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

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 *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 *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? Currently, this question has only been resolved in specific cases. In this paper we use graph neural networks and AI explainability techniques to discover mutation equivalence criteria for the previously unknown case of quivers of type $\tilde{D}_n$. Along the way we also show that even without explicit training to do so, our model captures structure within its hidden representation that allow us to reconstruct known criteria from type $D_n$, adding to the growing evidence that modern machine learning models are capable of learning abstract and general rules from mathematical data.