# Minimising Willmore Energy via Neural Flow

arXiv: https://arxiv.org/abs/2604.04321 · HTML: https://mlsanity.com/paper/2604.04321/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 42 references that are in the map.
- First posted on arXiv 6 Apr 2026.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Learning Size and Shape of Calabi-Yau Spaces](https://mlsanity.com/paper/2111.01436/) (2021)
- [Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks](https://mlsanity.com/paper/2201.06780/) (2022)
- [Machine Learned Calabi-Yau Metrics and Curvature](https://mlsanity.com/paper/2211.09801/) (2022)
- [Metric Flows with Neural Networks](https://mlsanity.com/paper/2310.19870/) (2023)
- [Physics-informed neural network solves minimal surfaces in curved spacetime](https://mlsanity.com/paper/2509.10866/) (2025)
- [Approximating High-Dimensional Minimal Surfaces with Physics-Informed Neural Networks](https://mlsanity.com/paper/2309.02589/) (2023)
- [A Machine Learning Approach to the Nirenberg Problem](https://mlsanity.com/paper/2602.12368/) (2026)
- [Neural and numerical methods for G2-structures on contact Calabi-Yau 7-manifolds](https://mlsanity.com/paper/2602.12438/) (2026)

## Built on by (most cited first, 4 of 4)

- [Minimal surfaces, Knots, and Neural Networks](https://mlsanity.com/paper/2605.26234/) (2026)
- [PINNs in More General Geometry](https://mlsanity.com/paper/2604.25020/) (2026)
- [Lean-Quantum: Toward AI-Assisted Formalization of Quantum Information](https://mlsanity.com/paper/2607.05492/) (2026)
- [A user's guide to PINNs in geometric analysis: lessons from the asymptotic Plateau problem](https://mlsanity.com/paper/2607.28733/) (2026)

## Abstract (arXiv)

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.