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

Sum Kyun Song, Bong Gyun Shin, Jae Yong Lee
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:115029-115067, 2026.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-song26x, title = {Discovering Ordinary Differential Equations with {LLM}-Based Qualitative and Quantitative Evaluation}, author = {Song, Sum Kyun and Shin, Bong Gyun and Lee, Jae Yong}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {115029--115067}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/song26x/song26x.pdf}, url = {https://proceedings.mlr.press/v306/song26x.html}, 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.} }
Endnote
%0 Conference Paper %T Discovering Ordinary Differential Equations with LLM-Based Qualitative and Quantitative Evaluation %A Sum Kyun Song %A Bong Gyun Shin %A Jae Yong Lee %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-song26x %I PMLR %P 115029--115067 %U https://proceedings.mlr.press/v306/song26x.html %V 306 %X 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.
APA
Song, S.K., Shin, B.G. & Lee, J.Y.. (2026). Discovering Ordinary Differential Equations with LLM-Based Qualitative and Quantitative Evaluation. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:115029-115067 Available from https://proceedings.mlr.press/v306/song26x.html.

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