用AI演化系统探索组合数学中的双射构造，发现新双射仍极困难。
Even with AI, Bijection Discovery is Still Hard: The Opportunities and Challenges of OpenEvolve for Novel Bijection Construction
- 用多LLM协作生成代码并演化，寻找组合结构间的显式双射
- 在三个迪克路径双射问题中，仅成功复现已知结果，开放问题未突破
- 提示需人类数学家参与，适合探索型研究者参考
进化式程序合成系统（如AlphaEvolve、OpenEvolve、ShinkaEvolve）为人工智能辅助数学发现提供了新范式。这些系统利用大型语言模型（LLMs）团队生成可读代码作为候选解，并通过演化过程逐步优化，超越单次生成能力。尽管现有应用多聚焦于边界证明（如球体堆积），该方法也适用于需显式构造解的数学问题。本文探讨OpenEvolve在组合双射发现中的应用，针对三个涉及迪克路径的问题展开实验，其中两个为已知双射，一个为开放问题。结果表明，当前前沿系统虽具潜力，但发现新研究级双射仍具挑战性，凸显人类数学家在闭环中的必要性。文中总结了可复用的经验教训，供领域内研究者参考。
原文摘要 · Abstract (English)
Evolutionary program synthesis systems such as AlphaEvolve, OpenEvolve, and ShinkaEvolve offer a new approach to AI-assisted mathematical discovery. These systems utilize teams of large language models (LLMs) to generate candidate solutions to a problem as human readable code. These candidate solutions are then 'evolved' with the goal of improving them beyond what an LLM can produce in a single shot. While existing mathematical applications have mostly focused on problems of establishing bounds (e.g., sphere packing), the program synthesis approach is well suited to any problem where the solution takes the form of an explicit construction. With this in mind, in this paper we explore the use of OpenEvolve for combinatorial bijection discovery. We describe the results of applying OpenEvolve to three bijection construction problems involving Dyck paths, two of which are known and one of which is open. We find that while systems like OpenEvolve show promise as a valuable tool for combinatorialists, the problem of finding novel, research-level bijections remains a challenging task for current frontier systems, reinforcing the need for human mathematicians in the loop. We describe some lessons learned for others in the field interested in exploring the use of these systems.
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