LLM-ODE: Data-driven Discovery of Dynamical Systems with Large Language Models

Amirmohammad Ziaei Bideh; Jonathan Gryak

cs.LGMar 21, 2026

LLM-ODE: Data-driven Discovery of Dynamical Systems with Large Language Models

Authors: Amirmohammad Ziaei Bideh, Jonathan Gryak

Organizations: The Graduate Center, CUNY New York, New York, USA · Queens College and The Graduate Center, CUNY New York, New York, USA

Abstract

Discovering the governing equations of dynamical systems is a central problem across many scientific disciplines. As experimental data become increasingly available, automated equation discovery methods offer a promising data-driven approach to accelerate scientific discovery. Among these methods, genetic programming (GP) has been widely adopted due to its flexibility and interpretability. However, GP-based approaches often suffer from inefficient exploration of the symbolic search space, leading to slow convergence and suboptimal solutions. To address these limitations, we propose LLM-ODE, a large language model-aided model discovery framework that guides symbolic evolution using patterns extracted from elite candidate equations. By leveraging the generative prior of large language models, LLM-ODE produces more informed search trajectories while preserving the exploratory strengths of evolutionary algorithms. Empirical results on 91 dynamical systems show that LLM-ODE variants consistently outperform classical GP methods in terms of search efficiency and Pareto-front quality. Overall, our results demonstrate that LLM-ODE improves both efficiency and accuracy over traditional GP-based discovery and offers greater scalability to higher-dimensional systems compared to linear and Transformer-only model discovery methods.

Figures & tables

Figure 1. Schematic overview of LLM-ODE. (1) Given trajectory data from an unknown dynamical system, the observations are decomposed into state variables. (2) A large language model acts as an evolutionary operator, guiding the evolution of symbolic equation populations toward higher-fitness candidates. (3) The final system of equations is selected from the Cartesian product of equation-level Pareto fronts.

Figure 3. System discovery rate as a function of search iterations across various NMSE thresholds

\lambda

. Overall, LLM-ODE methods achieve higher discovery rates across all iteration limits. LLM-ODE methods enjoy a faster convergence rate than PySR.

Figure 4. The number of points along the system Pareto front of GP-based methods during training. The error bars represent 95% confidence intervals across systems. LLM-ODE methods are more efficient at obtaining a diverse Pareto-front as compared to stochastic GP.

Figure 5. System pool sizes from which the system Pareto-front is computed.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Hyperparameter	Possible Values
Optimizer threshold	0.05, 0.1, 0.15
Optimizer alpha	0.025, 0.05, 0.075
Max iterations	20, 100
Polynomial degree	$1,2,\dots,10$
Basis functions	{ polynomials } { polynomials, sin, cos, exp} { polynomials, sin, cos, log, exp, sqrt, inverse }

Appendix

Table 2. Hyperparameter search space of SINDy.

Name	Train IV	Test IV	Equation
RC-circuit	$x_{0}=10.0$	$x_{0}=3.54$	$\dot{x}_{0}=0.303-0.361\,x_{0}$
Population growth (naive)	$x_{0}=4.78$	$x_{0}=0.87$	$\dot{x}_{0}=0.23\,x_{0}$
Population growth (carrying capacity)	$x_{0}=7.3$	$x_{0}=21.0$	$\dot{x}_{0}=0.79\,x_{0}\left(1-0.0135\,x_{0}\right)$
RC-circuit (nonlinear resistor)	$x_{0}=0.8$	$x_{0}=0.02$	$\dot{x}_{0}=-0.5+\dfrac{1}{1+1.65\,e^{-1.04\,x_{0}}}$
Velocity of a falling object	$x_{0}=0.5$	$x_{0}=73.0$	$\dot{x}_{0}=9.81-0.00212\,x_{0}^{2}$
Autocatalysis	$x_{0}=0.13$	$x_{0}=2.24$	$\dot{x}_{0}=-0.5\,x_{0}^{2}+2.1\,x_{0}$

Appendix

Table 3. The governing equations and the initial values of one-dimensional (

D=1

) dynamical systems.

Figure 6. The training trajectories of dynamical systems with

D=1

state variables.

Figure 7. The training trajectories of dynamical systems with

D=2

state variables.

Figure 8. The training trajectories of dynamical systems with

D=3

state variables.

Figure 9. The training trajectories of dynamical systems with

D=4

state variables.

Explore similar work

LLM-ACES: Closed-Loop Discovery of Dynamical Systems with LLM-Guided Adaptive Search
Jun 23, 2026Nikhil Abhyankar, Sha Li, Sanchit Kabra +3Nonlinear System Identification System Identification
Virginia Tech
Automatic Ordinary Differential Equations Discovery For Biological Systems Using Large Language Model Powered Agentic System
Jul 15, 2026David Krongauz, Arad Zulti, Eran Segal +1Dynamical Systems AI Agents for Scientific Discovery
Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel · Department of Molecular Cell Biology, Weizmann Institute of Science, Rehovot, Israel · Mohamed bin Zayed University of Artificial Intelligence, Abu Dhabi, United Arab Emirates +2
Neuro-Symbolic ODE Discovery with Latent Grammar Flow
Apr 17, 2026Karin Yu, Eleni Chatzi, Georgios KissasAutonomous Scientific Discovery Neurosymbolic Learning
Institute of Structural Engineering, ETH Zurich, Zurich, Switzerland · ETH AI Center, ETH Zurich, Zurich, Switzerland · Swiss Data Science Center, Zurich, Switzerland

Jun 23, 2026J/K move · Enter open · S save

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

Nikhil Abhyankar, Sha Li, Sanchit Kabra, Naren Ramakrishnan+2

Virginia Tech

Nonlinear System Identification System Identification Autonomous Scientific Discovery Large Language Model-Guided Scientific Discovery ODE Discovery Symbolic Regression

Open paper AskPDF arXiv

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.

Abstract

Explore similar work

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

Automatic Ordinary Differential Equations Discovery For Biological Systems Using Large Language Model Powered Agentic System

Neuro-Symbolic ODE Discovery with Latent Grammar Flow