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

Researchers introduce DoLQ, a new method that combines large language models with symbolic regression to discover ordinary differential equations from observational data. The approach integrates both qualitative physical reasoning and quantitative metrics through a multi-agent architecture, demonstrating superior performance over existing methods in recovering accurate symbolic equations.

DoLQ represents a meaningful advancement in scientific machine learning by addressing a critical gap in equation discovery systems. Traditional symbolic regression methods optimize purely for numerical accuracy, often producing equations that lack physical plausibility or violate domain constraints. This research bridges that divide by incorporating LLM-based qualitative evaluation alongside quantitative metrics, enabling systems to reason about whether discovered equations make physical sense.

The multi-agent architecture demonstrates how different AI components can collaborate effectively. The Sampler Agent generates candidates, the Parameter Optimizer refines numerical accuracy, and the Scientist Agent—powered by an LLM—conducts comprehensive evaluation by synthesizing both mathematical correctness and domain knowledge. This design pattern reflects growing recognition that complex scientific problems benefit from heterogeneous AI systems rather than monolithic approaches.

For the scientific computing community, DoLQ offers practical improvements in discovering governing equations for physical systems, which has applications across physics, engineering, biology, and chemistry. Higher success rates and more accurate symbolic recovery reduce the need for manual validation and refinement by domain experts, potentially accelerating the research cycle.

The work signals expanding capabilities at the intersection of symbolic AI and LLMs. As these systems become more reliable at reasoning about mathematical and physical constraints simultaneously, they may enable faster hypothesis generation in scientific discovery. Future developments likely focus on scaling these methods to higher-dimensional systems, handling noisy real-world data more robustly, and extending beyond differential equations to other mathematical structures.

• →DoLQ integrates LLM-based qualitative reasoning with quantitative metrics to discover physically plausible differential equations from data.
• →The multi-agent architecture separates sampling, optimization, and evaluation tasks to improve both accuracy and symbolic term recovery.
• →Superior performance over existing symbolic regression methods suggests LLMs can effectively validate domain knowledge constraints in scientific discovery.
• →The approach addresses a practical gap where mathematically accurate equations may be physically implausible without domain-aware evaluation.
• →Open-source availability enables broader adoption across scientific computing communities.