深度学习自动发现积分

近期深度学习领域的进展，特别是大型语言模型（LLM）的发展，已显示出AI解决复杂数学问题或编程挑战的能力。然而，基于广泛训练数据解决明确定义问题的能力，与人类科学家式的细致过程存在显著差异。今天高度发达的LLM几乎学习了人类几乎所有可获得的知识，基本上学习的是预测标记序列的能力。它们以类似撰写散文的方式生成数学推导并编写代码， lacks 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 $\int_{0}^{x} t^n dt = \frac{x^{n+1}}{n+1}$.