In-Context System Identification for Nonlinear Dynamics using Large Language Model

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. The pipeline begins with a baseline SINDy model fit using an adaptive library and then enters a GPT-4-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 forms 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, with a safeguard reverting to the baseline model if the LLM’s proposals underperform a trusted baseline by >5% error. We evaluate this LLM-aided SINDy on 63 dynamical system datasets (ODEBench) and compare against classical SINDy. The results show the LLM-loop consistently improves symbolic recovery: LLM-aided SINDy achieves higher equation similarity to ground truth and lower test RMSE than baseline SINDy for cases with complex dynamics. We also highlight diverse case studies, from exact equation recoveries of logistic growth and harmonic oscillators to partial-success surrogates and outright failures, to illustrate how the LLM contributions differ from manual SINDy. 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.