用神经网络发现平面着色新解，30年来首次突破。
Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?
- 将几何着色问题转为可微优化，用神经网络搜索可行解。
- 发现两种新型六色方案，破解30年未解的离散几何难题。
- 方法可推广至其他带约束的数学探索，适合算法与几何研究者。
我们展示神经网络如何推动数学发现，以哈迪格-尼尔森问题为例——这是一个长期悬而未决的离散几何与极值组合交叉问题，关注在平面上对点进行着色，避免出现单色单位距离点对。通过将这一具有硬约束的混合离散-连续几何着色问题重新表述为带有概率性、可微损失函数的优化任务，利用神经网络作为近似器，实现了基于梯度的可适配置探索，最终发现了两种全新的六色方案，首次在三十年内改进了该问题的非对角变体。本文揭示了实现这些结果的机器学习方法，并通过额外数值洞察展示了其更广泛适用性。
原文摘要 · Abstract (English)
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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