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Neural Operator based Multi-Field Reconstruction of Inner Solar Boundary State

This paper introduces a Local Neural Operator (LocalNO) framework to reconstruct a comprehensive multi-field solar magnetohydrodynamic state at 30 solar radii from limited radial velocity and magnetic field data, thereby providing essential boundary conditions for improved heliospheric modeling and solar wind prediction.

Original authors: Vignesh Kumar Pandian Sathia, Reza Mansouri, Dustin J. Kempton, Pete Riley, Rafal A. Angryk

Published 2026-08-25

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CC BY 4.0

Original authors: Vignesh Kumar Pandian Sathia, Reza Mansouri, Dustin J. Kempton, Pete Riley, Rafal A. Angryk

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Sun is not a static ball of fire; it is a churning, magnetic engine that constantly blows a stream of charged particles into space. This stream, known as the solar wind, flows outward from the Sun's surface, carrying magnetic fields and plasma across the entire solar system. As this wind travels, it shapes the space environment around Earth, influencing everything from satellite operations to the safety of astronauts. To predict how this wind will behave, scientists rely on complex computer models that simulate the physics of the Sun and the space around it. However, these models face a significant hurdle: they need a complete picture of the conditions right at the Sun's edge to start their calculations, but we can only directly measure a few specific parts of that picture. It is like trying to predict the weather of a whole continent when you only have temperature readings from a handful of cities; without the full context, the forecast often fails.

Researchers at Georgia State University and Predictive Sciences Inc. have developed a new way to fill in these missing pieces using a type of artificial intelligence designed to understand physical fields. Their work focuses on a specific layer of the solar atmosphere located thirty times the Sun's radius away from its surface. At this distance, the solar wind has already formed, but it is still close enough to the Sun that its behavior is tightly linked to the magnetic forces below. In this region, scientists can directly observe the speed of the wind moving outward and the strength of the magnetic field pointing outward. What they cannot easily see are the other crucial details: how the wind is swirling sideways, how the magnetic field is twisting, and how the density and pressure of the plasma are changing. The team set out to teach a computer to look at the two observable quantities and mathematically reconstruct the entire, complex state of the solar wind at that boundary.

To solve this, the researchers turned to a specialized form of machine learning called a neural operator. Unlike standard computer programs that learn to map one number to another, or even standard image-recognition systems that look at pixels, a neural operator learns to translate entire fields of data into other fields of data. Imagine trying to understand a landscape not by looking at individual trees, but by understanding how the wind moves across the whole forest. The researchers trained their system on a massive dataset of solar simulations that covered more than four complete solar cycles, spanning from 1975 to 2020. These simulations provided the "ground truth," showing the full, nine-part state of the solar wind, including velocity, magnetic fields, current, density, and pressure, all mapped across a spherical grid. The computer was then given only the radial velocity and radial magnetic field from these simulations and asked to predict the remaining seven variables.

The team tested several different approaches to see which one could best capture the physics of the solar wind. They compared their new method against other advanced models that rely on global patterns or simplified mathematical shortcuts. The results showed that the most effective approach was a "local neural operator." This model succeeded because it paid close attention to the immediate neighborhood of every point on the solar surface while still understanding the larger, global connections. The solar wind is a system where local details matter immensely; a small change in one area can ripple out and affect the whole structure. The researchers found that by focusing on these local interactions, their model could reconstruct the missing variables with high accuracy, capturing the sharp edges and subtle gradients that other models missed. They also discovered that the way they prepared the data for the computer was critical. By applying a specific mathematical transformation that preserved the direction of the flow while making the numbers easier for the computer to process, they significantly improved the model's ability to learn the complex relationships between the different physical quantities.

The study confirms that it is possible to infer a complete, multi-layered physical state from just two observable components, provided the right tools are used. The researchers demonstrated that their reconstructed data is not just a rough guess but a detailed, consistent picture that matches the physics of the solar wind. This success suggests that in the future, these AI-generated reconstructions could be used as the starting point for larger, more accurate simulations of the space weather that affects Earth. While the current work relies on simulated data rather than direct observations from space, the method offers a powerful new way to bridge the gap between what we can measure and what we need to know to protect our technology and explore the solar system. The findings indicate that by respecting the local structure of physical systems, artificial intelligence can become a vital partner in understanding the dynamic and often violent environment of our star.

Technical Summary: Neural Operator based Multi-Field Reconstruction of Inner Solar Boundary State

Problem Statement
Accurate specification of inner-boundary conditions is critical for heliospheric modeling and solar wind prediction. However, in practical applications, only a subset of interacting multi-field variables is directly observable. Specifically, at the inner boundary of 30 solar radii ( $30 R_\odot$ ), only the radial velocity ( $v_r$ ) and radial magnetic field ( $B_r$ ) are typically available. The challenge lies in reconstructing the complete, coupled magnetohydrodynamic (MHD) state from these partial observations. The target variables include non-radial velocity ( $v_\theta, v_\phi$ ), non-radial magnetic field ( $B_\theta, B_\phi$ ), current density components ( $J_r, J_\theta, J_\phi$ ), plasma density ( $\rho$ ), and pressure ( $p$ ). This task is characterized by strong nonlinear coupling, multi-field dependencies, and multi-scale spatial variations, making it a difficult inverse problem for standard data-driven approaches. Traditional methods like Convolutional Neural Networks (CNNs) often struggle with spherical geometries and long-range interactions, while standard regression models fail to capture the functional transformations between field spaces.

Methodology
The authors formulate the reconstruction task as a supervised operator learning problem, aiming to learn a mapping between infinite-dimensional function spaces rather than fixed-length vectors.

Data Preparation: The dataset comprises 907 samples derived from MAS (Magnetohydrodynamic Algorithm outside a Sphere) simulations, covering over four solar cycles (1975–2020) with a resolution of $109 \times 128$ on a spherical grid. To address the disparate physical ranges and skewed distributions of the target variables, a sign-preserving nonlinear transformation is applied. Specifically, a signed square-root transform is used for variables with narrow-tailed distributions (e.g., $v_\theta, B_\theta, J_r$ ), and a power transform ( $1/4$ ) is applied to pressure ( $p$ ). Following transformation, min-max normalization aligns all channels to a $[0, 1]$ scale.
Model Architecture: The study evaluates three neural operator families:
1. Spherical Fourier Neural Operator (SFNO): A global baseline using spectral convolution in Fourier space to capture long-range interactions via spherical harmonics.
2. Tensorized Fourier Neural Operator (TFNO): A parameter-efficient variant that factorizes spectral weight tensors.
3. Local Neural Operator (LocalNO): The proposed primary model. Unlike purely global spectral methods, LocalNO emphasizes localized interactions using differential and integral kernels combined with spherical convolution. This architecture is designed to respect the spherical manifold geometry without domain padding, balancing global coupling with local structural dependencies.
Training Configuration: Models are trained on a 90/10 split grouped by Carrington rotation to prevent data leakage. The proposed LocalNO utilizes a common operator backbone feeding into task-specific heads for correlated variables (velocity, magnetic field, current density, density/pressure).

Key Contributions

Problem Formulation: The work defines solar boundary reconstruction as a multi-output operator learning problem, utilizing two observed radial components to predict a nine-channel MHD target state.
Architectural Focus: It demonstrates the suitability of Local Neural Operators for this specific domain, arguing that the locality bias inherent in LocalNO is better matched to the structured boundary fields of solar physics than purely global spectral operators.
Ablation and Preprocessing Analysis: The study provides empirical evidence on the necessity of specific preprocessing steps (signed transformations) and optimal model capacity (factorization rank) for this multi-field reconstruction task.

Results
Experiments were conducted using multiple evaluation metrics, including Mean Squared Error (MSE), Peak Signal-to-Noise Ratio (PSNR), Universal Quality Index (UQI), and Anomaly Correlation Coefficient (ACC).

Performance Comparison: The LocalNO significantly outperformed both SFNO and TFNO baselines across all metrics. Notably, LocalNO achieved a mean ACC of 0.9874 and a mean MSE of 0.0040, compared to SFNO's 0.6307 ACC and 0.0870 MSE. The LocalNO also exhibited lower standard deviations across components, indicating more consistent reconstruction quality.
Rank Ablation: An ablation study on factorization rank revealed that a rank of 0.4 provided the optimal trade-off between compression and expressivity. Lower ranks (0.2–0.3) underfit the complex nonlinear relationships, while higher ranks (0.5–1.0) led to over-parameterization without corresponding gains in reconstruction fidelity.
Transformation Ablation: The signed square-root transformation yielded superior results compared to signed logarithmic transformations or no transformation, particularly in preserving the reconstruction of small-magnitude field values and maintaining directional consistency.
Visual Validation: Qualitative analysis of reconstructed fields (e.g., Carrington Rotations 1653 and 2113) showed that the LocalNO successfully captured the spatial structures and residuals were minimal compared to ground truth.

Significance and Claims
The paper claims that the proposed framework offers a viable path for generating complete inner-boundary conditions for downstream heliospheric modeling pipelines, such as WSA/ENLIL. By learning the operator that maps partial observations to a full multi-field state, the method provides a "holistic surrogate modeling framework." The authors emphasize that the success of the LocalNO suggests that locality-aware operator learning is a compelling approach for reconstructing coupled physical fields on spherical domains.

The authors modestly note limitations: the work is currently evaluated solely on simulation data (MAS) rather than real-world observations, does not account for radiative interactions, and is constrained by the spatial resolution of the available simulation data. The primary contribution is establishing the boundary mapping itself as a foundation for future integration with full inner-heliospheric simulation frameworks.

Technical Summary: Neural Operator based Multi-Field Reconstruction of Inner Solar Boundary State

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