Regression-based Physics Informed Neural Networks (Reg-PINNs) for Magnetopause Tracking

TL;DR

The magnetopause boundary is challenging to predict due to a trade-off between generalization in empirical models and accuracy in purely data-driven approaches. The authors introduce Regression-based Physics-Informed Neural Networks (Reg-PINN), which embed a physics-inspired regression loss into neural network training to combine empirical physics with data-driven fitting. Reg-PINN with the Shue model and a data-overfitting variant demonstrates approximately a 30% reduction in $RMSE$ on unseen magnetopause data and shows robustness across data splits and extreme conditions, highlighting the value of physics-informed regression beyond traditional ODE/PDE PINNs. This framework provides a flexible, generalizable approach for physics-guided forecasting in scientific boundary tracking and can be extended to other disciplines requiring integration of domain-specific empirical physics with neural networks.

Abstract

Previous research in the scientific field has utilized statistical empirical models and machine learning to address fitting challenges. While empirical models have the advantage of numerical generalization, they often sacrifice accuracy. However, conventional machine learning methods can achieve high precision but may lack the desired generalization. The article introduces a Regression-based Physics-Informed Neural Networks (Reg-PINNs), which embeds physics-inspired empirical models into the neural network's loss function, thereby combining the benefits of generalization and high accuracy. The study validates the proposed method using the magnetopause boundary location as the target and explores the feasibility of methods including Shue et al. [1998], a data overfitting model, a fully-connected networks, Reg-PINNs with Shue's model, and Reg-PINNs with the overfitting model. Compared to Shue's model, this technique achieves approximately a 30% reduction in RMSE, presenting a proof-of-concept improved solution for the scientific community.

Figures (7)

• Figure 1: Magnetopause crossing data visualization, including (a) distribution of crossing data for origin is the Earth position and (b) $D_p$ v.s. $B_z$ distribution.
• Figure 2: Parameters for fitting the empirical model. (a) ($\alpha$ & $r_0$) v.s. $D_p$, and (b) ($\alpha$ & $r_0$) v.s. $B_z$.
• Figure 3: Contour plot of different models, including (a) overfitting, (b) Vanilla NN, (c) Reg-PINN (Shue), and (d) Reg-PINN (Overfitting) along with ($D_p$ v.s. $B_z$) distribution for comprehensively visualization.
• Figure 4: RMSE for evaluating different physical parameters, including (a) $\theta$, (b) $D_p$, and (c) $B_z$, along with (d) Loss vs. Iteration for NN and Reg-PINN on this case.
• Figure 5: Selected model RMSE evaluated under the baseline applicable region