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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
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.
Method
The study evaluates LocalNO, combining locality-aware operator learning with preprocessing transformations and a multi-output architecture for correlated target fields.
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.
Takeaways & Limitations
The findings support locality-aware operator-learning surrogates for partial boundary-state reconstruction and future integration with heliospheric simulation pipelines.
Takeaways & Limitations
Evaluation uses simulations rather than real-world data, excludes radiative interactions, and does not address high-resolution simulation differences without additional preprocessing.
Abstract
from arXiv · showhide
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.
- 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.
- The reconstruction is challenging because target variables are nonlinear, strongly coupled, high-dimensional fields with multi-scale spatial dependence.
- Conventional CNN and MLP regressors may be sensitive to discretization, spherical geometry, resolution changes, or long-range interactions, motivating an operator-learning formulation.
- Local Neural Operators emphasize localized interactions while learning function-space mappings, matching boundary fields that combine local structure with broader physical coupling.
- 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.
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.
- Forward-simulation approaches evolve physical systems using governing laws and numerical solvers, including ballistic, kinematic, coronal, and MHD-based heliospheric models.
- MAS is a physics-based three-dimensional spherical MHD model that supplies the training data and contains the target variables used in this study.
- Inversion methods infer system parameters or states from observations and are used for surrogate modeling because numerical simulations can be computationally expensive and time-sensitive.
- Unlike much prior work on forecasting scalar or limited observables, this study reconstructs an entire multi-field boundary state from two observed spherical-surface components.
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.
- Physical surrogate models must preserve geometry and coupling because black-box regressors can produce low average error while violating physical consistency.
- Neural operators learn mappings between function spaces, making them suitable for physical problems whose inputs and outputs are fields defined over a domain.
- FNO, SFNO, TFNO, and geometry-informed neural operators use spectral or geometry-adapted mechanisms to model interactions across physical domains.
- LocalNO uses localized differential and integral kernel interactions to capture fine-scale structure, sharp gradients, and localized correlations.
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: Multi-field operator learning jointly predicts transverse velocity and magnetic fields, current-density components, density, and pressure from a shared underlying plasma state.
- D. Operator Learning for Multi-Field and Partial-State Reconstruction: Partial-state reconstruction treats radial velocity and radial magnetic field as observed boundary components and learns an operator for the missing fields.
- D. Operator Learning for Multi-Field and Partial-State Reconstruction: The study differs from prior work by combining heliospheric boundary reconstruction, multi-field operator learning, and LocalNO for localized and long-range spatial dependence.
- III. DATA PREPARATION: The dataset contains 907 samples from 616 Carrington rotations and 11 solar-wind components on a spherical surface grid.
- III. DATA PREPARATION: Sign-preserving nonlinear transformations expand small- and medium-valued regions while retaining directional information in vector quantities.
- III. DATA PREPARATION: Each output channel is normalized independently to [0, 1], then predictions are de-normalized and inverse-transformed to recover physical scales.
- III. DATA PREPARATION: A common Neural Operator maps input channels into latent channels, whose representation is passed to task-specific heads for correlated variables.
- III. DATA PREPARATION: The preprocessing pipeline stabilizes optimization and improves multi-field reconstruction quality.
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.
- The input fields are radial velocity and radial magnetic field at 30 solar radii.
- The target state contains non-radial velocity and magnetic-field components, three current-density components, density, and pressure.
- The model must infer missing physical quantities from limited observations across the spherical boundary.
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.
- The evaluated model families are SFNO, TFNO, and LocalNO.
- SFNO: SFNO provides a global spherical operator-learning baseline using Fourier-space kernels and spherical convolution.
- TFNO: TFNO factorizes spectral weight tensors to reduce learnable parameters while retaining the spectral operator structure.
- LocalNO: LocalNO combines localized differential and integral kernels with operator learning and spherical convolution without domain padding.
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.
- The comparison includes SFNO, TFNO, and LocalNO variants using spherical convolution without domain padding.
- All models use the same preprocessed dataset and evaluation protocol for a fair comparison.
- Evaluation reports PSNR, UQI, EMD, ACC, NNSE, and MSE using sample-wise and component-wise views.
- PSNR measures signal-to-noise reconstruction fidelity, UQI captures structural similarity, and MSE measures component-wise grid reconstruction error.
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: The rank ablation evaluates r ∈ {0.2, 0.3, 0.4, 0.5, 1.0}.
- Factorization Rank: Lower ranks improve parameter efficiency and regularization but can limit representation of complex coupling among solar hydrodynamic fields.
- Factorization Rank: Rank 0.4 yields the best overall trade-off across the evaluated metrics.
- Data Transformation: Signed square-root transformation improves data spread while preserving the sign of physically directional quantities.
- Data Transformation: The preprocessing transformation improves stability and reconstruction across multiple target components.
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.
- Baseline comparison: SFNO shows larger standard deviations, indicating less stable performance across reconstructed components.The passage links this instability to differing variable scales and spatial patterns.
- 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.
- Ablation studies: 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.
- 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.
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.
- Conclusion: 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.
- Conclusion: 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.
- 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.
- 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.