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

Jesse He, Helen Jenne, Herman Chau, Davis Brown, Mark Raugas, Sara C. Billey, Henry Kvinge
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:22434-22455, 2025.

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 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? 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 general rules from mathematical data.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-he25g, title = {Machines and Mathematical Mutations: Using {GNN}s to Characterize Quiver Mutation Classes}, author = {He, Jesse and Jenne, Helen and Chau, Herman and Brown, Davis and Raugas, Mark and Billey, Sara C. and Kvinge, Henry}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {22434--22455}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/he25g/he25g.pdf}, url = {https://proceedings.mlr.press/v267/he25g.html}, 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 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? 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 general rules from mathematical data.} }
Endnote
%0 Conference Paper %T Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes %A Jesse He %A Helen Jenne %A Herman Chau %A Davis Brown %A Mark Raugas %A Sara C. Billey %A Henry Kvinge %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-he25g %I PMLR %P 22434--22455 %U https://proceedings.mlr.press/v267/he25g.html %V 267 %X 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? 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 general rules from mathematical data.
APA
He, J., Jenne, H., Chau, H., Brown, D., Raugas, M., Billey, S.C. & Kvinge, H.. (2025). Machines and Mathematical Mutations: Using GNNs to Characterize Quiver Mutation Classes. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:22434-22455 Available from https://proceedings.mlr.press/v267/he25g.html.

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