Automated Discovery of Integral with Deep Learning
Xiaoxin Yin
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 $\int_{0}^{x} t^n dt = \frac{x^{n+1}}{n+1}$.