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

Recovering governing ordinary differential equations (ODEs) from data is a fundamental challenge in scientific modeling. Existing methods often treat this as a static inference problem, where equations are fit to a fixed dataset. However, limited data often leads to identifiability gaps, where multiple structurally distinct equations fit the same data equally well. This paper asks: can symbolic hypotheses themselves guide adaptive data acquisition to resolve these ambiguities and improve the discovery of true governing dynamics?

The authors introduce LLM-ACES (LLM-guided Active Closed-loop Equation Search), a framework that treats dynamical system discovery as a co-evolutionary process between hypothesis construction and data acquisition.

1. LLM-Guided Hypothesis Construction: Instead of generating final equations, an LLM proposes operator-level priors (e.g., specific unary and binary operators) that constrain the search space for a symbolic regression backend.
2. Candidate Population: The system fits multiple candidate equations within these constrained spaces, creating a population that represents uncertainty over the true dynamics.
3. Divergence-Driven Acquisition: The framework identifies regions of the state space where these candidate equations produce the most divergent predictions. It then queries an oracle (a simulator) to obtain trajectories in these informative regions.
4. Feedback Loop: New data is incorporated to re-score candidates, update an experience buffer, and refine the LLM's future operator priors, creating a closed-loop system.

LLM-ACES significantly outperforms state-of-the-art baselines across 122 ODE systems from the ODEBench and ODEBase datasets. It achieves the lowest median normalized mean squared error (NMSE) and demonstrates high symbolic accuracy (46.2% to 52.4%). The authors show that the framework is highly sample-efficient, often requiring only one-tenth of the data used by passive methods. Furthermore, the feedback-driven acquisition makes the model robust to noise, effectively filtering out spurious terms that might otherwise fit the data locally but fail to capture the true underlying physical structure.