# Minimal surfaces, Knots, and Neural Networks

arXiv: https://arxiv.org/abs/2605.26234 · HTML: https://mlsanity.com/paper/2605.26234/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 45 references that are in the map.
- First posted on arXiv 25 May 2026.

## Builds on (most cited first, 14 of 14)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017)
- [Neural Network Approximations for Calabi-Yau Metrics](https://mlsanity.com/paper/2012.15821/) (2020)
- [Learning Size and Shape of Calabi-Yau Spaces](https://mlsanity.com/paper/2111.01436/) (2021)
- [Machine learning Calabi-Yau metrics](https://mlsanity.com/paper/1910.08605/) (2019)
- [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)
- [A Physics-Informed Neural Network Framework For Partial Differential Equations on 3D Surfaces: Time-Dependent Problems](https://mlsanity.com/paper/2103.13878/) (2021)
- [Approximating High-Dimensional Minimal Surfaces with Physics-Informed Neural Networks](https://mlsanity.com/paper/2309.02589/) (2023)
- [Aspects of holographic entanglement using physics-informed-neural-networks](https://mlsanity.com/paper/2509.25311/) (2025)
- [A Machine Learning Approach to the Nirenberg Problem](https://mlsanity.com/paper/2602.12368/) (2026)
- [Minimising Willmore Energy via Neural Flow](https://mlsanity.com/paper/2604.04321/) (2026)
- [Machine Learning Gravity Compactifications on Negatively Curved Manifolds](https://mlsanity.com/paper/2501.00093/) (2024)

## Built on by (most cited first, 2 of 2)

- [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)

A recent conjecture by Joel Fine posits a relationship between the coefficients of the HOMFLY polynomial of a knot K in the 3-sphere S3, and the signed count of minimal surfaces in hyperbolic 4-space H4 meeting the sphere at infinity at K, with prescribed genus and self-intersection number. In this paper, we develop a novel machine learning framework based on Physics-Informed Neural Networks (PINNs) to solve the minimal surface equation in hyperbolic space. We utilise this framework to test Fine's Conjecture by constructing near-minimal surfaces bounding various families of knots in S3. Furthermore, we develop an algorithmic method to find self-intersections and compute their sign. For every knot analysed, the computationally discovered minimal surfaces and their self-intersection numbers perfectly align with the predictions of Fine's Conjecture, providing empirical evidence for it.