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

Forecasting N-Body Dynamics: A Comparative Study of Neural Ordinary Differential Equations and Universal Differential Equations Suriya R S, Prathamesh Dinesh Joshi, Rajat Dandekar, Raj Dandekar, Sreedath Panat Vizuara AI Labs 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 frame- work. Through robust modelling in the Julia programming language, our method uses the Scientific ML frameworks: Neural ordinary differential equations (NODEs) and Univer- sal differential equations (UDEs) to predict and forecast the system’s 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%. 1 Introduction The classical n-body problem in astrophysics seeks to predict the motion of a system of celestial objects that interact grav- itationally with each other. Although an analytical closed- form solution exists for a system of two objects, no such solution has been discovered for three or more objects. As a result, historically, numerical integration methods such as the Runge-Kutta method or leap-frog schemes have been used to simulate solutions. However, traditional solvers operate under the assumption that the physical model of an n-body system is perfectly known and complete. Therefore, this as- sumption limits our ability to apply it to a realistic scenario where the system might be subject to unmodeled physics. To address these challenges, Scientific Machine Learn- ing ( Scientific ML ) has emerged as a powerful paradigm where we shift our objective from just simulating a known physical model to discovering or correcting the governing equations directly from observational data. Scientific ML combines the expressive power of neural networks with the interpretability of differential equations. This approach has Copyright © 2026, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. been successfully implemented in various scientific disci- plines like fluid mechanics, circuit modelling, optics, gene expression, quantum circuits, and epidemiology (Baker et al. 2019; Dandekar, Rackauckas, and Barbastathis 2020; Dan- dekar et al. 2020; Abhijit Dandekar 2022; Ji et al. 2022; Bills et al. 2020; Lai et al. 2021; Nieves, Dandekar, and Rack- auckas 2024; Wang, Garnier, and Rea 2023; Ramadhan 2024; Rackauckas, Campin, and Ferrari; Sharma et al. 2023a,b; Aboelyazeed et al. 2023). Primarily, the progress in Scientific ML is driven by the following two frameworks: Neural Ordinary Differential Equations (NeuralODEs) (Chen et al. 2018; Dupont, Doucet, and Teh 2019; Massaroli et al. 2020; Yan et al. 2019), which learns the entire system dynamics through Neural Networks from data, and Universal Differential Equations (UDEs)(Rackauckas et al. 2020; Bolibar et al. 2023; Teshima et al. 2020; Bournez and Pouly 2020), which blends in the known physical laws with neural networks to learn only the unknown/unmodelled dynamics from data. While these frameworks are being used in astrophysics (Gupta, Srijith, and Desai 2022; Branca and Pallottini 2023; Origer and Izzo 2024), a thorough comparative analysis of their effectiveness in solving problems is yet to be determined. In this study, we try to understand the effectiveness and limitations of these two Scientific ML frameworks. Specifically, we aim to address the following questions in the context of the n-body problem: 1. Can the UDE framework be used to learn and recover the pairwise gravitational interaction term by replacing it with a neural network? 2. How does the predictive accuracy of NeuralODEs com- pare to that of UDEs when modelling the trajectories? 3. Can both NeuralODEs and UDEs be used to forecast the system’s trajectories in the long term? 4. Do UDEs, incorporating known physics, offer superior performance in forecasting over the purely data-driven NeuralODEs? We perform rigorous comparative analysis using advanced Scientific ML libraries to answer these questions. Our work provides critical insights into the effectiveness and limitations of these frameworks. Furthermore, we analyze the forecast- ing breakdown point as a metric to quantify the time horizon arXiv:2512.20643v1 [cs.LG] 12 Dec 202

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