This paper presents a comprehensive overview on the applications of artificial intelligence (AI) in mathematical research, highlighting the transformative role AI has begun to play in this domain. Traditionally, AI advancements have heavily relied on theoretical foundations provided by mathematics and statistics. However, recent developments in AI, particularly in reinforcement learning (RL) and large language models (LLMs), have demonstrated the potential for AI to contribute back to mathematics by offering flexible algorithmic frameworks and powerful inductive reasoning capabilities that support various aspects of mathematical research. This survey aims to establish a bridge between AI and mathematics, providing insights into the mutual benefits and fostering deeper interdisciplinary understanding. In particular, we argue that while current AI and LLMs may struggle with complex deductive reasoning, their "inherent creativity", the ability to generate outputs at high throughput based on recognition of shallow patterns, holds significant potential to support and inspire mathematical research. This creative capability, often overlooked, could be the key to unlocking new perspectives and methodologies in mathematics. Furthermore, we address the lack of cross-disciplinary communication: mathematicians may not fully comprehend the latest advances in AI, while AI researchers frequently prioritize benchmark performance over real-world applications in frontier mathematical research. This paper seeks to close that gap, offering a detailed exploration of AI fundamentals, its strengths, and its emerging applications in the mathematical sciences.

翻译：本文全面综述了人工智能（AI）在数学研究中的应用，重点阐述了AI在该领域已开始发挥的变革性作用。传统上，AI的进步在很大程度上依赖于数学和统计学提供的理论基础。然而，AI的最新发展，特别是在强化学习（RL）和大语言模型（LLMs）方面，已展现出AI通过提供灵活的算法框架和强大的归纳推理能力来回馈数学研究的潜力，这些能力支持着数学研究的各个方面。本综述旨在建立AI与数学之间的桥梁，提供对双方互益关系的见解，并促进更深入的跨学科理解。我们特别指出，尽管当前的AI和LLMs可能在处理复杂的演绎推理方面存在困难，但其“内在创造力”——即基于对浅层模式的识别以高吞吐量生成输出的能力——在支持和启发数学研究方面具有巨大潜力。这种常被忽视的创造性能力，可能是解锁数学新视角和方法的关键。此外，我们探讨了跨学科交流的不足：数学家可能未能完全理解AI的最新进展，而AI研究人员则常常优先考虑基准性能，而非前沿数学研究中的实际应用。本文旨在弥合这一差距，对AI的基础原理、其优势及其在数学科学中的新兴应用进行了详细探讨。