LLM4ED: Large Language Models for Automatic Equation Discovery

TL;DR

This paper presents a novel framework using large language models and natural language prompts to automatically discover physical equations from data, simplifying the process compared to traditional symbolic methods.

Contribution

It introduces a new LLM-based approach with iterative optimization strategies for equation discovery, enhancing stability and usability over existing methods.

Findings

Successfully discovers equations for nonlinear dynamic systems

Demonstrates stability and usability in experiments

Outperforms some state-of-the-art models

Abstract

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.