_u × φ_v) / |φ_u × φ_v|

Known minima:

• Genus 0: round sphere, W = 4π ≈ 12.566
• Genus 1: Clifford torus, W = 2π² ≈ 19.74
• Genus 2: Lawson surface ξ_{2,1} (conjectured), W ≈ 21.89

Output

Training creates:

• checkpoints/run_N/ - Model states (best_model.pt, latest_model.pt, epoch checkpoints)
• logs/run_N/training_history.json - Loss curves and metrics
• logs/run_N/loss_curves.png - Automatically generated after training

BibTeX Citation

@article{Hirst:2026qwi,
    author = "Hirst, Edward and Earp, Henrique N. S{\'a} and Silva, Tom{\'a}s S. R.",
    title = "{Minimising Willmore Energy via Neural Flow}",
    eprint = "2604.04321",
    archivePrefix = "arXiv",
    primaryClass = "math.DG",
    month = "4",
    year = "2026"
}