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

• URL: http://arxiv.org/abs/2501.18527v3
• Date: Thu, 05 Jun 2025 07:01:14 GMT
• Title: Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?
• Abstract summary: We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem.<n>We reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function.<n>This enables gradient-based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings.
• Score: 21.02602149305327
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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