The practice of mathematics involves discovering patterns and using these to formulate and prove conjectures, resulting in theorems. Since the 1960s, mathematicians have used computers to assist in the discovery of patterns and formulation of conjectures¹, most famously in the Birch and Swinnerton-Dyer conjecture², a Millennium Prize Problem³. Here we provide examples of new fundamental results in pure mathematics that have been discovered with the assistance of machine learning—demonstrating a method by which machine learning can aid mathematicians in discovering new conjectures and theorems. We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures. We outline this machine-learning-guided framework and demonstrate its successful application to current research questions in distinct areas of pure mathematics, in each case showing how it led to meaningful mathematical contributions on important open problems: a new connection between the algebraic and geometric structure of knots, and a candidate algorithm predicted by the combinatorial invariance conjecture for symmetric groups⁴. Our work may serve as a model for collaboration between the fields of mathematics and artificial intelligence (AI) that can achieve surprising results by leveraging the respective strengths of mathematicians and machine learning.

数学实践涉及发现模式并使用这些模式来制定和证明猜想，从而产生定理。自 1960 年代以来，数学家一直在使用计算机来帮助发现猜想¹的模式和公式，最著名的是 Birch 和 Swinnerton-Dyer 猜想²，千年奖问题³. 在这里，我们提供了在机器学习的帮助下发现的纯数学新基本结果的例子——展示了一种机器学习可以帮助数学家发现新猜想和定理的方法。我们提出了一个使用机器学习来发现数学对象之间潜在模式和关系的过程，通过归因技术来理解它们，并使用这些观察来指导直觉并提出猜想。我们概述了这个机器学习引导的框架，并展示了它在纯数学不同领域的当前研究问题中的成功应用，在每种情况下都展示了它如何为重要的开放问题带来有意义的数学贡献：代数和几何结构之间的新联系结，⁴ . 我们的工作可以作为数学和人工智能 (AI) 领域之间合作的模型，通过利用数学家和机器学习的各自优势，可以取得令人惊讶的结果。