Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes

TL;DR

This paper employs graph neural networks to analyze quiver mutation classes in cluster algebras, successfully identifying mutation equivalence criteria and revealing that models learn underlying mathematical structures.

Contribution

It introduces a novel application of GNNs to characterize mutation classes in quiver theory and demonstrates models can learn and reconstruct complex mathematical criteria.

Findings

GNNs can determine mutation equivalence in quivers.

Models learn and encode known mathematical criteria.

Unsupervised features reveal underlying mathematical structures.

Abstract

Machine learning is becoming an increasingly valuable tool in mathematics, enabling one to identify subtle patterns across collections of examples so vast that they would be impossible for a single researcher to feasibly review and analyze. In this work, we use graph neural networks to investigate \emph{quiver mutation} -- an operation that transforms one quiver (or directed multigraph) into another -- which is central to the theory of cluster algebras with deep connections to geometry, topology, and physics. In the study of cluster algebras, the question of \emph{mutation equivalence} is of fundamental concern: given two quivers, can one efficiently determine if one quiver can be transformed into the other through a sequence of mutations? In this paper, we use graph neural networks and AI explainability techniques to independently discover mutation equivalence criteria for quivers of type $\tilde{D}$. Along the way, we also show that even without explicit training to do so, our model captures structure within its hidden representation that allows us to reconstruct known criteria from type $D$, adding to the growing evidence that modern machine learning models are capable of learning abstract and parsimonious rules from mathematical data.