Toward Data-Driven Surrogates of the Solar Wind with Spherical Fourier Neural Operator

TL;DR

The paper introduces a data-driven surrogate for steady-state solar wind modeling using a Spherical Fourier Neural Operator trained on MAS MHD simulations, and benchmarks it against the HUX baseline. SFNO achieves comparable or improved accuracy, particularly in high-gradient regions, while offering grid-invariant, fast predictions suitable for real-time forecasting. The results highlight the potential of operator-learning methods in heliophysics, though they also reveal limitations of current evaluation metrics for capturing large-scale morphology. The work suggests future directions including physics-informed loss terms and incorporating additional solar wind variables to strengthen fidelity and utility.

Abstract

The solar wind, a continuous stream of charged particles from the Sun's corona, shapes the heliosphere and impacts space systems near Earth. Variations such as high-speed streams and coronal mass ejections can disrupt satellites, power grids, and communications, making accurate modeling essential for space weather forecasting. While 3D magnetohydrodynamic (MHD) models are used to simulate and investigate these variations in the solar wind, they tend to be computationally expensive, limiting their usefulness in investigating the impacts of boundary condition uncertainty. In this work, we develop a surrogate for steady state solar wind modeling, using a Spherical Fourier Neural Operator (SFNO). We compare our model to a previously developed numerical surrogate for this task called HUX, and we show that the SFNO achieves comparable or better performance across several metrics. Though HUX retains advantages in physical smoothness, this underscores the need for improved evaluation criteria rather than a flaw in SFNO. As a flexible and trainable approach, SFNO enables efficient real-time forecasting and can improve with more data. The source code and more visual results are available at https://github.com/rezmansouri/solarwind-sfno-velocity.

Figures (11)

• Figure 1: Radial velocity from a MAS simulation of Carrington Rotation 2293 (1 Jan–3 Feb 2025). (A) and (B) show velocity at the inner boundary ($30\,R_\odot$) and at 1 AU ($\sim236\,R_\odot$), respectively. (C) and (D) show slices across all radii at $90^\circ$ latitude and $0^\circ$ longitude. The spiral in (C) reflects solar rotation and shapes heliospheric structure. A and B represent spherical surfaces and are not drawn to scale.
• Figure 2: Equiangular projections of solar wind radial velocity in one instance of our dataset (MAS simulation of Carrington Rotation 2234, from 11 August 2020 to 7 September 2020). The first radius $r_0 = 30\,R_\odot$ serves as the input, and the remaining radii constitute the ground truth.
• Figure 3: Solar wind radial velocity at $R\approx49\ R\odot$ for Carrington Rotation 2293 from the MHD solution. The edge regions, detected using a Sobel filter, are outlined with dashed lines.
• Figure 4: 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.
• Figure 5: Training and test loss curves for the optimal SFNO configuration ($8\times256$). Both losses decrease consistently, with convergence reached after about the $160^{th}$ epoch, indicating stable generalization to the held-out test set.