In-Context System Identification for Nonlinear Dynamics Using Large Language Models

Abstract

Sparse Identification of Nonlinear Dynamics (SINDy) is a powerful method for discovering parsimonious governing equations from data, but it often requires expert tuning of candidate libraries. We propose an LLM-aided SINDy pipeline that iteratively refines candidate equations using a large language model (LLM) in the loop through in-context learning. The pipeline begins with a baseline SINDy model fit using an adaptive library and then enters a LLM-guided refinement cycle. At each iteration, the current best equations, error metrics, and domain-specific constraints are summarized in a prompt to the LLM, which suggests new equation structures. These candidate equations are parsed against a defined symbolic form and evaluated on training and test data. The pipeline uses simulation-based error as a primary metric, but also assesses structural similarity to ground truth, including matching functional forms, key terms, couplings, qualitative behavior. An iterative stopping criterion ends refinement early if test error falls below a threshold (NRMSE < 0.1) or if a maximum of 10 iterations is reached. Finally, the best model is selected, and we evaluate this LLM-aided SINDy on 63 dynamical system datasets (ODEBench) and march leuba model for boiling nuclear reactor. The results are compared against classical SINDy and show the LLM-loop consistently improves symbolic recovery with higher equation similarity to ground truth and lower test RMSE than baseline SINDy for cases with complex dynamics. This work demonstrates that an LLM can effectively guide SINDy's search through equation space, integrating data-driven error feedback with domain-inspired symbolic reasoning to discover governing equations that are not only accurate but also structurally interpretable.