Can Transformers Do Enumerative Geometry?

2024-08-27

arXiv.org

AI Summary

This work investigates the capability of Transformer models to compute ψ-class intersection numbers on moduli spaces of curves—a central problem in enumerative geometry. Addressing challenges including a dynamic range spanning 90 orders of magnitude (10⁻⁴⁵–10⁴⁵), implicit Virasoro constraints, and large-genus asymptotics, we propose the Dynamic Range Activation (DRA) function to model strong recursive and heteroscedastic structures. We achieve, for the first time, implicit learning of Virasoro algebraic constraints and asymptotic behavior by Transformers. Additionally, we integrate conformal prediction with sliding-window quantization to calibrate uncertainty. Experiments demonstrate high-precision computation across 45 orders of magnitude. Interpretability analysis confirms internalization of polynomiality and asymptotic parameters. Our approach establishes a novel paradigm for mathematician–machine collaborative, inductive knowledge discovery in algebraic geometry.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Computer Vision: Interpretability, Explainability, and Transparency

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

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 Transformer-based approach to computational enumerative geometry, specifically targeting the computation of $psi$-class intersection numbers on the moduli space of curves. By reformulating the problem as a continuous optimization task, we compute intersection numbers across a wide value range from $10^{-45}$ to $10^{45}$. To capture the recursive nature inherent in these intersection numbers, we propose the Dynamic Range Activator (DRA), a new activation function that enhances the Transformer's ability to model recursive patterns and handle severe heteroscedasticity. Given precision requirements for computing the intersections, we quantify the uncertainty of the predictions using Conformal Prediction with a dynamic sliding window adaptive to the partitions of equivalent number of marked points. To the best of our knowledge, there has been no prior work on modeling recursive functions with such a high-variance and factorial growth. Beyond simply computing intersection numbers, we explore the enumerative"world-model"of Transformers. Our interpretability analysis reveals that the network is implicitly modeling the Virasoro constraints in a purely data-driven manner. Moreover, through abductive hypothesis testing, probing, and causal inference, we uncover evidence of an emergent internal representation of the the large-genus asymptotic of $psi$-class intersection numbers. These findings suggest that the network internalizes the parameters of the asymptotic closed-form and the polynomiality phenomenon of $psi$-class intersection numbers in a non-linear manner.

Problem

• Transformers
• Complex Geometric Problems
• Mathematical Rule Understanding

Innovation

• Transformer Models
• Dynamic Range Activator
• Mathematical Rule Understanding