Prediction of Solar Coronal Structures Using Fourier Neural Operators Based on the Solar Photospheric Magnetic Field Observation

This paper constructs the structures of the solar corona (SC) using Fourier neural operators (FNO) based on solar photospheric magnetic field observation. The purpose is to learn the mapping between two infinite‐dimensional function spaces, which takes the photospheric magnetic field as input and the magnetohydrodynamic (MHD) solar wind plasma parameters as output, from a finite collection of input‐output pairs. The FNO‐SC model is established using MHD simulated results of 36 Carrington rotations (CRs) from 2008, 2009, and 2020. The performance of the FNO‐SC model is tested for 6 CRs during various phases of the solar activity such as descending, minimum, and ascending phases to generate the 3D structures of the SC. With the MHD simulations as references, the average structure similarity index measure (SSIM) value for the magnetic field topology from 1 to 3Rs is around 0.88. From 1 to 20Rs, the SSIM values for the number density and radial speed surpass 0.9. Relative to OMNI observations, the mean absolute percentage error for the radial speed generated from the FNO‐SC model does not exceed 0.25. These results indicate that the FNO‐SC model effectively captures the solar coronal structures typical of the periods investigated, by recovering the MHD simulations as well as the observations. The FNO‐SC model is further trained with enriched data from the maximum phase to assess the capability of modeling such a situation. The FNO‐SC model costs 48.7 s for a single CR prediction, and thus facilitates real‐time space weather forecasting.


Introduction
Solar eruptions, the primary drivers of adverse space weather events, propagate through the ambient solar wind.Thus, understanding the structures of the background solar wind is crucial for comprehending these disturbances' propagation in the solar corona and interplanetary space.Due to its first-principle nature, the magnetohydrodynamic (MHD) equations describe the physical laws from the Sun to Earth in a self-consistent manner.Advancements in computer technology have rendered MHD equations effective in investigating the threedimensional (3D) solar wind structures, providing a robust framework for understanding the physical mechanisms of solar-terrestrial space.Two primary methods are commonly utilized to solve MHD parameters in the solar wind system.These include numerical MHD models based on traditional numerical schemes such as the finite element, finite difference, finite volume, and discontinuous Galerkin methods.Using these methods, numerous comprehensive physics-based numerical models have been developed to study the background solar wind.Examples include the Space Weather Modeling Frame (SWMF) (Gombosi et al., 2018;Tóth et al., 2005Tóth et al., , 2012)), the Hybrid Heliospheric Modeling System (HHMS) (Detman et al., 2006(Detman et al., , 2012;;Smith et al., 2008;Wu et al., 2007), the Coronal and Heliospheric Model (CORHEL) (Linker et al., 1994(Linker et al., , 1999(Linker et al., , 2010;;Odstrcil, 2003;Riley et al., 2001), the Solar InterPlanetary-Conservative Element Solution Element (SIP-CESE) (Feng et al., 2007(Feng et al., , 2010(Feng et al., , 2012(Feng et al., , 2015;;Hu et al., 2008), the coronal model by an implicit high-order reconstructed discontinuous Galerkin (Liu et al., 2023), the 3D MHD model proposed by Hayashi and others (Hayashi, 2005(Hayashi, , 2013;;Hayashi et al., 2022) and many others.For an in-depth overview of existing 3D MHD models, see the first chapter of Feng (2020).Grounded in physical principles, these methods require temporal and spatial discretization.Thus, a trade-off between resolution and computational cost is essential: coarser grids increase speed at the cost of accuracy, whereas finer meshes enhance accuracy but are more time-consuming.
Another category relies on experimental data, supported by statistical or machine learning techniques.Empirical models, vital for predicting solar wind parameters, balance computational efficiency and cost by identifying relationships in historical data (Owens et al., 2008;Riley et al., 2006).The widely-used Wang-Sheeley-Arge (WSA) model characterizes solar wind speed by factoring in expansion and distance from coronal hole boundaries (Arge & Pizzo, 2000;Wang & Sheeley, 1990).Further, studies indicate a correlation between solar wind speed and coronal hole areas (Levine et al., 1977;Nolte et al., 1976;Sheeley & Harvey, 1981).Using a parametric empirical model simplifies forecasting and accelerates computations significantly compared to numerical MHD models based on traditional numerical schemes.
Recently, various machine learning-based methods have emerged as faster alternatives for uncovering potential patterns in data.The deep neural networks (DNN) WindNet, proposed by Upendran et al. (2020), can be trained with extreme ultraviolet (EUV) images of the solar corona from Atmospheric Imaging Assembly (AIA) (Lemen et al., 2012) predict solar wind speed near 1 Astronomy Unit (AU).Raju and Das (2021) construct a convolutional neural network-based deep-learning model using images from AIA for solar-wind prediction.Bailey et al. (2021) introduce a machine learning approach as an alternative to the WSA model for predicting solar wind speed near Earth.Yang and Shen (2021) introduce a 3D MHD solar wind model driven by boundary conditions that are trained with an artificial neural network from multiple observations.Issan and Kramer (2023) present a datadriven reduced-order model for forecasting heliospheric solar wind speeds.Asensio Ramos et al. (2023) provide further information on machine learning in solar physics.These mesh-independent machine learning methods parameterize solutions in neural networks within finite-dimensional spaces.The accuracy of these models depends on data discretization levels during training, and they are prone to overfitting in predictions (Li et al., 2020).Camporeale (2019) emphasizes the necessity and importance of combining physics-based and machine learning approaches, termed "gray box" methods.Physics-Informed Neural Networks (PINNs) merge neural networks with physics principles to address fluid dynamics and other physical system challenges (E & Yu, 2018;Raissi et al., 2019;Bar & Sochen, 2019).In PINNs, physical laws or domain expertise serve to regularize machine learning models.This structure enhances the algorithm's learning efficiency and its ability to generalize from limited data.Bard and Dorelli (2021) explore the use of PINNs to reconstruct full MHD solutions from partial samples, simulating space-time environments around spacecraft observations in 1D.Zhao et al. (2023) propose a mutually embedded perception model to construct the primitive variables in MHD equations for a specific CR.Johnson et al. (2023) propose a loss function based on the adjusted Ohm's law for ideal plasma and model the solar wind prediction as a multivariate time series task.Jarolim et al. (2023) provides a powerful new framework for modeling solar active region magnetic fields by using the PINN approach.Generally, PINNs substitute the linear span of local basis functions in numerical schemes with neural networks.Similar to the fact that numerical schemes need to be rerun for each new set of initial and boundary conditions, Physics-Informed Neural Networks (PINNs) also requires to be optimized for each given set of initial and boundary conditions (Cai et al., 2021).
Recently, the focus has expanded from learning mappings within neural networks in finite-dimensional Euclidean spaces to employing neural operators between function spaces.Although both neural networks and neural operators are categorized as "black box" models, neural operators are distinguished by their ability to learn mappings from any parametric dependence to the solution of partial differential equations (PDEs), thus modeling an entire family of PDEs.Li et al. (2020) develop a graph neural network with nodes in the spatial domain of the output function, facilitating direct learning of the kernel that approximates elliptic PDE solutions.Sun et al. (2022) combine the OMNI data from Lagrangian Point 1 (L1) with the EUV images from the Solar Dynamics Observatory satellite to predict the solar wind speed at L1.After recognizing the equivalence of convolution in Euclidean space and point-wise multiplication in Fourier space, the corresponding representation of PDE inputs and outputs in Fourier space markedly enhances the efficiency of training Fourier neural operators compared to traditional deep neural networks.This provides an expressive, efficient architecture with state-of-the-art prediction accuracy for PDE approximation.Li et al. (2021aLi et al. ( , 2021b) ) propose and employ the FNO to model turbulent flows, achieving zero-shot super-resolution.The FNO technique has been used to establish a surrogate model that solves MHD governing plasma transport in a fusion device (Gopakumar et al., 2023).Peng et al. (2024) demonstrate that the FNO can learn 3D Navier-Stokes equations for reconstructing urban microclimates.
This paper, inspired by successfully learning the mappings between function spaces in parametric PDEs, aims to develop a solar corona (SC) prediction model using FNO (hereafter called FNO-SC model) based on solar photospheric magnetic field observation.The FNO-SC model is able to predict the solar coronal plasma flow and magnetic field topology.The establishment of the model involves the following sections.Section 2 describes the setup of the solar wind model and the data generation process using a numerical MHD model.Section 3 presents the framework of the FNO-SC model.Section 4 provides the 3D solar coronal structures by using the FNO-SC model to demonstrate the feasibility.The conclusion and discussion can be found in Section 5. Our next task will incorporate physical constraints by following the concept of PINNs as mentioned above.

Problem Formulation and DataSet Description
In this section, by using a numerical MHD model, we first generate solar coronal data for density, pressure, velocity, and magnetic fields.Then we describe the procedures for preprocessing the numerical MHD model results for training and testing the FNO-SC model.

Problem Formulation
The MHD equations provide a framework for understanding the interactions between plasmas and magnetic field to predict the behavior of plasma in the solar wind.The MHD model involves solving Equation 1 in a Suncorotating coordinate system (for details, refer to Feng et al., 2021).
In Equation 1, U = (ρ, ρv, E, B) T = ρ, ρv x , ρv y , ρv z , E, B x , B y , B z ) T stands for the conserved variables, where ρ is the plasma density, v the velocity field, E the total energy and B the magnetic field.The initial-boundary conditions, U 0 (x) and U b (x, t), are set to constrain the MHD problems, where t represents a specific time within the interval (0, T ).The position vector x = (x,y,z) ∈ Ω ⊂ R 3 lies within the 3D computational domain Ω.
The flux dyad F, incorporating the total energy E = p γ 1 + 1 2 ρv 2 + B 2 2 , is defined as follows, where the ratio of specific heats γ is 1.05 and I is the identity matrix.The factor 1 ̅̅ μ √ 0 is absorbed in B with vacuum permeability μ 0 = 4 × 10 7 π H m 1 .For ease of expression, the flux F is represented by primitive variables T with pressure p.In the solar corona, the source term S = S(U,∇U) includes the Powell source term (Powell et al., 1999), solar gravity, solar rotation, coronal heating, and solar wind acceleration source terms defined by Space Weather where solar gravity g = GM s r 3 r, r is the heliocentric position vector, G is the gravitational constant, and M s is the mass of the Sun.The angular velocity Ω of the Sun's rotation along the z-axis, the centrifugal force Ω × (Ω × r), and the Coriolis force 2Ω × v reflect the influence of the Sun's rotation.The momentum S m and the energy source terms Q e are used to empirically describe coronal heating and solar wind acceleration (Feng et al., 2010(Feng et al., , 2014(Feng et al., , 2017;;Nakamizo et al., 2009).
where Q 1 = 1.5 × 10 10 J • m 3 • s 1 .The dissipation lengths for momentum L M and added energy L Q 1 and L Q 2 are set at 1R s corresponding to the solar radius R s .The added momentum's intensity is defined as M = M 0 C a with M 0 = 3.5 × 10 13 N • m 3 , and the added energy's heating intensity is . C′ a is generally obtained from empirical models like WSA, and specified by C′ a = 1.0 0.8 ⋅ e (θ b /(1.0)) )/ 1 + f s ) 2 9 here.This formula indicates that C′ a relates to the coronal magnetic field's expansion factor f s and the minimum angular distance θ b between the open magnetic field line's foot point and the nearest coronal hole's boundary.In this work, the expansion factor f s is set to ( R s R ss ) 2 B Rs B Rss , where B R s and B R ss represent the magnetic field strengths at the solar surface and source surface R ss = 2.5R s , respectively.
The governing Equation 1 is rendered dimensionless using the parameters ρ and Ω s = v s /R s .The constants mentioned are detailed in Table 1, and the solar wind parameters derived in this study are processed using the aforementioned dimensionless approach.
The solar coronal structures are numerically obtained by solving Equation 1.The initial fluid values are determined by the Parker solar wind model (Parker, 1958(Parker, , 1959(Parker, , 1963)).With the determination of temperature T, density ρ, and sound velocity a s on the solar surface, the Parker solar wind model is uniquely defined.The initial magnetic field is obtained by the potential field (PF) model (Altschuler & Newkirk, 1969;Schatten et al., 1969), which utilizes solar photospheric observations from the Global Oscillation Network Group (GONG), managed by the National Solar Observatory.GONG's ground-based observations offer extensive temporal coverage, exceeding two decades.This longer period of continuous data collection is beneficial for machine learning training.The magnetic field data files used here are zero point corrected products, which are downloaded from the website https://gong2.nso.edu and in the Flexible Image Transport System (FITS) format.Fixed boundary conditions are applied to the solar surface, and the outer boundary is determined using equivalent extrapolation.With initial-boundary conditions determined, training and testing data can now be derived from the numerical MHD model of the solar corona as detailed in Feng et al. (2021), which employs a six-component grid system to eliminate polar singularities.The physical positivity of the model is managed by a positivity-preserving Harten-Lax-van Leer Riemann solver.To enhance the convergence rate, the model adopts the implicit lower-upper symmetric Gauss-Seidel method, ensuring a diagonally dominant sparse Jacobian matrix.A globally solenoidality-preserving method is used to maintain a divergence-free magnetic field.
By using the numerical MHD model just mentioned above, we run the MHD code to obtain the coronal plasma and magnetic field {U(x,T)} 36 1 (CRs 2065-2092and CRs 2224-2231).

Training Data Set
The data set U 0 (x, 0) and U(x, T ) obtained in Section 2.1 can be seen as the values of the function on the point clouds Given the nature of coronal expansion, point clouds are denser near the Sun's surface and become sparser radially away from the Sun.Specifically, the radial interval ΔR(i) between the i-th and (i + 1)-th point is as follows, here, A = 0.01/log(1.09),and n r represents the number of grids in one radial direction.It's straightforward that The grid is uniformly divided into n θ and n ϕ parts in each θ (zenith)and ϕ (azimuth) direction.Specifically, n r = 93, n θ = 80, and n ϕ = 42 in this work.
The direct use of point cloud data sets U 0 (x, 0) and U(x, T ), each encompassing n r × n θ × n ϕ points, requires excessive memory allocation.To optimize this, it is crucial to convert the 3D data information into a collective of 2D data sets.This process is displayed in Figure 2. By slicing the computational grid at a constant azimuth ϕ = ϕ 0 , n ϕ semi-toroidal surfaces are extracted.Then, pairing two such surfaces, specifically those at ϕ = ϕ 0 and ϕ = ϕ 0 + π, forms a series of complete toroidal surfaces.As a result, the point cloud T is reinterpreted as 2 toroidal surfaces.To preserve the azimuthal information, it is necessary to rotate the coordinate system.This entails rotating the semi-toroidal surfaces such that ϕ = ϕ 0 and ϕ = ϕ 0 + π are aligned with ϕ = 0 and ϕ = π, respectively.These rotations yield n ϕ 2 sets of 2D data sets U 2D 0 ∪U 2D , each containing 2 × n r × n θ points.These data sets, now containing rotated variables ρ, v r , p, Bx and, Bz, are then used as training data.

Establishment of the FNO-SC Model and Metrics
With the data set U 2D 0 and U 2D available in Section 2.2, this section commences with a description of the iterative architecture of the FNO-SC model.Additionally, the objective function and optimization strategy are introduced.Finally, the matrices used to assess the performance of the FNO-SC model are specified.

Establishment of the FNO-SC Model
We now shift our attention to the FNO-SC model designed for constructing the coronal structures, which learns the mapping from any functional parametric dependence to the solution for a finite collection of the observed input-output pairs.Thus an entire family of PDEs instead of a single equation can be learned by the FNO.The mapping from initial conditions to solutions is explicitly represented as U 2D 0 → U 2D .In particular, the input data U 2D 0 for the FNO-SC model is characterized within an independent Banach space U 2D 0 ∈ R d in , where the input space dimension d in = n r × n θ × 7 encapsulates the variables in U 2D 0 , along with r and θ to denote coordinates.Output data U 2D is defined in an independent Banach space U 2D ∈ R d out .The output space dimension is delineated by d out = n r × n θ × 1, indicating that the training is for each of MHD variables separately.
Our goal is to approximate G + by creating a parametric map G : U 0 × Ψ → U or G ψ : U 0 → U with ψ ∈ Ψ a finite-dimensional parameter space.By selecting ψ + ∈ Ψ, we ensure G ψ = G(ψ + ) ≈ G + .This allows for obtaining the solution U ∈ U for any U 0 ∈ U .The operator G ψ approximates the highly non-linear operator G + using an iterative architecture as follows,

Space Weather
10.1029/2024SW003875 ZHAO AND FENG here, P encodes functions from the space R d in to R d 1 dimensions, while Q decodes them from R d L to R d out dimensions, with L representing the total number of Fourier layers, set to 4 in this work (Li et al., 2021a(Li et al., , 2021b)).G ψ , lifts the input functions from lower-dimensional space to higher-dimensional representations through a point-wise linear transformation P and projects them back in the last step using Q to augment expressiveness.The FNO-SC model across its architecture G ψ , optimizes parameters including those within the P and Q lifting and projection layers, as well as the W and R matrices in the Fourier layers v j , j = 1, …, L.
The Fourier layer v j+1 (x) = σ(Wv j (x) + Kv j (x)), ∀ x ∈ Ω, which maintains the tensor shape unchanged and takes values in R d v , consists of the Fourier integral operator K, point-wise linear weight matrices W : R d v → R d v , and the elementally non-linear activation function σ : R → R. Specifically, W ∈ R d v ×d v with each element Utilizing uniform meshes and regular domains allows the application of the fast Fourier transform algorithm (FFT) for K, parameterized by FFT F, inverse FFT F 1 , and weight matrix R.This is described as The frequency mode k allows for a Fourier series expansion.For a frequency mode k, Given the truncation of higher modes due to k Fourier modes, the operation by weight tensor The entire structure of the FNO-SC model is depicted in Figure 3.

Training
Similar to a classical finite-dimensional setting (Vapnik, 1998), the cost function l is naturally defined in infinite dimensions as U × U → R, aiming to find a minimizer for the problem Demonstrating the existence of minimizers in an infinite-dimensional setting remains a challenging problem.In this case, the solution operator is optimized by minimizing the empirical data loss on a given data pair, and the norm ‖ ⋅ ‖ U on the Banach space U is defined by square error loss across all possible inputs For the FNO-SC model, structured with four Fourier integral operator layers, we set frequency mode k as 24 (Li et al., 2021a(Li et al., , 2021b)).Thus, the total number of parameters to be learned in training is 9,489,281.The training process employs the Adam optimizer to train 200 epochs, beginning with an initial learning rate of 0.005 that is halved every 60 epochs.

Evaluation
To evaluate the FNO-SC model performance in the prediction of solar wind parameters in the solar corona, we perform five common statistical analyses for the results, including structure similarity index measure (SSIM), Pearson correlation coefficient (Pcc), coefficient of determination R 2 , mean absolute percentage error (MAPE), cosine similarity (Cosi), and F1-score.The SSIM excels at capturing the correlation of spatial structures within image data, making it suitable for evaluating visual and structural similarities represented by human vision (Osorio et al., 2022).Both the Pcc and Cosi measure the similarity between two vectors, focusing on directional consistency.Notably, Pcc normalizes data by subtracting the mean and dividing by the standard deviation, thereby rendering it sensitive to the magnitude of the data (Rodgers & Nicewander, 1988).This characteristic ensures Pcc's effectiveness in detecting linear relationships between variables (Benesty et al., 2009).Cosi normalizes the vector lengths to unity, which allows it to gauge the orientation similarity in a high-dimensional space without being constrained by the linearity of the relationship (Manning et al., 2008).R 2 quantifies how variations in one data set can be elucidated by another, acting as an indicator of model fit data (Kvålseth, 1985).Meanwhile, the MAPE determines prediction accuracy by computing the average percentage difference between predicted and actual values, providing a macroscopic insight into data accuracy (Hyndman & Koehler, 2006).Furthermore, the F1-score, by harmonizing precision and recall, addresses potential biases due to sample imbalances, offering a comprehensive evaluation of model accuracy (Powers, 2011).Specifically, the total number N of actual (predicted) values y i ŷi ) is defined.SSIM computes the mean structure similarity index between two images, considering local luminance, contrast, and structure (Avanaki, 2009;Zhou et al., 2004).These three factors, represented by mean μ y , variance σ y , and covariance σ y ŷ, are weighted and combined to reflect the differences or degree of similarity in structural and numerical information.
To better capture the similarity of the images, SSIM is calculated as the mean of all blocks defined by sliding windows.
When two images are identical, SSIM reaches its maximum value of 1.The further the value is from 1, the lower the structure similarity between the two images.The Pcc measures the linear correlation between two variables.It is defined as the quotient of covariance and standard deviation between two variables (Kowalski, 1972), with values ranging between +1 and 1.A value of +1 indicates a total positive linear correlation, 0 indicates no correlation, and 1 indicates a total negative linear correlation.

√
The coefficient of determination R 2 is a widely used regression score function.It is represented by the ratio of the deviation between predicted and true values to the true mean.When the prediction is completely accurate, R 2 can achieve its maximum value of 1.The closer R 2 is to 1, the better the model's fitting effect is evaluated.
Mean absolute percentage error (MAPE) is sensitive to relative errors, reflected in the ratio of absolute error to the true value.When MAPE reaches its minimum value of 0, it indicates complete consistency between the actual and predicted values.
Cosi computes similarity as the normalized dot product of the predicted and true values (Manning et al., 2008).It focuses on directional differences between two vectors, characterizing the trend consistency between predictions and labels.The value range of Cosi is between 1 and 1.A value closer to 1 indicates higher similarity, while a value near 1 suggests opposite directions.A value approaching 0 indicates no significant similarity or difference between the vectors.
The F1-score is the harmonic mean of precision and recall (Davis & Goadrich, 2006;Everingham et al., 2010;Flach & Kull, 2015) and is used in the binary classification problem of magnetic field polarity in this work.The terms "positive" and "negative" refer to the results of a prediction model, and the terms "true" and "false" refer to whether that prediction corresponds to the observation results.Precision measures the prediction model's ability to not label a negative sample as positive, and recall measures its ability to find all positive samples.

Experiments for Solar Descending, Minimum, and Ascending Phases
For the testing CRs 2199, 2210, 2236, 2238, 2100, and 2249, Figure 4 presents the magnetic field topology on the meridional plane spanning from 1 to 3R s .For each CR in Figure 4, magnetic field lines are plotted from the same position on the solar surface, enabling a more effective comparison of different structures extending into space.
The magnetic field pattern near the solar surface plays a crucial role in determining the (pseudo-)coronal streamers.Generally, magnetic field lines forming coronal streamer structures originate from coronal holes with opposite polarity.In contrast, magnetic field lines leading to pseudo-coronal streamer structures originate from coronal holes with the same polarity.The position of the coronal streamer coincides with the formation of the current sheet, attributed to the opposite radial magnetic field polarity on both sides of the streamer.Both the FNO-SC and the numerical MHD model results reveal helmet streamers of the closed magnetic field near the Sun, extending outward at nearly identical latitudes.The pseudo-coronal streamers, observed in both schemes, consistently align for each CR, thereby confirming their presence at specific latitudes and altitudes.A notable distinction between the numerical MHD model results and those of the FNO-SC model lies in the sharper cusps of the helmet streamers in the FNO-SC model's results.Additionally, the same magnetic field line of the closed magnetic field is positioned closer to the solar surface in the FNO-SC model compared to the numerical MHD model results.
To further assess the outcomes of the FNO-SC model, the SSIM is employed to quantify the similarity of magnetic field topology structures in meridional planes.Utilizing a uniformly distributed set of 40 fixed azimuth angles, the SSIM values for the magnetic field structure are depicted in Figure 5.The values are concentrated within the range of 0.85-0.92,underscoring the FNO-SC model's capability in constructing magnetic field topology structures.The distribution exhibits a maximum value of 0.918, a minimum value of 0.852, and a mean value of 0.880.For the meridional plane illustrated in Figure 4, the corresponding SSIM values are 0.871, 0.882, 0.915, 0.888, 0.891, and 0.884.The magnetic neutral line B r = 0 from the numerical MHD model, the FNO-SC model and the potential field source surface (PFSS) model are compared in Figure 6.It is evident that the magnetic neutral lines follow the same variation pattern with longitude and latitude, and the inflection points align closely.The metrics Cosi and Pcc, presented in Table 2, are notably higher than 0.8, confirming that the FNO-SC model effectively captures the trend of the magnetic neutral line near the source surface.

Space Weather
10.1029/2024SW003875 (IQR = Q3 Q1).Outliers exceeding the maximum and minimum observed values are denoted by round dots.It is noteworthy that the SSIM distribution in each case is uniform and compact, with all values surpassing 0.90, affirming the FNO-SC model's proficiency in reconstructing the fluid field structures.Upon scrutinizing outliers, the density SSIM values for CR 2236 (0.98513 and 0.98515) marginally fall below the minimum observation threshold of 0.9852.However, it is crucial to note that even this slightly lower value remains comparatively high, signifying a commendable outcome.In particular, for the case of CR 2249, five points surpass the maximum observation threshold of 0.9408, which indicates exceptionally outstanding performance.
To elucidate this distribution pattern clearly, synoptic maps of plasma number density N (10 5 cm 3 ) and radial speed Vr (km•s 1 ) are presented in Figure 9, depicting the results from both the numerical MHD model and the FNO-SC model.As seen from this figure, solar wind characterized by low density and high speed predominantly occurs in the coronal holes at high latitudes or polar regions, while solar wind with high density and low speed is primarily distributed at low latitudes.Tilt structures observed in various longitude ranges are associated with polar coronal holes extending toward lower latitudes under the influence of the magnetic field.Both the FNO-SC model and the numerical MHD model exhibit tilt structures with nearly identical latitudes extending toward the north and south poles, with the longitude of the inflection point being almost consistent in each case.This consistency reinforces the robustness and accuracy of the FNO-SC model in replicating the intricate distribution patterns observed in the numerical MHD model results.
The accuracy of the FNO-SC model is quantitatively measured at 2.5R s by Pcc, R 2 , MAPE, and Cosi, as presented in Figure 10.For all cases, these four indicators provide consistent evaluations.The optimal performance is observed in CR 2210, where the maximum values for Pcc, R 2 , and Cosi in number density (radial speed) are 0.974, 0.918, and 0.999 (0.983, 0.962, and 0.998), while the minimum value of MAPE is 0.027 (0.059).For number density, CR 2238 attains the minimum values of 0.866 for Pcc, 0.998 for Cosi, and 0.535 for R 2 , with simultaneous maximum values of 0.057 for MAPE.In the case of radial speed, CR 2249 exhibits the minimum values of 0.896 for Pcc, 0.988 for Cosi, and 0.658 for R 2 , along with the maximum value of 0.135 for MAPE.These results affirm that the FNO-SC model effectively captures the distribution of variables on the sphere.
Furthermore, the distribution of variables around 0.1 AU is investigated, which is commonly utilized as the inner boundary in heliospheric MHD models.Figure 11 illustrates the distribution of number density N (10 3 cm 3 ) and radial speed Vr (km•s 1 ) on a sphere at 20R s .The pattern of high-speed and low density in high latitude areas remains consistent with that near the source surface.Due to the tenfold increase in radial distance, To further validate the FNO-SC model, the radial speed Vr and polarity of the radial magnetic field Br near 20R s from the FNO-SC model and the numerical MHD model are compared with the corresponding OMNI observations.To this end, the radial velocity at 1 AU is mapped back to 20R s from the ballistic approximation (e.g., Yang et al., 2012) and Heliospheric Upwinding eXtrapolation (HUX) Technique (Riley & Issan, 2021).Figure 13 presents the outcomes for the six cases.The FNO-SC model effectively captures the ascending and descending trends of radial speed observed by OMNI as heliolongitude changes and demonstrates comparable performance to the numerical MHD model.It is worth noting that the difference between the maximum and minimum values observed by the model and OMNI varies at different CRs, and the numerical MHD model shows similar performance.Specifically, the MAPE measures the difference between the FNO-SC model, the numerical MHD model, and OMNI data, as shown in Table 3. Regarding the HUX model, the MAPE for the FNO-SC model ranges from 0.1515 to 0.2462, in comparison with those of the numerical MHD model exhibits a range of 0.1323 to 0.2426.In the context of the ballistic approximation, the FNO-SC model demonstrates a MAPE range from 0.1459 to 0.2457, whereas the numerical MHD model ranges from 0.1469 to 0.2804.Across the examined cases, the observed difference in performance between the two models does not exceed 0.08.The comparison of magnetic field polarity obtained by the FNO-SC and the numerical MHD model is assessed with the F1-score.The F1-score for the FNO-SC model varies from 0.5352 to 0.7590, whereas the numerical MHD model's F1-score is between 0.4471 and 0.8316.As usual, magnetic field polarity can be affected by waves, perturbations, and the input magnetic map (Lepri et al., 2008;Perri et al., 2023;Wang et al., 2019).

Experiment for Solar Maximum
This part evaluates the performance of the FNO-SC model for solar maximum phase by CRs 2141 and 2149.Figure 14 displays the magnetic field configurations for meridian planes and source surface.Although the topology of the solar maxima magnetic field is complex, the FNO-SC model can still capture some leading structures such as pseudo-coronal streamers on the meridian planes, and locate the extension position and trend of the magnetic neutral line.The distribution of number density and radial speed on the meridional planes are presented in Figure 15.These demonstrate that the FNO-SC model exhibits the same patterns and trends in number density and radial speed as the numerical simulation results with respect to the variation of heliocentric distance.
In practice, the FNO-SC model primarily captures the basic trends of number density and radial speed as functions of azimuth ϕ for a given radius, which means it can distinguish the forms of high-speed flow and low-speed flow.
As for the metrics associated with number density and radial speed, R 2 , Pcc, Cosi, and MAPE, do not achieve the levels of optimality for the solar minimum, descending, and ascending phases when using numerical simulations as reference.This observation is consistent with the conclusion drawn by Rahman et al. (2024) that the performance of the machine learning model during solar maxima does not match its efficacy during solar minima.
The declining performance of the FNO-SC model during solar maxima may stem from several factors.One potential reason is the inherent observational uncertainties in photospheric magnetic field data from the imperfect measurements (e.g., Jin et al., 2022;Sun et al., 2015;Wang et al., 2002).Meanwhile, the input magnetogram derived from various sources significantly influences numerical MHD simulations (Jian et al., 2015; Li  Here, the "+1" stands for a radial magnetic field away from the Sun and " 1" toward the Sun. et al., 2021aSun. et al., , 2021b;;Sachdeva et al., 2019).This effect becomes particularly pronounced during solar maxima, where the simulated results can vary with the specific input magnetograms utilized (Arge et al., 2024;Huang et al., 2024;Sachdeva et al., 2021Sachdeva et al., , 2023)).Another factor could be the significant gradients in strong magnetic fields, which pose challenges for numerical methods in terms of convergence and non-physical deviations, complicating high-quality numerical simulations (Gressl et al., 2014).Additionally, the global coronal study for solar maxima remains an area of ongoing research (e.g., Brchnelova et al., 2023).Due to the aforementioned factors, high-quality MHD numerical results cannot well be achieved during solar maxima for our training purpose.This may be the third possibility.Although facing such problems, we enriched our training data set with numerical MHD results from CRs 2129-2134 and 2154-2159 (hereafter called enriched data set), to see if the model's performance during solar maxima can be improved.
The model trained with the enriched data set is denoted as FNO-SC+.To quantitatively assess the impact of enriched data on model performance, we present the metrics in Figure 16 for the FNO-SC and FNO-SC+ models.
It is evident that for number density and radial speed at 2.5 and 20R s , the Cosi, R 2 , Pcc have increased, while the

Space Weather
10.1029/2024SW003875 ZHAO AND FENG MAPE has decreased, indicating an overall improvement in the model performance.However, this performance is not as good as that during solar descending, minimum, and ascending phases.
Finally, we compared the results of the proposed models with the observed data.Figure 17 shows the results from the FNO-SC, FNO-SC+, numerical MHD model, alongside a comparison of observed radial speed and magnetic field polarity from OMNI.Table 4 illustrates the MAPE and F1-score between the FNO-SC, FNO-SC+, numerical MHD model, and OMNI data.From Table 4, we can see that the FNO-SC trained with the enriched data set slightly improves.

Conclusions and Discussions
In this paper, we establish the FNO-SC model for 3D solar coronal study by using the FNO technique.In such a model, to address the memory bottleneck in processing 3D data, we recommend employing reduction techniques alongside a rotated coordinate system.This method compresses the 3D field data grids to a more manageable 2D data format, thus preserving essential details and facilitating training on a single CPU.The experimental results demonstrate that the FNO-SC model performs well in producing solar coronal structures.The modeled results indicate an average SSIM of 0.88 for the FNO-SC model in forecasting solar magnetic field patterns between 1 and 3R s , compared with the numerical MHD model.The prediction of complex coronal features, such as pseudo-streamers, highlights the FNO-SC model's proficiency in capturing the intricate variations along heliocentric distances within the solar corona.This suggests the model's capability to precisely predict the emergence of two smaller or shorter ordinal streamers in the lower corona, evolving into a pseudo-streamer at higher altitudes.Such proficiency suggests the model's potential to significantly enhance our understanding of phenomena in the corona.For the radial speed and number density spanning from 1 to 20R s in the meridional, all SSIM values consistently surpass 0.90.Close to the source surface, the Cosi in number density and radial speed remain above 0.98.Additionally, the Cosi of the magnetic neutral line is consistently above 0.9.At approximately 20R s , the Cosi in number density and radial speed is more than 0.97.Compared with OMNI observations, the difference in MAPE between the FNO-SC and MHD model for radial speed does not exceed 0.08.This paves the way for high-quality AI-aided solar coronal simulations.In comparison, utilizing the FNO-SC for a single Carrington rotation prediction takes merely 48.7 s.This represents a significant speedup, at 556.6X, in contrast to the MHD model employed here (Feng et al., 2021).This undoubtedly makes the FNO-SC more advantageous for real-time forecasting.
There is a lot of room for improvement.First, the input photospheric magnetic field data contains some observational uncertainties.Solar magnetic observation data suffer from imperfect measurements (e.g., Jin et al., 2022;Sun et al., 2015;Wang et al., 2002).Meanwhile, as evidenced by former studies (Gressl et al., 2014;Jian et al., 2015;Li et al., 2021aLi et al., , 2021b;;Sachdeva et al., 2019), the input magnetogram, derived from various sources, significantly influences numerical MHD simulations.Similarly, the machine learning model can somehow yield different results with different solar-surface observations.The impact of the input magnetogram on the proposed FNO-SC model will be investigated in the future.Second, the current training data is restricted to the numerical MHD model results for solar corona during solar minimum, obtained by using numerical schemes.The addition Here, the "+1" stands for a radial magnetic field away from the Sun and " 1" toward the Sun.(Rosofsky & Huerta, 2023) to make it possible to acquire more effective and precise solar wind parameters.
Figure 1 displays the monthly mean and 13-month smoothed sunspot numbers (SILSO World Data Center, 2007-2023), covering 2007 to 2023.Solar from November 2008 to November 2019, peaking in February 2014.Solar activity increases for 62 months and then declines over the next 70 months.Solar cycle 25 begins in December 2019.As marked by orange circles in Figure 1, CRs 2065-2092 and CRs 2224-2231 situated in the years 2008, 2009, and 2020, are typically considered around the solar minima of Solar cycles 24 and 25.In the next subsection, we describe how to generate our training data set required for the setup of the FNO-SC model.

Figure 1 .
Figure 1.Time series of 13-month smoothed monthly sunspot number for 2007-2023.The orange circles stand for Carrington rotations near the years 2008, 2009, and 2020, which are utilized for training the model.The gray squares represent the monthly average sunspot numbers, with their corresponding Carrington rotations indicated in the figure.These gray squares are used for testing the model's performance.

Figure 2 .
Figure 2. Illustration of the training data.For each photospheric magnetic field observation B, 3D data U 0 and U are defined on the point cloud T , while 2D training data sets U 2D 0 and U 2D are generated by slicing and rotating the 3D data.

Figure 3 .
Figure 3. Schematic of the FNO-SC model for solving the MHD equations with the solar photospheric magnetic observation.The box in the right column shows the sequence of a Fourier layer, which is the core of the FNO-SC model.The Fourier layer adopts an encoder-decoder structure and consists of Fourier transform F, linear transform on the truncated Fourier modes R, and inverse Fourier transform F 1 .
of the F1-score is 1 and its worst is 0. In this work, precision and recall contribute equally to the F1This section evaluates the performance of the FNO-SC model for eight testingCRs: 2199CRs:  , 2210CRs:  , 2236CRs:  , 2238CRs:  ,  2100CRs:  , 2249CRs:  , 2141CRs:  , and 2149.These eight CRs, marked by gray squares in Figure1, are almost around solar descending, minimum, ascending, and maximum phases.Specifically, CR 2199 (from 30 December 2017, to 27 January 2018) and CR 2210 (from 26 October to 23 November 2018) are around the descending phase of Solar cycle 24.CR 2236 (lasting from 5 October to 1 November) and CR 2238 (from 28 November to 25 December) are near the solar minimum of Solar cycle 24 in 2020.Additionally, CR 2100 ( 9August to 5 September 2010) and CR 2249 ( 24September to 21 October 2021) are around the ascending phase of Solar cycles 24 and 25.CRs 2141 (31 August to 28 September 2013) and 2149 (7 April to 4 May 2014) are around the solar maximum phase of Solar cycle 24.The computational work was carried out at the National Key Scientific and Technological Infrastructure project "Earth System Science Numerical Simulator Facility" (EarthLab) on a single node with two Hygon C86 7185 32-core processors operating at 2.0 GHz.The FNO-SC model costs about 282 s per epoch, finishing the training by spending almost 15 hr in total.Predicting solar coronal structures for one CR takes about 48.7 s.

Figure 5 .
Figure 5.The structure similarity index measure (SSIM) of the magnetic field topology (B) from the FNO-SC model and the numerical MHD model across 40 uniformly distributed meridian planes for testing Carrington rotations 2210, 2199, 2236, 2238, 2100, and 2249.The minimum SSIM value of 0.85 proves the ability of the FNO-SC model to model magnetic field configurations.

Figure 8 .
Figure 8. Box plots of the SSIM (structure similarity index measure) for number density N (log 10 cm 3 ) and radial speed Vr (km •s 1 ), comparing the FNO-SC and the numerical MHD models across testing Carrington rotations 2210, 2199, 2236, 2238, 2100, and 2249.The analyses are performed on the meridional plane at 40 uniformly distributed fixed longitudes.

Figure 9 .
Figure 9. Synoptic maps of number density N(10 5 cm 3 , left two columns) and radial speed Vr (km•s 1 , right two columns) obtained from the FNO-SC model (first and third columns) and the numerical MHD model (second and fourth columns).These maps correspond to the surface at 2.5R s for testing Carrington rotations (a) 2210, (b) 2199, (c) 2236, (d) 2238, (e) 2100, and (f) 2249.

Figure 10 .Figure 11 .Figure 12 .
Figure10.The composite diagram compares the FNO-SC model to the numerical MHD model in terms of the number density N (cm 3 ) and radial speed Vr (km•s 1 ) at 2.5R s .The cluster bar represents Cosi (cosine similarity) on the primary axis, complemented by line charts for Pcc (Pearson correlation coefficient), R 2 (coefficient of determination), and MAPE (mean absolute percentage error) on the secondary axis.

Figure 13 .
Figure 13.Line chart comparing radial speed and radial magnetic field polarities at 20R s among the mapped temporal profiles of OMNI, the numerical MHD model, and the FNO-SC model for testing Carrington rotations (a) 2210, (b) 2199, (c) 2236, (d) 2238, (e) 2100, and (f) 2249.Here, the "+1" stands for a radial magnetic field away from the Sun and " 1" toward the Sun.

Figure 16 .
Figure16.The composite diagram compares the FNO-SC and FNO-SC+ models with the numerical MHD model in terms of the number density N (cm 3 ) and radial speed Vr (km•s 1 ) at 2.5R s (the left column) and 20R s (the right column).The cluster bar represents Cosi (cosine similarity) on the primary axis, complemented by line charts for Pcc (Pearson correlation coefficient), R 2 (coefficient of determination), and MAPE (mean absolute percentage error) on the secondary axis.

Figure 17 .
Figure17.Line chart comparing radial speed and radial magnetic field polarities at 20R s among the mapped temporal profiles of OMNI, the numerical MHD model, and the FNO-SC model for testing Carrington rotations 2141 (the first and third columns) and 2149 (the second and fourth columns).Here, the "+1" stands for a radial magnetic field away from the Sun and " 1" toward the Sun.

Table 1
Parameters Used for Nondimensionalizing MHD Equations

Table 2
Comparison of Cosi (Cosine Similarity) and Pcc (Pearson Correlation Coefficient) Values Among the Results of the FNO-SC Model, Numerical MHD Model, and PFSS Model

Table 3
Evaluation of Radial Speed and Magnetic Field Using the FNO-SC and the Numerical MHD Models, Compared to the Mapped Temporal Profiles of OMNI With Ballistic Approximation and Heliospheric Upwinding eXtrapolation (HUX) Technique Figure 14.Magnetic field topology B on the meridional planes for (a) ϕ = 236° 56°, testing CR 2141 and (b) ϕ = 281° 101°, testing CR 2149, spanning from 1 to 3R s .They are derived from the FNO-SC model (the first column) and the numerical MHD model (the second column).The synoptic maps in the third column depict the magnetic neutral line Br = 0 at 2.5R s , obtained from the FNO-SC model (red dash), numerical MHD model (green solid) and PFSS model (black dash dot).

Table 4
Evaluation of Radial Speed and Magnetic Field Using the FNO-SC and the Numerical MHD Models for CRs 2141 and 2149, Compared to the Mapped Temporal Profiles of OMNI With Ballistic Approximation and Heliospheric Upwinding eXtrapolation (HUX) Technique MHD numerical results for solar corona during solar maximum also be helpful.Also, observational data from other sources, such as the Parker Solar Probe (PSP) and Solar Orbiter (SO), can be utilized to refine the model training through data restrictions.Third, the current model training is solely data-driven.The next step entails integrating physical information into the model and creating physics-based neural operators of