An Empirical Investigation of Neural ODEs and Symbolic Regression for Dynamical Systems

2026-01-28

AI Summary

This work addresses the challenge of modeling complex dynamical systems and discovering their governing equations from noisy, sparse observations. To this end, it proposes a novel framework that integrates Neural Ordinary Differential Equations (Neural ODEs) with Symbolic Regression. By leveraging the strong extrapolation capabilities of Neural ODEs under dynamically similar conditions, the method generates high-fidelity trajectory data, which in turn enriches the input for Symbolic Regression and enables efficient recovery of interpretable analytical governing equations. Experimental results demonstrate that, using only 10% of the original data, the approach accurately reconstructs the complete governing equations for two out of three real-world systems and yields a highly accurate approximation for the third, substantially enhancing the ability to uncover physical laws from limited and noisy measurements.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Accurately modelling the dynamics of complex systems and discovering their governing differential equations are critical tasks for accelerating scientific discovery. Using noisy, synthetic data from two damped oscillatory systems, we explore the extrapolation capabilities of Neural Ordinary Differential Equations (NODEs) and the ability of Symbolic Regression (SR) to recover the underlying equations. Our study yields three key insights. First, we demonstrate that NODEs can extrapolate effectively to new boundary conditions, provided the resulting trajectories share dynamic similarity with the training data. Second, SR successfully recovers the equations from noisy ground-truth data, though its performance is contingent on the correct selection of input variables. Finally, we find that SR recovers two out of the three governing equations, along with a good approximation for the third, when using data generated by a NODE trained on just 10% of the full simulation. While this last finding highlights an area for future work, our results suggest that using NODEs to enrich limited data and enable symbolic regression to infer physical laws represents a promising new approach for scientific discovery.

Problem

• Neural ODEs
• Symbolic Regression
• Dynamical Systems
• Differential Equations
• Scientific Discovery

Innovation

• Neural ODEs
• Symbolic Regression
• Dynamical Systems
• Equation Discovery
• Data Extrapolation