Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915326231314432 |
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| author | Mundinger, Konrad Zimmer, Max Kiem, Aldo Spiegel, Christoph Pokutta, Sebastian |
| author_facet | Mundinger, Konrad Zimmer, Max Kiem, Aldo Spiegel, Christoph Pokutta, Sebastian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18527 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Neural Discovery in Mathematics: Do Machines Dream of Colored Planes? Mundinger, Konrad Zimmer, Max Kiem, Aldo Spiegel, Christoph Pokutta, Sebastian Machine Learning Combinatorics 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. |
| title | Neural Discovery in Mathematics: Do Machines Dream of Colored Planes? |
| topic | Machine Learning Combinatorics |
| url | https://arxiv.org/abs/2501.18527 |