The practice of mathematics involves discovering patterns and using these to  数学实践涉及到发现模式并将其用于  formulate and prove conjectures, resulting in theorems. Since the 1960s,  提出并证明猜想，从而得出定理。自20世纪60年代以来，  mathematicians have used computers to assist in the discovery of patterns and  数学家们用计算机来帮助发现模式和规律  formulation of conjectures1, most famously in the Birch and Swinnerton-Dyer  猜测公式1，最著名的是Birch和Swinnerton Dyer  conjecture2, a Millennium Prize Problem3. Here we provide examples of new  猜想2，千年奖问题3。这里我们提供了新的  fundamental results in pure mathematics that have been discovered with the  纯数学中的基本结果是用  assistance of machine learning—demonstrating a method by which machine learning  机器学习辅助：演示一种机器学习方法  can aid mathematicians in discovering new conjectures and theorems. We propose a  可以帮助数学家发现新的猜想和定理。我们建议  process of using machine learning to discover potential patterns and relations  使用机器学习发现潜在模式和关系的过程  between mathematical objects, understanding them with attribution techniques and  在数学对象之间，使用归因技术和  using these observations to guide intuition and propose conjectures. We outline this  利用这些观察来引导直觉并提出推测。我们概述了这一点  machine-learning-guided framework and demonstrate its successful application to  机器学习指导框架，并演示其在以下方面的成功应用：  current research questions in distinct areas of pure mathematics, in each case  在每种情况下，纯数学不同领域的当前研究问题  showing how it led to meaningful mathematical contributions on important open  展示了它如何在重要的开放式教学中产生有意义的数学贡献  problems: a new connection between the algebraic and geometric structure of knots,  问题：节点的代数结构和几何结构之间的新联系，  and a candidate algorithm predicted by the combinatorial invariance conjecture for  以及由组合不变性猜想预测的候选算法  symmetric groups4. Our work may serve as a model for collaboration between the  对称组4.我们的工作可以作为团队之间协作的模型  fields of mathematics and artificial intelligence (AI) that can achieve surprising results  数学和人工智能（AI）领域，可以取得令人惊讶的结果  by leveraging the respective strengths of mathematicians and machine learning.  通过利用数学家和机器学习各自的优势。  One of the central drivers of mathematical progress is the discovery  数学进步的核心动力之一是发现  of patterns and formulation of useful conjectures: statements that  模式和有用猜想的表述：声明  are suspected to be true but have not been proven to hold in all cases.  怀疑为真，但尚未证明在所有情况下都成立。  Mathematicians have always used data to help in this process—from  数学家总是使用数据来帮助这一过程，从  the early hand-calculated prime tables used by Gauss and others that  高斯和其他人使用的早期手工计算素数表  led to the prime number theorem5, to modern computer-generated  导致素数理论5，现代计算机生成  data1,5 in cases such as the Birch and Swinnerton-Dyer conjecture2.  在Birch和Swinnerton-Dyer猜想2等情况下，数据1,5。  The introduction of computers to generate data and test conjectures  引入计算机生成数据和测试猜想  afforded mathematicians a new understanding of problems that were  给数学家们提供了一个新的理解问题的方法  previously inaccessible6, but while computational techniques have  以前是不可能的6，但计算技术已经  become consistently useful in other parts of the mathematical process7,8,  在数学过程7、8的其他部分中变得一致有用，  artificial intelligence (AI) systems have not yet established a  人工智能（AI）系统尚未建立  similar place. Prior systems for generating conjectures have either  类似的地方。先前用于生成猜测的系统具有以下两种类型：  contributed genuinely useful research conjectures9 via methods that  通过以下方法贡献了真正有用的研究猜想9：  do not easily generalize to other mathematical areas10, or have demonstrated  不容易推广到其他数学领域10，或已证明  novel, general methods for finding conjectures11 that have  用于发现具有以下特征的猜想11的新的通用方法  not yet yielded mathematically valuable results.  尚未产生数学上有价值的结果。  AI, in particular the field of machine learning12–14, offers a collection  人工智能，特别是机器学习领域12-14，提供了一个集合  of techniques that can effectively detect patterns in data and has  能够有效检测数据中的模式的技术  increasingly demonstrated utility in scientific disciplines15. In mathematics,  在科学学科中越来越显示出实用性15.在数学中，  it has been shown that AI can be used as a valuable tool by  已经证明，人工智能可以通过以下方式作为一种有价值的工具：  finding counterexamples to existing conjectures16, accelerating calculations17,  找到现有猜想的反例16、加速计算17，  generating symbolic solutions18 and detecting the existence  生成符号解18并检测其存在性  of structure in mathematical objects19. In this work, we demonstrate  在这项工作中，我们演示了  that AI can also be used to assist in the discovery of theorems and conjectures  人工智能也可以用来帮助发现定理和猜想  at the forefront of mathematical research. This extends work  处于数学研究的前沿。这扩展了工作  using supervised learning to find patterns20–24 by focusing on enabling  通过专注于启用，使用监督学习查找模式20–24  mathematicians to understand the learned functions and derive useful  数学家理解所学函数并推导出有用的  mathematical insight. We propose a framework for augmenting the  数学洞察力。我们提出了一个框架来增强  standard mathematician’s toolkit with powerful pattern recognition  具有强大模式识别功能的标准数学家工具包  and interpretation methods from machine learning and demonstrate  以及机器学习和演示的解释方法  its value and generality by showing how it led us to two fundamental  它的价值和一般性，通过展示它如何引导我们实现两个基本点：  new discoveries, one in topology and another in representation theory.  新发现，一个在拓扑学，另一个在表示理论。  Our contribution shows how mature machine learning methodologies  我们的贡献展示了机器学习方法的成熟程度  can be adapted and integrated into existing mathematical workflows  可以调整并集成到现有的数学工作流中  to achieve novel results.  以获得新颖的结果。  Guiding mathematical intuition with AI  用人工智能引导数学直觉  A mathematician’s intuition plays an enormously important role in  数学家的直觉在以下方面起着极其重要的作用：  mathematical discovery—“It is only with a combination of both rigorous  数学发现—“它只是两者的结合，严谨  formalism and good intuition that one can tackle complex mathematical  形式主义和良好的直觉可以解决复杂的数学问题  problems”25. The following framework, illustrated in Fig. 1,  图1所示的以下框架，  describes a general method by which mathematicians can use tools  描述数学家使用工具的一般方法  from machine learning to guide their intuitions concerning complex  从机器学习到指导他们关于复杂的直觉  mathematical objects, verifying their hypotheses about the existence  数学对象，验证其关于存在的假设  of relationships and helping them understand those relationships. We  并帮助他们理解这些关系。我们  propose that this is a natural and empirically productive way that these  提出这是一种自然且经验性的生产方式