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

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.

@article{mundinger2025_2501.18527,
  title={ Neural Discovery in Mathematics: Do Machines Dream of Colored Planes? },
  author={ Konrad Mundinger and Max Zimmer and Aldo Kiem and Christoph Spiegel and Sebastian Pokutta },
  journal={arXiv preprint arXiv:2501.18527},
  year={ 2025 }
}