Discovering Ordinary Differential Equations with LLM-Based Qualitative and Quantitative Evaluation

TL;DR

This paper introduces DoLQ, a novel LLM-based multi-agent framework for discovering ordinary differential equations that combines qualitative and quantitative evaluation to improve accuracy and physical plausibility.

Contribution

It presents a multi-agent approach leveraging LLMs for qualitative and quantitative evaluation, enhancing symbolic regression in differential equation discovery.

Findings

DoLQ outperforms existing methods in success rates.

It more accurately recovers symbolic terms of ground truth equations.

The approach demonstrates robustness on multi-dimensional ODE benchmarks.

Abstract

Discovering governing differential equations from observational data is a fundamental challenge in scientific machine learning. Existing symbolic regression approaches rely primarily on quantitative metrics; however, real-world differential equation modeling also requires incorporating domain knowledge to ensure physical plausibility. To address this gap, we propose DoLQ, a method for discovering ordinary differential equations with LLM-based qualitative and quantitative evaluation. DoLQ employs a multi-agent architecture: a Sampler Agent proposes dynamic system candidates, a Parameter Optimizer refines equations for accuracy, and a Scientist Agent leverages an LLM to conduct both qualitative and quantitative evaluations and synthesize their results to iteratively guide the search. Experiments on multi-dimensional ordinary differential equation benchmarks demonstrate that DoLQ achieves superior performance compared to existing methods, not only attaining higher success rates but also more accurately recovering the correct symbolic terms of ground truth equations. Our code is available at https://github.com/Bon99yun/DoLQ.