Physics-informed Attention-enhanced Fourier Neural Operator for Solar Magnetic Field Extrapolations

TL;DR

NLFFF extrapolation is computationally expensive when solved with traditional iterative methods. This work introduces PIANO, a Physics-informed Attention-enhanced Fourier Neural Operator, which handles multimodal inputs by lifting 2D boundary data and scalar vectors, refines scalar features with Efficient Channel Attention and Dilated Convolution, and enforces physics via divergence-free and force-free losses within a two-phase training regime. On the ISEE NLFFF dataset, PIANO achieves state-of-the-art accuracy across magnetic-field components and heights while maintaining strong physics consistency, outperforming baselines such as FNO, GLFNO, UFNO, GeoFNO, GNOT, FNOMIO, and PINO. The approach promises faster, reliable NLFFF extrapolations suitable for real-time space-weather analysis and large-scale parametric studies.

Abstract

We propose Physics-informed Attention-enhanced Fourier Neural Operator (PIANO) to solve the Nonlinear Force-Free Field (NLFFF) problem in solar physics. Unlike conventional approaches that rely on iterative numerical methods, our proposed PIANO directly learns the 3D magnetic field structure from 2D boundary conditions. Specifically, PIANO integrates Efficient Channel Attention (ECA) mechanisms with Dilated Convolutions (DC), which enhances the model's ability to capture multimodal input by prioritizing critical channels relevant to the magnetic field's variations. Furthermore, we apply physics-informed loss by enforcing the force-free and divergence-free conditions in the training process so that our prediction is consistent with underlying physics with high accuracy. Experimental results on the ISEE NLFFF dataset show that our PIANO not only outperforms state-of-the-art neural operators in terms of accuracy but also shows strong consistency with the physical characteristics of NLFFF data across magnetic fields reconstructed from various solar active regions. The GitHub of this project is available https://github.com/Autumnstar-cjh/PIANO

Figures (8)

• Figure 1: Overall pipeline of NLFFF. (a) NASA SDO mission; (b) Observed photospheric magnetogram; (c) the boundary conditions $\mathbf{B}^{\text{obs}}$ from an active region in magnetogram for NLFFF to obtain $\mathbf{B}$ extrapolated along the height.
• Figure 2: Architecture of PIANO. The input consists of observed magnetic field $\mathbf{B}^{\text{obs}}$ and scalar parameters. $\mathbf{B}^{\text{obs}}$ is processed through the lifting layer $P_1$ to get feature $v_0$. Scalar parameters are processed by the lifting layer $P_2$ and an ECA with DC block to get the feature $u_0$. The aggregated features $z_0$ are passed through a series of Fourier layers. The projection layer $Q$ maps the $z_{n+1}$ to magnetic fields $\mathbf{B}_x, \mathbf{B}_y, \mathbf{B}_z$. The model includes $\mathcal{L}_{\text{div}}$ and $\mathcal{L}_{\text{ff}}$, ensuring physics consistency in the magnetic fields.
• Figure 3: Active regions type distribution of the dataset. The top subfigure illustrates the counts of different types in the training set, which has 143 samples from 2010 to 2014. The bottom subfigure shows the distribution of types in the test set, which has 27 samples from 2015 to 2016. The majority of the data belongs to the $\beta$ and $\beta\gamma$ types, with smaller counts for others.
• Figure 4: Comparison of the visualization and error maps of $\mathbf{B}_x$, $\mathbf{B}_y$ and $\mathbf{B}_z$ across different neural operator models. Subfigures (a), (c), and (e) depict the visualizations of $\mathbf{B}_x$, $\mathbf{B}_y$ and $\mathbf{B}_z$, respectively, at different heights (2 Mm, 4 Mm, 8 Mm, 12 Mm, 16 Mm) for the ground truth (GT) and other neural operator models. Subfigures (b), (d), and (f) show the corresponding error maps, highlighting the prediction errors for each model compared to the ground truth.
• Figure 5: Variation of divergence-free loss and force-free loss with height for different types of solar active regions. Subfigures (a), (c), (e) and (g) show the divergence-free loss as a function of height for $\beta$, $\beta\delta$, $\beta\gamma$, and $\beta\gamma\delta$, respectively. Subfigures (b), (d), (f) and (h) depict the corresponding force-free loss for the same types. Both losses decrease with height across all configurations, with higher losses observed at lower heights, indicating greater complexity in the magnetic field structures at lower heights.