# Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?

arXiv: https://arxiv.org/abs/2501.18527 · HTML: https://mlsanity.com/paper/2501.18527/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 59 references that are in the map.
- First posted on arXiv 30 Jan 2025.

## Builds on (most cited first, 23 of 23)

- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [Erdos Goes Neural: an Unsupervised Learning Framework for Combinatorial Optimization on Graphs](https://mlsanity.com/paper/2006.10643/) (2020)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [On scientific understanding with artificial intelligence](https://mlsanity.com/paper/2204.01467/) (2022)
- [Constructions in combinatorics via neural networks](https://mlsanity.com/paper/2104.14516/) (2021)
- [Budgeted Training: Rethinking Deep Neural Network Training Under Resource Constraints](https://mlsanity.com/paper/1905.04753/) (2019)
- [PatternBoost: Constructions in Mathematics with a Little Help from AI](https://mlsanity.com/paper/2411.00566/) (2024)
- [Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers](https://mlsanity.com/paper/2410.08304/) (2024)
- [Estimating Canopy Height at Scale](https://mlsanity.com/paper/2406.01076/) (2024)
- [Is deep learning a useful tool for the pure mathematician?](https://mlsanity.com/paper/2304.12602/) (2023)
- [Geometry-Informed Neural Networks](https://mlsanity.com/paper/2402.14009/) (2024)
- [The signature and cusp geometry of hyperbolic knots](https://mlsanity.com/paper/2111.15323/) (2021)
- [Finding Increasingly Large Extremal Graphs with AlphaZero and Tabu Search](https://mlsanity.com/paper/2311.03583/) (2023)
- [Refutation of Spectral Graph Theory Conjectures with Monte Carlo Search](https://mlsanity.com/paper/2207.03343/) (2022)
- [Neural Parameter Regression for Explicit Representations of PDE Solution Operators](https://mlsanity.com/paper/2403.12764/) (2024)

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

- [Has the Creativity of Large-Language Models peaked? An analysis of inter- and intra-LLM variability](https://mlsanity.com/paper/2504.12320/) (2025)
- [Geometric Algorithms for Neural Combinatorial Optimization with Constraints](https://mlsanity.com/paper/2510.24039/) (2025)
- [The Agentic Researcher: A Practical Guide to AI-Assisted Research in Mathematics and Machine Learning](https://mlsanity.com/paper/2603.15914/) (2026)
- [Self-Supervised Combinatorial Optimization with Constraints via Frank-Wolfe](https://mlsanity.com/paper/2609.25728/) (2026)

## Abstract (arXiv)

We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with coloring the plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function. This enables gradient-based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings, providing the first improvement in thirty years to the off-diagonal variant of the original problem. Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights.