Recent advancements in the realm of deep learning, particularly in the development of large language models (LLMs), have demonstrated AI's ability to tackle complex mathematical problems or solving programming challenges. However, the capability to solve well-defined problems based on extensive training data differs significantly from the nuanced process of making scientific discoveries. Trained on almost all human knowledge available, today's sophisticated LLMs basically learn to predict sequences of tokens. They generate mathematical derivations and write code in a similar way as writing an essay, and do not have the ability to pioneer scientific discoveries in the manner a human scientist would do. In this study we delve into the potential of using deep learning to rediscover a fundamental mathematical concept: integrals. By defining integrals as area under the curve, we illustrate how AI can deduce the integral of a given function, exemplified by inferring $\int_{0}^{x} t^2 dt = \frac{x^3}{3}$ and $\int_{0}^{x} ae^{bt} dt = \frac{a}{b} e^{bx} - \frac{a}{b}$. Our experiments show that deep learning models can approach the task of inferring integrals either through a sequence-to-sequence model, akin to language translation, or by uncovering the rudimentary principles of integration, such as $\int_{0}^{x} t^n dt = \frac{x^{n+1}}{n+1}$.

翻译：近年来深度学习领域的最新进展，特别是大型语言模型（LLMs）的发展，已经证明了人工智能解决复杂数学问题或编程挑战的能力。然而，基于海量训练数据解决明确问题的能力，与进行科学发现这一微妙过程存在本质区别。当前精密的LLMs在几乎所有人类知识上训练，本质上学会了预测令牌序列——它们生成数学推导和编写代码的方式与撰写文章类似，因此不具备人类科学家那种开拓性科学发现的能力。本研究深入探索了利用深度学习重新发现基础数学概念——积分的潜力。通过将积分定义为曲线下的面积，我们展示了AI如何推断给定函数的积分，例如推导出$\int_{0}^{x} t^2 dt = \frac{x^3}{3}$和$\int_{0}^{x} ae^{bt} dt = \frac{a}{b} e^{bx} - \frac{a}{b}$。实验表明，深度学习模型可通过两种方式完成积分的推断任务：一是采用类似语言翻译的序列到序列模型，二是通过发现积分基本原理（如$\int_{0}^{x} t^n dt = \frac{x^{n+1}}{n+1}$）。