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

TL;DR

This paper compares Neural ODEs and Symbolic Regression in modeling complex dynamical systems, showing how they can complement each other for better scientific discovery from limited and noisy data.

Contribution

It demonstrates the effectiveness of Neural ODEs for extrapolation and Symbolic Regression for equation recovery, highlighting a new approach for scientific modeling.

Findings

NODEs can extrapolate effectively under similar dynamics.

SR recovers governing equations from noisy data.

Combining NODEs and SR aids in discovering physical laws.

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.