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

Vignesh Kumar Pandian Sathia; Reza Mansouri; Dustin J. Kempton; Pete Riley; Rafal A. Angryk

cs.LGAug 24, 2026

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

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

Organizations: Department of Computer Science, Georgia State University Atlanta, USA · Predictive Sciences Inc. San Diego, USA

Abstract

The Solar wind is a continuous flow of charged particles emanating from the solar surface and governed by complex, interacting magnetohydrodynamic processes. Accurate specification of inner-boundary conditions is essential for heliospheric modeling and solar-wind prediction. In many practical applications, only a subset of interacting multi-field variables is directly available, but for a comprehensive view of solar wind prediction and downstream magnetohydrodynamic simulations, a more complete boundary state is required. In this work, we study the problem of learning the multi-field multi-scale solar magnetohydrodynamic state at 30 solar radii ( $R_\odot$ ) using operator learning. Specifically, given the radial velocity and radial magnetic field, we aim to reconstruct the non-radial velocity and magnetic field components, radial and non-radial current density, thermodynamic density, and pressure components. This mapping is highly nonlinear, spatially coupled, and multi-scale, making it a challenging task for data-driven scientific machine learning. To address this problem, we employ a Local Neural Operator (LocalNO) that learns mappings between input and output function spaces while retaining locality and resolution-awareness. Unlike conventional regression models and autoencoder models, neural operators are better suited for learning structured field-to-field transformations arising from physical systems. The resulting predictions along with inputs are intended to serve as boundary condition variables for future inner-heliospheric modeling pipelines.

Figures & tables

Fig. 1 : Data distribution. Top row (A) represents the original data, bottom row (B) represents the transformed data with signed square root transformation for variables

v_{\theta},v_{\phi},\mathbf{B}_{\theta},\mathbf{B}_{\phi},\mathbf{J}_{r},\mathbf{J}_{\phi},\rho

, and power transform (power

=1/4

) for pressure

p

. No transformation is needed for

\mathbf{J}_{\theta}

Fig. 2 : Model Architecture of the proposed methodology employing a common Neural Operator that takes input channels and maps them into latent channels. The output of common NO is fed into task specific heads for correlated variables.

Fig. 3 : Component-wise distribution of evaluation metrics for rank ablation. Left is the Mean Square Error (MSE) of the 9 components to measure point-wise reconstruction error. Right is the Peak Signal-to-Noise Ratio (PSNR) to measure the signal normalized reconstruction fidelity.

Rank	ACC		EMD		MSE		NNSE		PSNR		UQI
Rank	Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.
0.2	0.9833	0.0241	0.0097	0.0194	0.0070	0.0192	0.9676	0.0430	37.3505	9.9128	0.8873	0.1108
0.3	0.9830	0.0256	0.0095	0.0203	0.0058	0.0169	0.9673	0.0445	37.5967	10.2752	0.8836	0.1149
0.4	0.9874	0.0205	0.0060	0.0121	0.0040	0.0114	0.9755	0.0364	38.6972	10.0915	0.9071	0.0929
0.5	0.9874	0.0203	0.0063	0.0127	0.0043	0.0126	0.9746	0.0381	38.0046	9.4680	0.9081	0.0914
1.0	0.9787	0.0338	0.0105	0.0215	0.0078	0.0218	0.9602	0.0549	37.0086	11.1103	0.8786	0.1204

TABLE I : Comparison of model performance when trained across factorization rank when using Tucker factorization. The metric values are aggregated over the components. For MSE and EMD [ 40 ] metrics, lower values are better, and for ACC, NNSE [ 41 ] , PSNR, and UQI, higher values are better.

Transformation	ACC		EMD		MSE		NNSE		PSNR		UQI
Transformation	Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.
None	0.9823	0.0390	0.0072	0.0169	0.0052	0.0173	0.9686	0.0644	39.6712	10.6426	0.8607	0.1530
Square-root	0.9874	0.0205	0.0060	0.0121	0.0040	0.0114	0.9755	0.0364	38.6972	10.0915	0.9071	0.0929
Natural Log	0.9806	0.0407	0.0081	0.0193	0.0062	0.0207	0.9642	0.0728	39.1683	10.7892	0.8527	0.1603

TABLE II : Comparison of model performance for ablation study across data transformation strategies. The metric values are aggregated over the components. For MSE and EMD metrics, lower values are better, and for others higher values are better.

Panel of Figure 6 — Fig. 4 : Component-wise reconstruction of solar MHD state variables. In each subfigure, columns represent the target variables. The top row shows the ground truth (GT), the middle row shows the model prediction (P), and the bottom row shows the residual computed as $GT-P$ .

Model	Params	ACC		EMD		MSE		NNSE		PSNR		UQI
Model	(in M)	Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.	Mean	Std.
SFNO	31.65	0.6307	0.4542	0.0416	0.1106	0.0870	0.3197	0.7651	0.2274	28.6258	11.4363	0.5662	0.4028
TFNO	412.92	0.9488	0.0753	0.0150	0.0337	0.0194	0.0555	0.9200	0.0937	31.8203	8.4401	0.8035	0.1882
LocalNO	14.82	0.9874	0.0205	0.0060	0.0121	0.0040	0.0114	0.9755	0.0364	38.6972	10.0915	0.9071	0.0929

TABLE III : Comparison of model performance with respect to baseline models. The values are aggregated over components. lower values are preferred for MSE and EMD, higher value is preferred for the remaining metrics.

Magnetohydrodynamics

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Transfer Learning PDE Operator Learning Neural Operators Data-Efficient Learning Magnetohydrodynamics

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Real-time reconstruction of magnetohydrodynamic equilibria is essential for plasma shaping, stability assessment and feedback control in magnetic confinement fusion. However, Grad-Shafranov equilibrium calculations remain largely device-specific and iterative, limiting their use in latency-constrained control settings. Existing neural approaches can accelerate individual equilibrium predictions, but they do not generally provide reusable models across changing plasma boundaries or tokamak geometries. Here we show that equilibrium reconstruction can be recast as a cross-device operator learning problem. We develop a domain-specific neural operator framework that maps geometry and profile parameters directly to the poloidal flux field, replacing repeated solve-on-demand computation with amortized operator inference. Using the analytically tractable Solov'ev family as a controlled Grad-Shafranov testbed, we generate equilibria across eight geometrically distinct tokamak-like configurations and benchmark five neural operator architectures under four transfer-learning strategies. Single-geometry pretraining gives poor transfer to unseen devices, whereas multi-geometry pretraining enables data-efficient adaptation. The Wavelet Neural Operator gives the strongest cross-geometry performance, reaching mean relative L2 errors below 4% with 100 labelled target equilibria and below 2% with full fine-tuning. The predicted magnetic fields satisfy the divergence-free constraint to numerical precision, and four architectures achieve millisecond or sub-millisecond inference. These results identify neural operator pretraining as a route towards reusable, real-time equilibrium inference across fusion device configurations.

Abstract

Magnetohydrodynamics

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