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

2026-05-08

AI Summary

This study addresses the problem of discovering governing ordinary differential equations (ODEs) from observational data, balancing physical plausibility with mathematical fidelity. To this end, the authors propose DoLQ, a novel method that introduces large language models (LLMs) into the equation discovery task for the first time, employing a multi-agent collaborative framework. Within this framework, a Sampler Agent generates candidate equations, a Parameter Optimizer refines their coefficients, and a Scientist Agent leverages an LLM to integrate domain knowledge–driven qualitative assessments with data-driven quantitative evaluations, iteratively guiding the search process. Experimental results demonstrate that DoLQ significantly outperforms existing approaches on multidimensional ODE benchmarks, achieving higher success rates and recovering the true symbolic structure of the underlying equations with greater accuracy.

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.

Problem

• ordinary differential equations
• symbolic regression
• scientific machine learning
• physical plausibility
• domain knowledge

Innovation

• symbolic regression
• ordinary differential equations
• large language models
• multi-agent architecture
• scientific machine learning