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

TL;DR

This paper shows how neural networks can be used to explore and discover new solutions in complex mathematical problems, specifically in plane coloring, leading to novel findings in a long-standing open problem.

Contribution

It introduces a neural network-based optimization approach to tackle the Hadwiger-Nelson problem, resulting in the first new six-colorings in thirty years.

Findings

Discovered two new six-colorings for the Hadwiger-Nelson problem.

Demonstrated neural networks can effectively explore geometric coloring configurations.

Provided broader numerical insights into the problem's solution space.

Abstract

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.