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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

TL;DR

This work tackles forecasting the classical $N$-body problem under limited data. It compares two Scientific ML frameworks, Neural ODEs and Universal Differential Equations, implemented in Julia SciML. With noisier data, UDEs learn the unknown pairwise interactions while preserving known physics, achieving accurate forecasts with as little as $20\%$ of data, whereas Neural ODEs require about $90\%$ in noiseless cases. The results highlight the data-efficiency and interpretability advantages of physics-informed modeling for scientific forecasting and point to avenues for applying these methods to real observational data and more complex gravitational regimes.

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%.

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

TL;DR

This work tackles forecasting the classical -body problem under limited data. It compares two Scientific ML frameworks, Neural ODEs and Universal Differential Equations, implemented in Julia SciML. With noisier data, UDEs learn the unknown pairwise interactions while preserving known physics, achieving accurate forecasts with as little as of data, whereas Neural ODEs require about in noiseless cases. The results highlight the data-efficiency and interpretability advantages of physics-informed modeling for scientific forecasting and point to avenues for applying these methods to real observational data and more complex gravitational regimes.

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%.
Paper Structure (16 sections, 5 equations, 10 figures, 2 tables)

This paper contains 16 sections, 5 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Dataset generation
  4. Neural Ordinary Differential Equation (NODE)
  5. Universal Differential Equations (UDEs)
  6. Results
  7. Case 1: Training on complete dataset
  8. Case 3: Training on 80% of the dataset and forecasting
  9. Case 5: Training on 20% of the dataset and forecasting
  10. Forecasting Breakdown Point Analysis
  11. Discussion and Conclusion
  12. Appendix
  13. Hyperparameter Details
  14. Additional Forecasting Results
  15. Case 2: Training on 90% of dataset and forecasting
  16. ...and 1 more sections

Figures (10)

  • Figure 1: Neural ODE results for Case 1 (100% training) across different noise levels for body 1.
  • Figure 2: UDE results for Case 1 (100% training) across different noise levels for body 1.
  • Figure 3: Neural ODE results for Case 3 (80% training) across different noise levels for body 1.
  • Figure 4: UDE results for Case 3 (80% training) across different noise levels for body 1.
  • Figure 5: Neural ODE results for Case 5 (20% training) across different noise levels for body 1.
  • ...and 5 more figures