EPDE-LLM: Equation Discovery with Large Language Models

We formulate differential equation discovery as a code-generation task for Large Language Models (LLMs). This repository introduces a compact, physics-preserving textual representation for fields and their derivatives, and integrates LLMs as oracles within an Evolutionary PDE Discovery (EPDE) meta-learning loop.

Overview

Discovering the underlying equations from observed data is a fundamental scientific challenge. Traditional methods can be computationally expensive or require significant expert input. EPDE-LLM combines the power of evolutionary search (via the EPDE framework) with the symbolic reasoning and code-generation capabilities of modern LLMs to efficiently and accurately identify governing equations.

Features

• LLM as an Oracle: Uses an LLM to propose physically plausible candidate terms for the PDE within an evolutionary algorithm.
• Physics-Preserving Representation: A novel text-based representation for encoding field data and derivatives that guides the LLM towards physically meaningful solutions.
• Meta-Learning Loop: Tight integration with the EPDE framework for robust and efficient equation discovery.
• Extensible: Designed to work with various open-source and proprietary LLMs (e.g., GPT, LLaMA, Claude).

Prerequisites

• Python 3.9
• Conda or pip
• An API key for the LLM service you wish to use (e.g., OpenAI, Anthropic) - if using a proprietary model

Installation

1. Clone the repository:

   git clone --recursive https://github.com/ITMO-NSS-team/EPDE_LLM_NeurIPS2026.git

2. Create and activate a Conda environment:

   conda create -n epde_llm python=3.9 -y
   conda activate epde_llm

3. Install the required dependencies:

   pip install torch==2.3.1+cu118 torchvision==0.18.1+cu118 torchaudio==2.3.1+cu118 -f https://download.pytorch.org/whl/cu118/torch_stable.html
   pip install -r requirements.txt

4. Set your LLM API key:

   Create creds.py file with API key in it. Use creds_example.py as an example.

Basic usage

The following example shows how to use EPDE-LLM.

Choose your approach

There are three main approaches to experiment with:

Option 1. LLM as equation discovery tool:

Uses generative symbolic reasoning where the LLM directly proposes equation structures based on the data representation.

cd pipeline
python pipeline_main.py

Option 2. EPDE baseline experiments:

The EPDE framework optimizes equation structures through evolutionary principles, treating each equation as an individual subject to mutation and crossover.

cd epde_experiments
python burgers.py 
# Other examples: kdv_sindy.py, wave.py, etc.

Option 3. EPDE-LLM framework experiments:

A hybrid approach where the LLM first generates an initial population of candidate equations, which is then refined by the EPDE algorithm using its evolutionary operations for optimal convergence.

cd epde_llm
python epde_llm_main.py

License

This project is licensed under the MIT License - see the LICENSE file for details.