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

TL;DR

This study shows that both Neural ODEs and UDEs can be used effectively for both prediction as well as forecasting of the CWDE, and introduces the forecasting breakdown point - the time at which forecasting fails for both Neural ODEs and UDEs.

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.

Figures (18)

• Figure 1: Comparison of the Neural ODE approximation for the Chandrasekhar’s white dwarf model. The training of the Neural ODE was performed with varying noise added to the synthetic data in the full solution domain. These training datasets encompassed the values for $\varphi$ and $\varphi'$ at the 100 equally spaced $\eta$ points with varied noise addition. Each figure shows the results for the different training sets: (a) No-noise Data (synthetic data) obtained numerically from the White Dwarf ordinary differential equation \ref{['WhiteDwarf_ODE']}. (b) Moderate-noise dataset with a standard deviation of $7 \%$. (c) High-noise dataset with a standard deviation of $35 \%$.
• Figure 2: Comparison of the UDE approximation for the Chandrasekhar's white dwarf equation. The training of the UDE model was performed with varyng noise added to the synthetic data in the full solution domain. These training datasets encompassed the values for $\varphi$ and $\varphi'$ of the 100 equally spaced $\eta$ points with varied noise addition: (a) No-noise data (synthetic data) obtained numerically from the White Dwarf ordinary differential equation \ref{['WhiteDwarf_ODE']}. (b) Moderate-noise dataset with standard deviation of $7\%$. (c) High-noise dataset with standard deviation of $35\%$.
• Figure 3: Comparison of the approximated missing term in the Chandrasekhar's white dwarf UDE model for the different training datasets: (a) No-noise dataset (synthetic data) set encompassing the numerically obtained values for $\varphi$ and $\varphi'$ within the solution domain (0, $\eta_\infty$). (b) Moderate-noise dataset with standard deviation of $7\%$ added directly to the synthetic data. (c) High-noise dataset with standard deviation of $35\%$ added directly to the synthetic data.
• Figure 4: Comparison of the Neural ODE approximation and forecasting for the Chandrasekhar’s white dwarf model. The training of the Neural ODE was performed with varying noise added to the synthetic data. These training data subsets encompassed the values for $\varphi$ and $\varphi'$ with varied noise levels added to the first 90 equally spaced $\eta$ points of the solution domain. The forecasted $\varphi$ corresponding to the remaining $10 \%$ of the $\eta$ points are shown against the testing data. Each figure shows the results for the different datasets: (a) No-noise Data (synthetic data) obtained numerically from the White Dwarf ordinary differential equation \ref{['WhiteDwarf_ODE']}. (b) Moderate-noise dataset with a standard deviation of $7 \%$. (c) High-noise dataset with a standard deviation of $35 \%$.
• Figure 5: Comparison of the UDE approximation and forecasting for the Chandrasekhar’s white dwarf model. The training of the UDE was performed with varying noise added to the synthetic data. These training data subsets encompassed the values for $\varphi$ and $\varphi'$ with varied noise addition for the first 90 equally spaced $\eta$ points of the solution domain. The forecasted $\varphi$ corresponding to the remaining $10 \%$ of the $\eta$ points are shown against the testing data. Each figure shows the results for the different datasets: (a) No-noise data (synthetic data) obtained numerically from the White Dwarf ordinary differential equation \ref{['WhiteDwarf_ODE']}. (b) Moderate-noise dataset with a standard deviation of $7 \%$. (c) High-noise dataset with a standard deviation of $35 \%$.