Autoregressive Surrogate Modeling of the Solar Wind with Spherical Fourier Neural Operator

TL;DR

This work addresses the computational burden of 3D MHD solar wind simulations by introducing an autoregressive surrogate based on the Spherical Fourier Neural Operator (SFNO) to predict the radial velocity field. Trained on MAS MHD data, the model propagates predictions outward through iterative radial steps, improving long-range accuracy over single-step surrogates and comparing favorably with the HUX baseline. Key contributions include the first autoregressive SFNO surrogate for steady-state solar wind velocity, a detailed SFNO-based modeling pipeline on spherical grids, and an empirical demonstration of superior or comparable performance to a physics-based solver, with open-source code for replication. This approach enables flexible, data-driven, real-time forecasting and uncertainty quantification for space-weather applications.

Abstract

The solar wind, a continuous outflow of charged particles from the Sun's corona, shapes the heliosphere and impacts space systems near Earth. Accurate prediction of features such as high-speed streams and coronal mass ejections is critical for space weather forecasting, but traditional three-dimensional magnetohydrodynamic (MHD) models are computationally expensive, limiting rapid exploration of boundary condition uncertainties. We introduce the first autoregressive machine learning surrogate for steady-state solar wind radial velocity using the Spherical Fourier Neural Operator (SFNO). By predicting a limited radial range and iteratively propagating the solution outward, the model improves accuracy in distant regions compared to a single-step approach. Compared with the numerical HUX surrogate, SFNO demonstrates superior or comparable performance while providing a flexible, trainable, and data-driven alternative, establishing a novel methodology for high-fidelity solar wind modeling. The source code and additional visual results are available at https://github.com/rezmansouri/solarwind-sfno-velocity-autoregressive.

Figures (11)

• Figure 1: Parker Solar Probe WISPR view at $\approx18R_\odot$, August 2021. The image captures the solar wind and coronal streamers (left) from within the Sun's atmosphere, with the Milky Way visible on the right, looking away from the Sun. Streaks are due to energetic particle impacts (Credit: NASA/Johns Hopkins APL/Naval Research Laboratory).
• Figure 2: Equiangular projections of radii for one instance (MAS simulation of CR 2285, 2–29 June 2024). The first radius $r_0 = 30\,R_\odot$ is the input; the remaining 139 radii are the ground truth. The physical area grows rapidly with radius, so scale increases substantially at larger distances.
• Figure 3: Autoregressive prediction approach. The model predicts a fixed number of radii (predictive horizon) per step, feeding the last predicted radius back as input for the next prediction. This repeats until the full radial range is covered. During training, teacher forcing uses ground truth data as input to stabilize learning.
• Figure 4: Solar wind radial velocity at $\approx49\ R\odot$ for Carrington Rotation 2285 from the MHD solution. The edge regions, detected using a Sobel filter, are outlined with dashed lines.
• Figure 5: Cross-validation results showing the MSE across five folds for SFNO architectures with varying depths and hidden channel sizes; for example, $4\times64$ denotes a model with 4 layers and 64 hidden channels.