  Equation discovery is aimed at directly extracting physical laws from data
and has emerged as a pivotal research domain. Previous methods based on
symbolic mathematics have achieved substantial advancements, but often require
the design of implementation of complex algorithms. In this paper, we introduce
a new framework that utilizes natural language-based prompts to guide large
language models (LLMs) in automatically mining governing equations from data.
Specifically, we first utilize the generation capability of LLMs to generate
diverse equations in string form, and then evaluate the generated equations
based on observations. In the optimization phase, we propose two alternately
iterated strategies to optimize generated equations collaboratively. The first
strategy is to take LLMs as a black-box optimizer and achieve equation
self-improvement based on historical samples and their performance. The second
strategy is to instruct LLMs to perform evolutionary operators for global
search. Experiments are extensively conducted on both partial differential
equations and ordinary differential equations. Results demonstrate that our
framework can discover effective equations to reveal the underlying physical
laws under various nonlinear dynamic systems. Further comparisons are made with
state-of-the-art models, demonstrating good stability and usability. Our
framework substantially lowers the barriers to learning and applying equation
discovery techniques, demonstrating the application potential of LLMs in the
field of knowledge discovery.
