A comparative study of NeuralODE and Universal ODE approaches to solving Chandrasekhar White Dwarf equation

TL;DR

This paper compares Neural ODEs and Universal Differential Equations in modeling the Chandrasekhar White Dwarf Equation, demonstrating their effectiveness in prediction and forecasting, and introducing a new forecasting breakdown metric.

Contribution

It systematically applies Neural ODEs and UDEs to an important astrophysical ODE, providing insights into their forecasting capabilities and hyperparameter optimization.

Findings

Both Neural ODEs and UDEs effectively model CWDE.

Introduction of the forecasting breakdown point metric.

Hyperparameter optimization improves model performance.

Abstract

In this study, we apply two pillars of Scientific Machine Learning: Neural Ordinary Differential Equations (Neural ODEs) and Universal Differential Equations (UDEs) to the Chandrasekhar White Dwarf Equation (CWDE). The CWDE is fundamental for understanding the life cycle of a star, and describes the relationship between the density of the white dwarf and its distance from the center. Despite the rise in Scientific Machine Learning frameworks, very less attention has been paid to the systematic applications of the above SciML pillars on astronomy based ODEs. Through robust modeling in the Julia programming language, we show that both Neural ODEs and UDEs can be used effectively for both prediction as well as forecasting of the CWDE. More importantly, we introduce the forecasting breakdown point - the time at which forecasting fails for both Neural ODEs and UDEs. Through a robust hyperparameter optimization testing, we provide insights on the neural network architecture, activation functions and optimizers which provide the best results. This study provides opens a door to investigate the applicability of Scientific Machine Learning frameworks in forecasting tasks for a wide range of scientific domains.