Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?
This work addresses a long-standing open problem in mathematics, offering a new computational approach for geometric discovery, though it is incremental in advancing a specific variant.
The paper tackled the Hadwiger-Nelson problem in discrete geometry by reformulating it as an optimization task using neural networks, resulting in the discovery of two novel six-colorings that improved upon a variant of the problem for the first time in thirty years.
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.