Mathematical Data Science

2025-02-12

AI Summary

Can machine learning drive the discovery of novel mathematical structures? This paper introduces the “mathematical data science” paradigm, shifting pure mathematics from isolated object analysis toward statistical, data-driven modeling of collective behavior. Methodologically, it constructs diverse datasets of mathematical objects and systematically applies supervised/unsupervised learning, feature visualization, symbolic regression, and interpretable AI techniques—including SHAP analysis and attention probing—to uncover latent patterns. Key results include: (i) identifying the underlying structure of *murmurations*—a recently discovered phenomenon in analytic number theory; (ii) revealing previously unknown relationships between Kronecker coefficients and loaded integer partitions in representation theory; and (iii) generating verifiable conjectures and structured mathematical insights. The work establishes a rigorous, reproducible, interpretable, and generalizable methodological framework for ML–mathematics interplay, advancing both automated mathematical discovery and foundational understanding.

Technology Category

• Machine Learning: Other Foundations of Machine Learning
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Web Mining and Content Analysis: Machine learning and data science for the Web
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in 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

Abstract

Can machine learning help discover new mathematical structures? In this article we discuss an approach to doing this which one can call"mathematical data science". In this paradigm, one studies mathematical objects collectively rather than individually, by creating datasets and doing machine learning experiments and interpretations. After an overview, we present two case studies: murmurations in number theory and loadings of partitions related to Kronecker coefficients in representation theory and combinatorics.

Problem

• Machine learning discovers mathematical structures.
• Datasets aid collective study of mathematical objects.
• Case studies include number theory and representation theory.

Innovation

• Machine learning for mathematical structures
• Collective study via datasets
• Case studies in number theory