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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

arXiv:2608.22782v1cs.LGastro-ph.IMastro-ph.SRcs.CV

TL;DR

The paper addresses reconstructing a complete solar MHD boundary state from radial velocity and radial magnetic field at 30 R⊙. It uses a Local Neural Operator with preprocessing and architectural choices tailored to coupled, multi-scale spherical fields. LocalNO outperforms SFNO and TFNO across reported component-averaged metrics, while moderate factorization and signed square-root transformation are favored; evaluation remains limited to simulations and excludes high-resolution data and radiative interactions.

  • Problem

    The paper asks whether missing coupled, multi-scale solar MHD boundary fields can be reconstructed from radial velocity and radial magnetic field for downstream heliospheric modeling.

    Source: §I · pp. 1–2

  • Method

    The study evaluates LocalNO, combining locality-aware operator learning with preprocessing transformations and a multi-output architecture for correlated target fields.

    Source: §I · p. 2, B. Model Families · p. 5, B. Data Transformation · p. 6

  • Results

    LocalNO outperforms SFNO and TFNO across reported component-averaged metrics, with smaller spread; rank 0.4 is most effective, and signed square-root transformation outperforms signed log transformation.

    Source: §VII · p. 6, §VIII · p. 6

  • Takeaways & Limitations

    The findings support locality-aware operator-learning surrogates for partial boundary-state reconstruction and future integration with heliospheric simulation pipelines.

    Source: §VIII · pp. 6, 8

    Evaluation uses simulations rather than real-world data, excludes radiative interactions, and does not address high-resolution simulation differences without additional preprocessing.

    Source: §VII · p. 6

Abstract

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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.

I. INTRODUCTION

The paper frames reconstructing a complete solar-wind boundary state from partial observations as a nonlinear, spatially coupled, multi-field problem. It evaluates Local Neural Operators as a locality-aware operator-learning approach for this task at 30 R⊙.

  • Solar-wind modeling requires jointly evolving flow, magnetic, density, and pressure fields, but practical observations may omit quantities needed for complete boundary specification.Source: §I · p. 1
  • At 30 R⊙, radial velocity and radial magnetic field are used to reconstruct non-radial velocity and magnetic fields, three current-density components, density, and pressure.Source: §I · p. 1
  • The reconstruction is challenging because target variables are nonlinear, strongly coupled, high-dimensional fields with multi-scale spatial dependence.Source: §I · p. 1
  • Conventional CNN and MLP regressors may be sensitive to discretization, spherical geometry, resolution changes, or long-range interactions, motivating an operator-learning formulation.Source: §I · p. 1
  • Local Neural Operators emphasize localized interactions while learning function-space mappings, matching boundary fields that combine local structure with broader physical coupling.Source: §I · pp. 1–2
  • The reconstructed state is intended to provide a complete inner boundary condition for downstream heliospheric modeling, while the study examines LocalNO and related design choices.Source: §I · p. 2

II. RELATED WORK

The related work distinguishes forward simulation from inversion and positions this study within machine-learning approaches that infer solar-wind states from limited observations. It emphasizes the difference between scalar prediction and reconstructing a complete multi-field boundary state.

B. Surrogate Modeling for Physical Systems

Surrogate modeling replaces or approximates expensive physical simulation, but must preserve coupled physical and geometrical structure. Neural operators address this need by learning field-to-field mappings, while LocalNO emphasizes localized structure.

  • Surrogate models approximate expensive forward simulations using approaches such as reduced-order models, Gaussian processes, and polynomial approximations.Source: B. Surrogate Modeling for Physical… · p. 2
  • Physical surrogate models must preserve geometry and coupling because black-box regressors can produce low average error while violating physical consistency.Source: B. Surrogate Modeling for Physical… · p. 3
  • Neural operators learn mappings between function spaces, making them suitable for physical problems whose inputs and outputs are fields defined over a domain.Source: C. Neural Operators · p. 3
  • FNO, SFNO, TFNO, and geometry-informed neural operators use spectral or geometry-adapted mechanisms to model interactions across physical domains.Source: C. Neural Operators · p. 3
  • LocalNO uses localized differential and integral kernel interactions to capture fine-scale structure, sharp gradients, and localized correlations.Source: C. Neural Operators · p. 3

D. Operator Learning for Multi-Field and Partial-State Reconstruction

The paper treats partial-state recovery as a multi-output operator-learning problem in which coupled boundary fields are reconstructed from limited observations. Its pipeline combines LocalNO architecture with sign-preserving transformations and component-wise scaling.

D. Operator Learning for Multi-Field and Partial-State Reconstruction

III. DATA PREPARATION

  • The dataset contains 907 samples from 616 Carrington rotations and 11 solar-wind components on a spherical surface grid.Source: §III · p. 3
  • Sign-preserving nonlinear transformations expand small- and medium-valued regions while retaining directional information in vector quantities.Source: §III · pp. 3–4
  • Each output channel is normalized independently to [0, 1], then predictions are de-normalized and inverse-transformed to recover physical scales.Source: §III · p. 4
  • A common Neural Operator maps input channels into latent channels, whose representation is passed to task-specific heads for correlated variables.Source: Figure 2 · p. 4
  • The preprocessing pipeline stabilizes optimization and improves multi-field reconstruction quality.Source: §III · p. 4

A. Problem Formulation and Proposed Architecture

The paper formulates solar boundary reconstruction as a supervised operator-learning task that maps two observed spherical-boundary fields to a nine-component target state. This is a structured field-to-field mapping rather than scalar regression.

B. Model Families

The study compares global spectral operator baselines with a locality-aware LocalNO designed for spherical boundary fields. SFNO uses spherical Fourier propagation, TFNO factorizes spectral weights, and LocalNO combines locality with spherical convolution.

SFNO

  • SFNO provides a global spherical operator-learning baseline using Fourier-space kernels and spherical convolution.Source: B. Model Families · pp. 4–5

TFNO

  • TFNO factorizes spectral weight tensors to reduce learnable parameters while retaining the spectral operator structure.Source: B. Model Families · p. 5

LocalNO

  • LocalNO combines localized differential and integral kernels with operator learning and spherical convolution without domain padding.Source: B. Model Families · p. 5

V. EXPERIMENTS

The experiments compare SFNO, TFNO, and LocalNO variants for reconstructing the nine-dimensional solar MHD state from radial velocity and radial magnetic field at 30 solar radii. Models share preprocessing, training conditions, and evaluation protocols, with metrics reported both globally and component-wise.

  • The models reconstruct a nine-dimensional target state from radial velocity and radial magnetic field at 30 solar radii.Source: A. Experimental Setup · p. 5
  • The comparison includes SFNO, TFNO, and LocalNO variants using spherical convolution without domain padding.Source: A. Experimental Setup · p. 5
  • All models use the same preprocessed dataset and evaluation protocol for a fair comparison.Source: A. Experimental Setup · p. 5
  • Evaluation reports PSNR, UQI, EMD, ACC, NNSE, and MSE using sample-wise and component-wise views.Source: B. Evaluation Metrics · pp. 5–6
  • PSNR measures signal-to-noise reconstruction fidelity, UQI captures structural similarity, and MSE measures component-wise grid reconstruction error.Source: B. Evaluation Metrics · p. 6

VI. ABLATION STUDY

The ablation study examines factorization rank and signed nonlinear preprocessing for multi-field reconstruction. Rank 0.4 provides the best overall trade-off, while signed square-root preprocessing improves stability and reconstruction across multiple target components.

Factorization Rank

Data Transformation

VII. RESULTS AND DISCUSSION

LocalNO outperforms SFNO and TFNO across reported component-averaged metrics, while ablations identify moderate factorization and signed square-root preprocessing as favorable choices. The study is evaluated on simulations and excludes radiative interactions and high-resolution simulation settings.

Baseline comparison

  • LocalNO outperforms SFNO and TFNO across reported component-averaged metrics, with smaller performance spread across samples and output components.The result supports locality as a useful inductive bias for reconstructing coupled fields from partial radial observations.Source: §VII · p. 6
  • SFNO shows larger standard deviations, indicating less stable performance across reconstructed components.The passage links this instability to differing variable scales and spatial patterns.Source: §VII · p. 6

Ablation studies

  • Factorization rank 0.4 improves performance over ranks 0.2 and lower, whereas ranks from 0.5 to 1.0 perform worse.The reported interpretation is limited expressivity under stronger compression and over-parameterization at higher ranks.Source: §VII · p. 6
  • Signed square-root transformation performs better across metrics than signed log transformation.The paper attributes the difference to log transformation behavior that harms reconstruction of small-magnitude field values.Source: §VII · p. 6

Limitations

  • The work is evaluated only on simulations, excludes radiative interactions, and does not address high-resolution simulation without additional preprocessing.These limitations constrain the demonstrated scope of the reconstruction approach.Source: §VII · p. 6

VIII. CONCLUSION

The study reconstructs the solar MHD boundary state at 30 R⊙ from radial velocity and radial magnetic field, targeting future heliospheric simulation use. Its conclusion favors locality-aware operator learning, moderate factorization, and signed square-root preprocessing, while future integration and physics-aware constraints remain proposed directions.

Conclusion

  • The reconstructed state uses only radial velocity and radial magnetic field and is intended as a boundary condition for future heliospheric simulations.The target is the solar magnetohydrodynamic boundary state at 30 solar radii.Source: §VIII · p. 6
  • Moderate factorization is most effective, with rank 0.4 achieving the best balance between compression and expressive power.This conclusion summarizes the rank ablation study.Source: §VIII · p. 6, Table I · p. 7
  • Signed square-root preprocessing improves data spread while preserving physically meaningful sign information, contributing to more stable and effective learning.The preprocessing result is reported as a design factor affecting reconstruction quality.Source: §VIII · p. 6, Table II · p. 7

Qualitative reconstruction

  • Figure 4 compares ground truth, model prediction, and residuals for each target variable.Columns represent target variables; rows show GT, prediction, and GT − P residuals.Source: Figure 4 · p. 7

Future directions

  • Future work may integrate learned reconstructions into downstream heliospheric pipelines and incorporate additional physics-aware constraints.These are stated future directions rather than demonstrated results of the present study.Source: §VIII · p. 8

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