LLM-ACES: Closed-Loop Discovery of Dynamical Systems with LLM-Guided Adaptive Search

2026-06-23

AI Summary

This work addresses the challenge of insufficient identifiability in recovering ordinary differential equations (ODEs) of dynamical systems from limited observational data, which often leads to erroneous governing equations. The authors propose LLM-ACES, a novel framework that uniquely integrates large language models (LLMs) with active learning: LLMs generate symbolic hypotheses and operator priors, while an uncertainty-driven active sampling strategy adaptively acquires trajectory data, forming a closed-loop optimization mechanism that iteratively refines equation discovery. Evaluated on 122 systems from ODEBench and ODEBase, the method achieves median normalized mean squared errors (NMSE) lower by several orders of magnitude using only one-tenth of the data required by existing approaches, with symbolic recovery accuracies of 46.2% and 52.4%, respectively, demonstrating substantially improved accuracy and robustness under sparse and noisy data conditions.

Abstract

Recovering governing Ordinary Differential Equations (ODEs) from data is a central challenge in modeling dynamical systems across scientific domains. Existing approaches cast discovery as a static inference problem over fixed datasets, assuming that the observed trajectories are sufficiently informative. However, dynamical systems evolve over large state spaces, and limited data can make multiple equations observationally indistinguishable, leading to identifiability gaps and the recovery of incorrect governing equations. To address this, we introduce LLM-ACES, or LLM-guided Active Closed-loop Equation Search, a closed-loop framework that jointly optimizes symbolic hypothesis construction and adaptive data acquisition. In LLM-ACES, a large language model (LLM) proposes operator priors that partition the large search space into distinct regions, within which candidate equations are fit to the observed data. The disagreement among these candidates guides the acquisition of informative trajectories, creating a feedback loop that iteratively refines both the hypothesis space and the discovered dynamics. On 122 ODE systems spanning ODEBench and ODEBase, LLM-ACES achieves the lowest median NMSE, outperforming state-of-the-art baselines by several orders of magnitude while achieving a high symbolic accuracy of 46.2% and 52.4%, respectively. Our analysis further shows that LLM-ACES is sample-efficient, achieving better performance with one-tenth the data. Furthermore, LLM-ACES's feedback-driven data acquisition makes it robust to noise and recovers the correct symbolic structure, while baselines introduce spurious terms that fit the data locally but obscure the true governing relationships.

Problem

• dynamical systems
• governing equations
• identifiability
• ordinary differential equations
• data-driven discovery

Innovation

• LLM-guided discovery
• active learning
• closed-loop optimization
• symbolic regression
• dynamical systems identification