LLM4ED: Large Language Models for Automatic Equation Discovery

🌈 Abstract

The paper introduces a new framework that utilizes natural language-based prompts to guide large language models (LLMs) in automatically mining governing equations from data. The key points are:

• The framework first generates diverse equations in string form using the generation capability of LLMs, and then evaluates the generated equations based on observations.
• Two alternately iterated optimization strategies are proposed - self-improvement based on historical samples and their performance, and evolutionary search guided by LLMs.
• Experiments on both partial differential equations (PDEs) and ordinary differential equations (ODEs) demonstrate that the framework can effectively discover governing equations to reveal underlying physical laws.
• The framework substantially lowers the barriers to learning and applying equation discovery techniques, showcasing the application potential of LLMs in knowledge discovery.

🙋 Q&A

[01] Introduction

1. What are the main challenges in traditional equation discovery methods?

• Traditional equation discovery methods based on symbolic mathematics often require the design and implementation of complex algorithms.
• They rely heavily on prior physical knowledge, which constrains their applicability to uncover more intricate representational forms.

2. How does the proposed framework address these challenges?

• The framework utilizes natural language-based prompts to guide LLMs in automatically generating and optimizing equations.
• This eliminates the need for manually crafting intricate programs for equation generators and optimizers, making the process more accessible.
• The framework is parametric-free during optimization, leveraging the generation and reasoning capabilities of LLMs.

3. What are the key components of the proposed framework?

• Equation generation using LLMs
• Evaluation of generated equations based on observations
• Two optimization strategies:
  1. Self-improvement based on historical samples and their performance
  2. Evolutionary search guided by LLMs

[02] Related Works

1. What are the main phases in traditional symbolic equation discovery methods?

• Generation: Equations are transformed into expression trees based on context-free grammars.
• Evaluation: The performance of the discovered equations is assessed in terms of fit to data and complexity.
• Optimization: Algorithms like genetic programming, gradient descent, and reinforcement learning are used to optimize the equations.

2. How does the proposed framework differ from traditional symbolic equation discovery methods?

• The proposed framework utilizes natural language-based prompts to guide LLMs in the generation and optimization of equations, eliminating the need for manually designed algorithms.
• This significantly streamlines the process and makes it more accessible to researchers, allowing them to focus on the evaluation aspect where domain expertise is crucial.

3. How have LLMs been applied in optimization problems?

• LLMs have been used as direct optimizers in a self-improvement manner, iteratively refining candidate solutions based on problem definitions and historical results.
• LLMs have also been combined with evolutionary search methods, where prompts are used to guide LLMs in executing evolutionary algorithms to enhance existing solutions.

[03] Methods

1. How does the proposed framework generate the initial equation population?

• The initial equations are generated through LLMs based on prompts that include a symbol library and problem descriptions.
• Constraints can be established using natural language to prevent the generation of equations that violate specified conditions.

2. How are the generated equations evaluated?

• For PDEs, the constants in the equations are determined using sparse regression.
• For ODEs, the constants are optimized using the BFGS algorithm.
• A score function is designed to evaluate the performance of the generated equations, considering both the fit to data and the complexity of the equations.

3. What are the two optimization strategies employed in the framework?

1. Self-improvement: LLMs are used to perform local modifications to the historical elite equations based on their performance.
2. Evolutionary search: LLMs are guided to execute crossover and mutation operations on the elite equations to generate more diverse equation combinations.

4. How do the two optimization strategies work together?

• The alternating iterative approach of the two strategies effectively strikes a balance between exploration and exploitation.
• The self-improvement method focuses on local modifications, while the evolutionary search enables global search, complementing each other to improve the optimization process.

[04] Results

1. How did the framework perform on the PDE discovery tasks?

• The framework was able to accurately identify the correct structure of the equations for various canonical nonlinear PDE systems, while maintaining minimal coefficient errors.
• Compared to fixed candidate set methods, the framework reduced the dependence on prior knowledge, enabling the discovery of more complex equation forms.

2. How did the different optimization strategies compare in the PDE discovery tasks?

• The alternating iterative approach combining self-improvement and evolutionary search outperformed the individual strategies, achieving the highest frequency of discovering the correct equations.
• The self-improvement method exhibited higher optimization efficiency in some systems but was more prone to converging to local optima, while the evolutionary search demonstrated superior global optimization capability.

3. How did the framework perform on the ODE discovery tasks?

• On a comprehensive benchmark of 16 one-dimensional ODEs, the framework achieved comparable or better performance compared to state-of-the-art symbolic regression methods.
• The percentage of equations with R-squared greater than 0.99 on the training set was 93.75%, and on the test set with a new initial condition, it was 68.75%.
• The framework's performance improved as the capability of the large language model increased, demonstrating the impact of the model's capacity on the generation and optimization of equations.