Forecasting N-Body Dynamics: A Comparative Study of Neural Ordinary Differential Equations and Universal Differential Equations

TL;DR

This paper compares Neural ODEs and Universal Differential Equations for predicting n-body system dynamics, demonstrating that UDEs are significantly more data-efficient and effective in forecasting with limited data.

Contribution

It introduces a comparative analysis of NODEs and UDEs within Scientific ML for n-body problems, highlighting UDEs' superior data efficiency and forecasting accuracy.

Findings

UDEs require only 20% of data for accurate forecasting.

NODEs need about 90% of data to achieve similar accuracy.

UDEs outperform NODEs in data efficiency and forecasting reliability.

Abstract

The n body problem, fundamental to astrophysics, simulates the motion of n bodies acting under the effect of their own mutual gravitational interactions. Traditional machine learning models that are used for predicting and forecasting trajectories are often data intensive black box models, which ignore the physical laws, thereby lacking interpretability. Whereas Scientific Machine Learning ( Scientific ML ) directly embeds the known physical laws into the machine learning framework. Through robust modelling in the Julia programming language, our method uses the Scientific ML frameworks: Neural ordinary differential equations (NODEs) and Universal differential equations (UDEs) to predict and forecast the system dynamics. In addition, an essential component of our analysis involves determining the forecasting breakdown point, which is the smallest possible amount of training data our models need to predict future, unseen data accurately. We employ synthetically created noisy data to simulate real-world observational limitations. Our findings indicate that the UDE model is much more data efficient, needing only 20% of data for a correct forecast, whereas the Neural ODE requires 90%.