Surface Flux Transport Modeling using Physics Informed Neural Networks (2409.01744v2)

Abstract: Studying the magnetic field properties on the solar surface is crucial for understanding the solar and heliospheric activities, which in turn shape space weather in the solar system. Surface Flux Transport (SFT) modeling helps us to simulate and analyse the transport and evolution of magnetic flux on the solar surface, providing valuable insights into the mechanisms responsible for solar activity. In this work, we demonstrate the use of machine learning techniques in solving magnetic flux transport, making it accurate. We have developed a novel Physics-Informed Neural Networks (PINN)-based model to study the evolution of Bipolar Magnetic Regions (BMRs) using SFT in one-dimensional azimuthally averaged and also in two-dimensions. We demonstrate the efficiency and computational feasibility of our PINN-based model by comparing its performance and accuracy with that of a numerical model implemented using the Runge-Kutta Implicit-Explicit (RK-IMEX) scheme. The mesh-independent PINN method can be used to reproduce the observed polar magnetic field with better flux conservation. This advancement is important for accurately reproducing observed polar magnetic fields, thereby providing insights into the strength of future solar cycles. This work paves the way for more efficient and accurate simulations of solar magnetic flux transport and showcases the applicability of PINN in solving advection-diffusion equations with a particular focus on heliophysics.

• The paper demonstrates that integrating physical equations into neural network training significantly enhances flux conservation over traditional numerical schemes.
• The PINNs method achieves superior accuracy with errors between 0.2% and 1.3%, outperforming conventional RK-IMEX and Explicit schemes.
• Case studies using 1D, 2D simulations and real magnetogram data validate the robustness of the approach in modeling solar magnetic flux evolution.

Surface Flux Transport Modelling using Physics Informed Neural Networks

This paper presents an innovative approach to Surface Flux Transport (SFT) modeling on the solar surface through Physics-Informed Neural Networks (PINNs). This emerging technique, applied here to assess the transport and evolution of magnetic flux, offers an appealing alternative to traditional numerical solutions. By embedding the governing equations directly within the neural network training process, PINNs resolve the dependencies on computational grids, thus promising enhanced accuracy and computational efficiency.

Introduction

Understanding the dynamic processes of the solar surface and their significant influence on solar and heliospheric activities is critical. SFT modeling, which simulates the transport and evolution of magnetic flux, provides valuable insights into the mechanisms driving solar activity, instrumental in forecasting future solar cycles.

Numerical Approaches vs. PINNs

The authors contrast traditional numerical schemes, such as the Explicit and Runge-Kutta IMplicit-EXplicit (RK-IMEX) methods, with PINNs. The paper highlights the numerical schemes’ demand for extensive grid-based calculations, presenting challenges in terms of computational expense and accuracy, especially regarding flux conservation. In contrast, PINNs inherently mitigate these issues by leveraging automatic differentiation, thereby enhancing resolution independence and reducing numerical diffusion.

Analytical Validation

The authors validate their PINNs and numerical codes through an analytical form based on prior works \cite{DeVore1984}. The validation demonstrates that PINNs outperform RK-IMEX and Explicit schemes on error metrics, particularly in maintaining flux conservation with errors between 0.2% and 1.3%. This superiority is paramount for precise modeling of solar cycles, as PINNs minimize flux loss crucial for accurate predictions of polar magnetic field strengths.

Case Studies

1D SFT Simulation

In the 1D scenario, the paper explores the evolution of magnetic fields by incorporating meridional flow, diffusion, decay, and source terms. The source term is modeled to represent solar activity cycles. The obtained butterfly diagrams from PINNs method exhibit close alignment with those from the numerical RK-IMEX scheme. However, notable differences in polarity inversion times at latitudinal boundaries suggest PINNs’ better precision in tracking flux transport.

2D SFT Simulation

For the 2D scenario, the authors introduce Bipolar Magnetic Regions (BMRs) and evolve them using both methods. The PINNs method elucidates superior grid-independence, showing a mean absolute error of ~1% when compared to the RK-IMEX method. This analysis asserts the efficacy of PINNs in simulating complex spatial flux distribution that involves less numerical diffusion.

Real Magnetogram Data Application

An intriguing case paper incorporates real data from SOHO-MDI for AR 7978, where both methods are tested on solar magnetogram observations. The results underscore the robustness of PINNs in reconstructing observed magnetic flux evolutions with lower flux errors compared to RK-IMEX. This demonstrates the applicability of PINNs in practical astrophysical simulations, bridging the gap between theoretical modeling and observational data.

Conclusion

The research conclusively demonstrates that PINNs offer a highly accurate and computationally efficient method for SFT modeling. By overcoming the resolution dependencies and maintaining superior flux conservation, PINNs provide a robust framework for predicting future solar cycles. The authors suggest the potential of PINNs in various heliophysical applications, emphasizing the need for further exploration of PINNs in diverse initial conditions and parameter optimizations.

The findings open pathways for integrating PINNs in ensemble modeling and the Green's function approach for broader applications, making substantial contributions to advancing solar and space weather forecasts. Future work will likely delve into optimizing these methods for more extensive datasets and more complex solar dynamics, aiming to refine the predictive capabilities of magnetic field models on the solar surface.

Overall, this paper establishes a viable, advanced alternative to traditional numerical methods in astrophysical modeling, positioning PINNs as a promising avenue for future research and applications in solar physics.