用AI辅助搜索数学构造，突破30年未解难题。
PatternBoost: Constructions in Mathematics with a Little Help from AI
- 交替使用经典搜索与Transformer生成新构造种子。
- 找到多个极值组合学问题的最优解，含30年悬案反例。
- 适合数学构造、自动推理方向研究者参考。
我们提出PatternBoost，一种灵活的数学构造发现方法。算法分两个阶段：第一阶段用经典搜索生成大量优质构造；第二阶段用最佳构造训练Transformer模型，再以模型采样结果作为新种子返回第一阶段，循环迭代。本文详细介绍该方法，并展示其在极值组合学多个问题中的应用。性能因问题而异，但许多场景下表现优异。利用该方法，我们找到了多个长期悬而未决问题的最佳已知解，包括一个持续30年的猜想的反例构造。
原文摘要 · Abstract (English)
We introduce PatternBoost, a flexible method for finding interesting constructions in mathematics. Our algorithm alternates between two phases. In the first ``local'' phase, a classical search algorithm is used to produce many desirable constructions. In the second ``global'' phase, a transformer neural network is trained on the best such constructions. Samples from the trained transformer are then used as seeds for the first phase, and the process is repeated. We give a detailed introduction to this technique, and discuss the results of its application to several problems in extremal combinatorics. The performance of PatternBoost varies across different problems, but there are many situations where its performance is quite impressive. Using our technique, we find the best known solutions to several long-standing problems, including the construction of a counterexample to a conjecture that had remained open for 30 years.
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