Dynamical System Discovery from Specifications and Partial Observations

Dynamical system discovery is often framed as fitting ordinary differential equations to observed trajectories. In many applications, however, the desired model is defined not only by prediction error but also by a *task specification*: a natural-language description of task-critical mechanisms that the model must represent. In postprandial glucose modeling, for example, a meal response simulator must represent how ingested carbohydrates enter the bloodstream over time; a post-meal decision support model must distinguish meal absorption from delayed insulin action; and a physiological model must expose hepatic regulation. These task-critical mechanisms are often not measured, creating a mismatch between what the data observe and what the model must represent. We study this setting as *task-specified dynamical system discovery* under *specification-induced partial observability*. We propose a framework that treats the state specification itself as an object of discovery. Given a task description and partially observed time series, the framework searches over observed and latent states, analytic dynamics, observation mappings, and causal initialization rules. A proposer large language model (LLM) constructs and revises candidate models through a graph meta model that represents dependencies. A deterministic verifier checks model validity and executable task requirements, while a separate LLM critic supplies scientific feedback for revision. Continuous parameters are fitted on training trajectories using collocation initialization and rollout refinement. Validation performance guides candidate selection and acceptance of pruning. Experiments demonstrate that compared to baselines, our method more reliably recovers task-critical mechanisms while achieving better prediction in unseen, held-out interventions.