How can Transformers model and learn enumerative geometry? What is a robust procedure for using Transformers in abductive knowledge discovery within a mathematician-machine collaboration? In this work, we introduce a new paradigm in computational enumerative geometry in analyzing the $\psi$-class intersection numbers on the moduli space of curves. By formulating the enumerative problem as a continuous optimization task, we develop a Transformer-based model for computing $\psi$-class intersection numbers based on the underlying quantum Airy structure. For a finite range of genera, our model is capable of regressing intersection numbers that span an extremely wide range of values, from $10^{-45}$ to $10^{45}$. To provide a proper inductive bias for capturing the recursive behavior of intersection numbers, we propose a new activation function, Dynamic Range Activator (DRA). Moreover, given the severe heteroscedasticity of $\psi$-class intersections and the required precision, we quantify the uncertainty of the predictions using Conformal Prediction with a dynamic sliding window that is aware of the number of marked points. Next, we go beyond merely computing intersection numbers and explore the enumerative "world-model" of the Transformers. Through a series of causal inference and correlational interpretability analyses, we demonstrate that Transformers are actually modeling Virasoro constraints in a purely data-driven manner. Additionally, we provide evidence for the comprehension of several values appearing in the large genus asymptotic of $\psi$-class intersection numbers through abductive hypothesis testing.

翻译：Transformer如何建模和学习枚举几何？在数学家-机器协作的溯因知识发现中，使用Transformer的稳健流程是什么？本工作通过分析曲线模空间上的$\psi$类相交数，提出了计算枚举几何的新范式。我们将枚举问题表述为连续优化任务，基于底层量子Airy结构开发了用于计算$\psi$类相交数的Transformer模型。在有限亏格范围内，该模型能够回归横跨极大数值范围（从$10^{-45}$到$10^{45}$）的相交数。为捕捉相交数递归行为提供合适的归纳偏置，我们提出了新型激活函数——动态范围激活器（DRA）。针对$\psi$类相交数严重的异方差性和所需精度要求，我们采用动态滑动窗口的保形预测方法量化预测不确定性，该窗口能感知标记点数量。进一步地，我们超越单纯计算相交数，探索了Transformer的枚举"世界模型"。通过系列因果推断与相关性可解释性分析，证明Transformer实际上以纯数据驱动方式建模Virasoro约束条件。此外，通过溯因假设检验，我们为Transformer理解$\psi$类相交数在大亏格渐近式中出现的若干数值提供了证据。