# Machine-Learning Mathematical Structures

arXiv: https://arxiv.org/abs/2101.06317 · HTML: https://mlsanity.com/paper/2101.06317/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 117 references that are in the map.
- First posted on arXiv 15 Jan 2021.

## Builds on (most cited first, 27 of 27)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [Universal Approximation with Deep Narrow Networks](https://mlsanity.com/paper/1905.08539/) (2019)
- [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [Learning to Unknot](https://mlsanity.com/paper/2010.16263/) (2020)
- [Detecting Symmetries with Neural Networks](https://mlsanity.com/paper/2003.13679/) (2020)
- [Neural Network Approximations for Calabi-Yau Metrics](https://mlsanity.com/paper/2012.15821/) (2020)
- [Machine learning Calabi-Yau metrics](https://mlsanity.com/paper/1910.08605/) (2019)
- [Machine Learning CICY Threefolds](https://mlsanity.com/paper/1806.03121/) (2018)
- [Learning selection strategies in Buchberger's algorithm](https://mlsanity.com/paper/2005.01917/) (2020)
- [The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning](https://mlsanity.com/paper/1812.02893/) (2018)
- [Quiver Mutations, Seiberg Duality and Machine Learning](https://mlsanity.com/paper/2006.10783/) (2020)
- [The world as a neural network](https://mlsanity.com/paper/2008.01540/) (2020)
- [Machine learning for complete intersection Calabi-Yau manifolds: a methodological study](https://mlsanity.com/paper/2007.15706/) (2020)
- [Machine Learning Calabi-Yau Four-folds](https://mlsanity.com/paper/2009.02544/) (2020)
- [Learning Algebraic Structures: Preliminary Investigations](https://mlsanity.com/paper/1905.02263/) (2019)
- [Inception Neural Network for Complete Intersection Calabi-Yau 3-folds](https://mlsanity.com/paper/2007.13379/) (2020)
- [Machine-Learning the Sato--Tate Conjecture](https://mlsanity.com/paper/2010.01213/) (2020)
- [Machine Learning Lie Structures & Applications to Physics](https://mlsanity.com/paper/2011.00871/) (2020)
- [Graph Laplacians, Riemannian Manifolds and their Machine-Learning](https://mlsanity.com/paper/2006.16619/) (2020)
- [Disentangling a Deep Learned Volume Formula](https://mlsanity.com/paper/2012.03955/) (2020)
- [Explore and Exploit with Heterotic Line Bundle Models](https://mlsanity.com/paper/2003.04817/) (2020)
- [Machine-Learning Dessins d'Enfants: Explorations via Modular and Seiberg-Witten Curves](https://mlsanity.com/paper/2004.05218/) (2020)
- [Machine-Learning Number Fields](https://mlsanity.com/paper/2011.08958/) (2020)
- [Machine Learning meets Number Theory: The Data Science of Birch-Swinnerton-Dyer](https://mlsanity.com/paper/1911.02008/) (2019)
- [Machine Learning String Standard Models](https://mlsanity.com/paper/2003.13339/) (2020)

## Built on by (most cited first, 25 of 25)

- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks](https://mlsanity.com/paper/2306.17844/) (2023)
- [Machine Learning Calabi-Yau Hypersurfaces](https://mlsanity.com/paper/2112.06350/) (2021)
- [Neurons on Amoebae](https://mlsanity.com/paper/2106.03695/) (2021)
- [Cluster Algebras: Network Science and Machine Learning](https://mlsanity.com/paper/2203.13847/) (2022)
- [Machine learning detects terminal singularities](https://mlsanity.com/paper/2310.20458/) (2023)
- [Explainable AI Insights for Symbolic Computation: A case study on selecting the variable ordering for cylindrical algebraic decomposition](https://mlsanity.com/paper/2304.12154/) (2023)
- [Machine learning the dimension of a Fano variety](https://mlsanity.com/paper/2309.05473/) (2023)
- [A Triumvirate of AI Driven Theoretical Discovery](https://mlsanity.com/paper/2405.19973/) (2024)
- [The R-mAtrIx Net](https://mlsanity.com/paper/2304.07247/) (2023)
- [Interpretable Graph Networks Formulate Universal Algebra Conjectures](https://mlsanity.com/paper/2307.11688/) (2023)
- [Symbolic Equation Solving via Reinforcement Learning](https://mlsanity.com/paper/2401.13447/) (2024)
- [Machine Learning Algebraic Geometry for Physics](https://mlsanity.com/paper/2204.10334/) (2022)
- [Abstraction, Reasoning and Deep Learning: A Study of the "Look and Say" Sequence](https://mlsanity.com/paper/2109.12755/) (2021)
- [From the String Landscape to the Mathematical Landscape: a Machine-Learning Outlook](https://mlsanity.com/paper/2202.06086/) (2022)
- [Learning a performance metric of Buchberger's algorithm](https://mlsanity.com/paper/2106.03676/) (2021)
- [Machine-Learning the Classification of Spacetimes](https://mlsanity.com/paper/2201.01644/) (2022)
- [Predicting the cardinality and maximum degree of a reduced Gr"obner basis](https://mlsanity.com/paper/2302.05364/) (2023)
- [Applying language models to algebraic topology: generating simplicial cycles using multi-labeling in Wu's formula](https://mlsanity.com/paper/2306.16951/) (2023)
- [Deep learning complete intersection Calabi-Yau manifolds](https://mlsanity.com/paper/2311.11847/) (2023)
- [Constrained Neural Networks for Interpretable Heuristic Creation to Optimise Computer Algebra Systems](https://mlsanity.com/paper/2404.17508/) (2024)
- [SC-Square: Future Progress with Machine Learning?](https://mlsanity.com/paper/2209.04361/) (2022)
- [Testing Transformer Learnability on the Arithmetic Sequence of Rooted Trees](https://mlsanity.com/paper/2512.01870/) (2025)
- [Murmurations: a case study in AI-assisted mathematics](https://mlsanity.com/paper/2603.09680/) (2026)
- [A penalised Saito functional for heuristic search of free line arrangements](https://mlsanity.com/paper/2604.02995/) (2026)

## Abstract (arXiv)

We review, for a general audience, a variety of recent experiments on extracting structure from machine-learning mathematical data that have been compiled over the years. Focusing on supervised machine-learning on labeled data from different fields ranging from geometry to representation theory, from combinatorics to number theory, we present a comparative study of the accuracies on different problems. The paradigm should be useful for conjecture formulation, finding more efficient methods of computation, as well as probing into certain hierarchy of structures in mathematics.