LLM-Guided ODE Discovery and Parameter Inference from Small-Cohort Aggregate Data
Authors: Hanning Yang, Meropi Karakioulaki, Lennart Purucker, Tim Litwin, Cristina Has, Moritz Hess
Organizations: Institute of Medical Biometry and Statistics, Faculty of Medicine and Medical Center, University of Freiburg, Germany · Department of Dermatology, Medical Faculty and Medical Center, University of Freiburg, Germany · Prior Labs, University of Freiburg, Germany
Abstract
Mechanistic modeling via ordinary differential equations (ODEs) provides interpretable descriptions of complex dynamics and enables inference of underlying mechanisms, which is particularly valuable in clinical settings. However, in rare diseases, both the structure and parameters of the model are typically unknown, while individual-level data is scarce, noisy, heterogeneous, and subject to privacy constraints. In such settings, population-level summary statistics provide a practical privacy-preserving data representation, while capturing heterogeneity further requires modeling parameters as distributions rather than fixed values. Yet no existing method jointly discovers ODE structure and refines parameter distributions solely from summary statistics. We present AgentODE, an end-to-end framework that addresses this gap. An LLM proposes candidate ODE structures, while a tool-augmented inference agent iteratively refines parameter distributions through a diagnosis--update loop, operating on population-level summary statistics alone. We evaluate AgentODE on three benchmark problems across different fields and two clinical datasets, including the rare disease recessive dystrophic epidermolysis bullosa (RDEB), with only 231 observations across 46 patients. AgentODE recovers functionally consistent ODE structures across all settings, and experiments on RDEB demonstrates that in sparse and noisy data settings reasoning from summary statistics promotes mechanistically principled structure discovery, whereas baselines with individual-level data access recover implausible structures despite better predictive performance. AgentODE opens new possibilities for mechanistic modeling of rare diseases directly from population-level summary statistics, where data scarcity and privacy constraints have traditionally limited such analyses.
Automatic scientific discovery has long been a goal of computational scholars - a machine that can discover nature's secrets on its own, moving computational systems beyond data-fitting tools toward the generation and refinement of mechanistic models of the universe. Recent advances in symbolic regression (SR) and large-language-model (LLM)-based agents suggest that such systems can recover equations from data, incorporate domain priors, and automate parts of the research workflow. However, most existing approaches either focus on narrow equation-discovery benchmarks or broad end-to-end automation pipelines, while biological systems remain comparatively underexplored. Here, we introduce the MEDA system, an LLM- and SR-powered agentic framework for discovering ordinary-differential-equation (ODE) models of biological and biologically inspired dynamical systems. MEDA retrieves background knowledge, defines admissible variables, generates mechanistic constraints, proposes candidate ODEs, and fits and evaluates them. We evaluate it across canonical model retrieval, reasoning-based extrapolation to unseen variants, and open-ended discovery, with and without experimental data. Across these settings, MEDA recovered the correct state variables, achieved strong structural recovery in retrieval and extrapolation tasks, and produced biologically plausible discovery-oriented models. Ablation and robustness analyses show that knowledge-guided formalization and mechanistic constraints are load-bearing components, whereas numerical fitting alone can preserve trajectory-compatible but biologically incorrect equations.
Rare disease diagnosis depends on expert reasoning that is scarce and difficult to transfer. Large language models rank the correct disease first in only 35.4% of benchmark cases and often rely on learned phenotype-disease associations rather than reusable diagnostic reasoning strategies. We developed liteOdyssey through Policy Iteration with Human Feedback, a process in which model failures and expert corrections are iteratively consolidated into a clinician-gated, natural-language policy executed by a language model. Across 1,243 public benchmark cases spanning 722 rare diseases, liteOdyssey ranked the correct disease first in 59.3% of cases versus 26.5% without the policy, with comparable gains in cases involving diseases excluded from policy development. The same policy transferred across model families and sizes without retraining. Adaptation of the policy to the Undiagnosed Diseases Network (UDN) improved diagnostic accuracy among 515 UDN patients, with gains confirmed by blinded physician adjudication. These results show that expert reasoning can be externalized into an inspectable and revisable natural-language policy that generalizes across rare diseases, transfers across model backbones, and adapts to a real-world patient cohort.
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.