发表机构
Princeton University; University of Texas at Austin; Institute for Advanced Study（普林斯顿大学; 德克萨斯大学奥斯汀分校; 高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过受控案例研究AI智能体发现科学公式，提出PI+双学生工作流，成功再发现hydrotope公式并发现新的六点振幅表达式。
AI 中文摘要
我们通过一个受控的案例研究，考察AI智能体如何发现并验证科学公式。该案例聚焦于hydrotope，这是一个最近发现的几何公式，它将非线性表面波散射的不同多项式部分整合为一个全局表达式。这个问题具有欺骗性的难度：简单的公式可以在单个频率区域内成立，但全局结果必须识别这些区域的边界，并组合指数级数量的潜在活跃项。我们重构了该公式最初通过人机协作发现的过程，并分析了18次单提示再发现运行，这些运行在无提示和两种形式的人类指导下进行：一种错误提示代表不正确的先验，一种正确提示代表领域知识。只有四次运行在所有运动学区域（即单个多项式形式适用的区域）中恢复了公式，而大多数不成功的运行找到了正确的区域多项式，但未能组合它们或测试其完整域。在我们的实验中，传统和LLM辅助的符号回归以及标准机器学习回归器同样未能恢复全局公式。在这些失败模式的指导下，我们测试了一个PI+双学生工作流，其中协调的首席智能体为两个研究智能体分配互补的分析和数值任务，并独立评估其结果。PI+双学生团队成功再发现了完整的hydrotope公式，而相同的工作流应用于更困难的三负波数问题，发现了六点振幅$A_6$的新的独立验证解析表达式。
英文摘要
We study how AI agents discover and validate scientific formulas using a controlled case study of the hydrotope, a recently discovered geometric formula that combines the different polynomial pieces of nonlinear surface-wave scattering into one global expression. This problem is deceptively difficult: simple formulas can hold within individual frequency regions, but the global result must identify their boundaries and combine exponentially many potentially active terms. We reconstruct how the formula was originally discovered through human--agent collaboration and analyze 18 single-prompt rediscovery runs under no hint and two forms of human guidance: a false hint representing an incorrect prior and a true hint representing domain-informed insight. Only four recover the formula across all kinematic chambers (i.e., regions in which a single polynomial form applies), while most unsuccessful runs find correct chamber polynomials but fail to combine them or test their full domain. Conventional and LLM-assisted symbolic regression and standard machine-learning regressors likewise fail to recover the global formula in our experiments. Guided by these failure modes, we test a PI$+$two-student workflow in which a coordinating lead agent assigns complementary analytic and numerical tasks to two research agents and independently evaluates their results. The PI$+$two-student team successfully rediscovers the complete hydrotope formula, while the same workflow applied to the harder three negative wavenumber problem discovers a new independent verified analytic expression for the six-point amplitude $A_6$.
Comments22+26 pages, 10 figures
