2025/01/30 by Mundinger, Konrad, Zimmer, Max, Kiem, Aldo +2 · 2 citations
#Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG)
paper · doi:10.48550/arxiv.2501.18527
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.