Enhancing Solar Radiation Storm Forecasting with Machine Learning and Physics Models at Korea Space Weather Center ^†

Abstract

Solar radiation storms, caused by high-energy solar energetic particles (SEPs) released during solar flares or coronal mass ejections (CMEs), have a substantial impact on the Earth’s environment. These storms can disrupt satellite operations, interfere with high-frequency (HF) communications, and increase the radiation exposure of high-altitude flights. To reduce these effects, the Korea Space Weather Center (KSWC) monitors and forecasts solar radiation storms using satellite data and predictive models. This paper introduces the space weather forecasting methods employed by the KSWC and the analysis approach for satellite data from GOES, SDO, the LASCO coronagraph, and STEREO. We introduce a predictive model for solar radiation storms, which is composed of two key components: (1) a machine learning model, which is trained using solar flare and CME data obtained from satellite observations, and (2) a physics-based model that incorporates the mechanisms of SEP generation through CMEs approaching the Earth. The machine learning model primarily forecasts the peak intensity of solar radiation storms based on real-time solar activity data, while the physics-informed model enhances the interpretability and understanding of the machine learning model’s predictions. The effectiveness and operability of this approach have been tested at the KSWC.

1. Introduction

Physical processes extending from the solar corona to the heliosphere accelerate particles, which are referred to as solar energetic particle (SEP) events. These processes include wave–particle interactions, magnetic reconnection associated with solar flares, and acceleration following shocks driven by coronal mass ejections (CMEs); e.g., [1,2,3]. SEP events can cause significant damage to the Earth, including impacts on electronics and satellites, biological harm to aircraft crews on polar routes, and damage to modern technological systems; e.g., [4,5]. To mitigate such effects, numerous models have been developed to analyze or predict SEP events; e.g., [6].

This paper aims to introduce the current status of the SEP prediction models available at the Korea Space Weather Center (KSWC). At the KSWC, we have operated and developed both machine learning-based models and physics-informed models.

The Korea Space Weather Center (KSWC) has operated an SEP prediction model based on machine learning (ML). The ML model is trained using observational data, including physical phenomena such as solar flares and CMEs, without prior knowledge of the underlying physics. Such a model has been employed based on the correlation between the observed SEP flux and solar flares or CMEs; e.g., [7,8,9]. Indeed, it has been observed that the SEP flux is correlated with the solar flare parameters (e.g., X-ray peak flux, source location, and impulsive time), e.g., [10], or CME parameters (e.g., speed, angular width, and location), e.g., [11]. The KSWC’s model predicts the maximum flux level of the SEPs based on physical processes, including solar flares and Earth-directed CMEs (see Section 2 for more details).

In contrast to ML models, the physics-informed model incorporates prior knowledge of the underlying physics of particle acceleration. In particular, in collisionless shocks driven by CMEs, the particles are accelerated by converging plasma waves near the shock surface through wave–particle interactions known as diffusive shock acceleration (DSA) e.g., [12,13]. Additionally, it has been theoretically demonstrated that the interaction between the waves and the shock structure influences the particle acceleration efficiency following shocks; e.g., [14,15,16]. At the KSWC, we have developed a physics-informed model that includes the aforementioned physical processes associated with collisionless shocks [17]. In the model, considering the wave–shock interactions, particle acceleration based on DSA is modeled to compute the SEP flux.

The organization of this paper is as follows. Section 2 describes the structure of the ML model and its performance. Section 3 provides a brief introduction to the physics used in the physics-informed model and its performance. A brief summary is given in Section 4.

2. Machine Learning Model

2.1. Model Structure

When the flux of 10 MeV protons ($J_{\mathrm{S}\mathrm{E}\mathrm{P}}$) is greater than $10 {\mathrm{c}\mathrm{m}}^{-2}{\mathrm{s}}^{-1}{\mathrm{s}\mathrm{r}}^{-1}$, the event is identified as an SEP event. In accordance with the correlations between the SEP flux and parameters associated with solar flares and CMEs, a multiple linear regression model was employed to derive the relationship between the SEP flux and the observed parameters. Figure 1 shows the structure of the ML model used in this work. Five input parameters were used (i.e., three parameters from solar flares and two parameters from CMEs), which were weakly correlated with each other. The parameters of the solar flares included the X-ray flux of the flare (i.e., flare strength), the location of the flare, and its growth timescale (i.e., flare rise time). The characteristics of the CMEs were parameterized by their speed and angular width. The model was trained by minimizing the cost function, defined as the mean squared error between the prediction and the observation. Observational data for the solar flares were obtained from the GOES geostationary satellite. Data for the SEP events from 1976 to 2024 were used for the training and testing datasets. During the same period, CME data were obtained from the LASCO coronagraph.

The model’s accuracy can be influenced by the choice of ML algorithms. For instance, it has been reported that empirical models based on linear regression can exhibit lower false-alarm rates compared to those using other algorithms, such as support vector machines and extreme gradient boosting, depending on the dataset and feature selection [6]. In this work, we adopted a fixed model structure based on linear regression and focused on optimizing the training strategy and input parameter combinations to enhance the model’s performance.

Solar flares can be classified into different groups based on the X-ray flux in the 1–8 Angstrom range ($F_X$). The following two strong classes of solar flares are mainly relevant to SEP production: (1) M class: ${10}^{-5} \mathrm{W} {\mathrm{m}}^{-2}\le F_X<{10}^{-4} \mathrm{W} {\mathrm{m}}^{-2}$; and (2) X class: ${10}^{-4} \mathrm{W} {\mathrm{m}}^{-2}\le F_X$. The characteristics of CMEs were analyzed using observational data from the LASCO coronagraph. Figure 2 shows the strategy for model training based on the physical processes. We trained four different models, defined as follows:

(a) Model 1: The model trained using solar flare data, including both M and X class flares but without CME data.

(b) Model 2: The model trained using solar flare data, including only M class flares but without CME data.

(c) Model 3: The model trained using M class solar flare data with CME data.

(d) Model 4: The model trained using both M and X class solar flares with CME data.

2.2. Results

The trained ML model provides the following empirical relation for the expected maximum SEP flux, in units of ${\mathrm{c}\mathrm{m}}^{-2}{\mathrm{s}}^{-1}{\mathrm{s}\mathrm{r}}^{-1}$:

$$J_{\mathrm{S}\mathrm{E}\mathrm{P},\mathrm{m}\mathrm{a}\mathrm{x}}\approx 30.64\times {\left(F_X\times ∆T\right)}^{1.325},$$

(1)

where $∆T$ represents the X-ray flare rise time. Here, the SEP flux considers only particles with energy greater than 10 MeV. This empirical relation highlights the correlation between the X-ray flux, the duration of the flare, and the production of SEP events. We confirm that this relation is consistent with previous findings; e.g., [18]. Figure 3 shows an example of the SEP prediction obtained by the ML model. Based on two solar flare events detected by the X-ray flux from the GOES-16 satellite, the model predicts the maximum SEP flux. The model has an operability for SEP prediction with a lead time of 12 h, which is reasonable for space weather forecasts within a 12 h period.

To evaluate the model’s performance, the mean absolute error (MAE) and root mean squared error (RMSE) are defined as follows:

$$\mathrm{M}\mathrm{A}\mathrm{E}={\displaystyle \frac{1}{N}}\sum _i\left|P^{\left(i\right)}-O^{\left(i\right)}\right|,$$

(2)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{\sum _i{\displaystyle \frac{{\left(P^{\left(i\right)}-O^{\left(i\right)}\right)}^2}{N}}},$$

(3)

where $P^{\left(i\right)}$ and $O^{\left(i\right)}$ stand for the predicted and observed SEP fluxes, and $N$ represents the number of test datasets. The annual performance of the ML model during 2024 is presented in

Table 1. Evaluation of the results of the ML model prediction for the year 2024.

	Lead Time = 1d	Lead Time = 2d	Lead Time = 3d
MAE	0.058	0.095	0.111
RMSE	0.189	0.252	0.281

. Depending on the flare location, the time evolution of the SEP flux can differ due to the structure of the interplanetary magnetic field between the Sun and the Earth. For instance, events occurring on the western side of the Sun produce a rapid enhancement of SEP flux within a lead time of one day, as shown in the example in Figure 3. In contrast, events occurring on the eastern side are less likely to produce an immediate enhancement of the SEP flux near the Earth. However, the SEP flux could be enhanced as a consequence of particle acceleration following a shock propagating through the interplanetary medium. In such cases, the SEP flux may be maximized when the CME reaches the Earth (i.e., typically corresponding to a lead time of two–three days). Because a substantial fraction of SEP enhancements due to solar flares are from events located on the western side of the Sun, both the MAE and RMSE increase as the lead time increases.

3. Physics-Informed Model Based on Particle Acceleration Following a Shock

3.1. Model Based on the One-Dimensional Fokker–Planck Equation

The physics-informed model particularly considers the physics of particle acceleration following shocks. The supersonic flow motions associated with CMEs drive the shocks as they propagate toward the Earth. The shock structure shows discontinuities in the fluid quantities, including the density ($\rho$), temperature ($T$), and magnetic field ($B$). The shock compression ratio is defined as the ratio between the downstream and upstream densities, given by $r\equiv {\rho }_2/{\rho }_1$. Additionally, the strength of the shock is typically parameterized by the Alfvén Mach number $M_A=U_1/V_{A1}$, where $U_1$ and $V_{A1}\propto B_1/\sqrt{{\rho }_1}$ are the flow speed and Alfvén speed measured upstream of the shock. The Mach number is also correlated to the compression ratio, given by

$$r={\displaystyle \frac{\gamma +1}{\gamma -1+2\beta /M_A^2}},$$

(4)

where $\gamma =5/3$ is the adiabatic index and $\beta$ is the ratio between the thermal and magnetic energy. Typical values of $\beta$ and $M_A$ for CME-driven shocks are $\beta ~ 0.1-1$ and $M_A ~ 1-4;$ e.g., [19].

In the shock rest frame, we employed the structure of a one-dimensional planar shock to build an SEP model based on the particle acceleration following shocks (see Figure 4). Under the test-particle regime, it is assumed that the dynamical feedback of shock-accelerated particles on the shock structure is negligible, meaning it does not alter the shock structure. Figure 2 shows the schematic diagram of the physics-informed model. In this paper, we solve the Fokker–Planck equation, particularly for the system containing fully isotropized downstream turbulence, see also [17,20,21]. To model the evolution of the particle distribution in position and momentum space through the wave–particle interactions near the shock surface, we solve the one-dimensional Fokker–Planck equation, incorporating particle advection and diffusion in an open field geometry with the magnetic scale length $L_B\approx B_1{\left(\partial B/\partial x\right)}^{-1}$:

$$\begin{array}{cc}{\displaystyle \frac{\partial f\left(x,p,t\right)}{\partial t}}& \\ & +{\displaystyle \frac{\partial }{\partial x}}\left(U\left(x\right)+U_W\left(x\right)-{\displaystyle \frac{\kappa \left(x,p,t\right)}{L_B}}\right)f\left(x,p,t\right)\hfill \\ & -{\displaystyle \frac{\partial }{\partial x}}\left(\kappa \left(x,p,t\right){\displaystyle \frac{\partial f\left(x,p,t\right)}{\partial x}}\right)\hfill \\ & -\left({\displaystyle \frac{\partial \left(U\left(x\right)+U_W\left(x\right)\right)}{\partial x}}-{\displaystyle \frac{U\left(x\right)+U_W\left(x\right)}{L_B}}\right){\displaystyle \frac{\partial }{\partial p}}\left({\displaystyle \frac{p}{3}}f\left(x,p,t\right)\right)\hfill \\ & -s\left(x,p,t\right)=0, \hfill \end{array}$$

(5)

where $s\left(x,p,t\right)~\delta \left(p-p_{min}\right)$ and $f\left(x,p,t\right)$ represent the source function and the particle distribution function, respectively. $p_{min}$ represents the minimum momentum for undergoing the DSA mechanism at the shock surface. The second and third terms of Equation (5) describe the particle distribution diffusion in position space with the appropriate diffusion coefficient $\kappa \left(x,p,t\right)$. The fourth term accounts for the momentum diffusion of the particle distribution, which is significant in describing the particle acceleration at the shock via the wave–particle interactions in a converging flow. As a consequence of the wave–particle interactions, the particle distribution follows a power-law form (i.e.,$f\left(x,p,t\right)\propto p^{-q})$. The spectral index $q=3r/(r-1)$ is associated with the shock compression ratio. The corresponding coefficient for momentum diffusion is $\kappa \left(p\right)\propto p^{q-3}$, in accordance with the plasma instabilities driven by the interaction between shock-accelerated particles and thermal background particles; e.g., [12,22,23]. While the wave speed in the upstream and downstream depends on the shock and wave interaction, this paper considers the upstream waves toward the shock surface (i.e., $U_{W1} ~-(U_1-V_{A1})$), which is generally applicable in converging flow. Additionally, the downstream waves are assumed to be isotropized (i.e., $U_{W2} ~ 0$); e.g., [24].

While cross-field diffusion, e.g., [25], is not implemented in this one-dimensional model, the wave drift effect near the shock, e.g., [14], is explicitly considered by incorporating the wave speeds (i.e., $U_{W1}$, $U_{W2}$) into the Fokker–Planck equation. This allows the model to account for the influence of upstream wave dynamics on particle acceleration. We expect that the combination of cross-field diffusion and the wave drift effect could further influence the production and propagation of SEPs, although this is beyond the scope of the present one-dimensional model.

Considering the gyroradius of the charged particles along the background magnetic field, the maximum momentum $p_{max}$ is determined by the size of the upstream magnetic field, $p_{max}c/e{B}_1\approx L_B$. If particles move beyond the distance $L_B$ from the shock position, they may escape into the ambient medium; e.g., [26]. The particle distribution function at an arbitrary timestep $t=t_0$, including the free escape boundary, is given by

$$f\left(x=0,p,t=t_0\right)\propto {\left({\displaystyle \frac{p}{p_{min}}}\right)}^{-q}\exp \left(-{\displaystyle \frac{p}{p_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\right)\exp \left\{-q{\int }_{p_0}^p{\displaystyle \frac{d{p}^{’}}{p^{’}\left(e^{\eta \left(p^{’}\right)}-1\right)}}\right\},$$

(6)

where the term $\eta \left(p\right)$ represents the integral describing the distance traveled by the particle before escaping from the shock region:

$$\eta \left(p\right) ~{\int }_0^{L_B}{\displaystyle \frac{U_{1}dx}{{\kappa }_1(x,p,t=t_0)}}.$$

(7)

Here, $U_1$ and ${\kappa }_1$ represent the upstream flow speed and diffusion coefficient, respectively.

3.2. Results

Using the physics-informed model, we estimate the expected SEP flux from the CME-driven shock. From the momentum spectrum, we primarily compute the SEP flux as a function of the energy ($F\left(E\right)$):

$$F\left(E\right)\approx {\displaystyle \frac{1}{4\pi s^2}}{\displaystyle \frac{dN}{dE}}={\displaystyle \frac{1}{4\pi s^2}}4\pi p^{2}f\left(p\right){\displaystyle \frac{dp}{dE}}{A}_{CME}{U}_{1,CME},$$

(8)

where $A_{CME}\approx R_{CME}^2{\Theta }_{CME}^2$ is the area of the observed CME-driven shock, with $R_{CME}/R_s\approx 21.5$ being the radius in units of solar radius, and ${\Theta }_{CME}$ is the angular width. $s$ represents the distance between the shock and the observer, and $U_{1,CME}$ is the upstream flow velocity of the observed CME-driven shock. We then calculate the observable flux by considering the solid angle of the CME-driven shock (${\Omega }_{CME}\approx {\Theta }_{CME}^2$), which is defined as follows:

$$J_{\mathrm{S}\mathrm{E}\mathrm{P}}\left(E>E_0\right)\approx \underset{E>E_0}{\int }{\displaystyle \frac{F\left(E\right)}{{\Omega }_{CME}}}dE,$$

(9)

where $E_0=10$ MeV is the threshold for SEP events.

Figure 5a shows the time evolution of the particle energy spectrum produced by CME-driven shocks. Spectral flattening is observed as a consequence of particle acceleration following the shock. Because the maximum wavelength of the waves near the shock is finite, the energy spectrum is unlikely to extend to infinite energy. Beyond the initial evolution stage (i.e., $t<5 \mathrm{h}\mathrm{r}$), the energy spectrum reaches saturation. Notably, a bump is observed at $t ~ 5 \mathrm{h}\mathrm{r}$, as shown by the magenta line in Figure 5a, due to the lack of longer-wavelength modes to scatter particles with energies greater than $E ~ {10}^{2.5} \mathrm{M}\mathrm{e}\mathrm{V}$. The accumulated flux of particles beyond 10 MeV is shown in Figure 5b. While the model slightly overestimates the particle flux during the initial evolution stage, the total SEP flux expected to be observed by geostationary satellites is generally consistent with observations. This result suggests that a substantial fraction of the SEP flux can be explained by one-dimensional physics (i.e., diffusive shock acceleration (DSA)) across the shock surface via wave–particle interactions.

According to the average performance of the physics-based model for 2024 (

Table 2. Evaluation results of the physics-based model prediction for the year 2024.

	Lead Time = 1d	Lead Time = 2d	Lead Time = 3d
MAE	0.095	0.177	0.231
RMSE	0.224	0.379	0.485

), its prediction accuracy is slightly lower than that of the ML model. To enhance its accuracy, future work should consider coupling it with solar wind models like ENLIL [27] to represent non-uniform shock structures. Moreover, modeling the nonlinear evolution of shock-driven plasma waves and their role in wave–particle interactions will be crucial. These remain important areas for further investigation.

4. Summary

This paper introduces the current status of the SEP prediction models available at the Korea Space Weather Center (KSWC). At the KSWC, we developed and operated both machine learning (ML) models and physics-informed models. The ML model has the advantage of predicting the maximum SEP flux based on the empirical relations between the SEP flux and the parameters of solar flares and CMEs. The physics-informed model, on the other hand, provides explainable results based on shock acceleration theory in CME-driven shocks. By operating both models simultaneously, we are able to predict SEP events in the presence of strong solar flares and associated CMEs with a lead time of approximately 12 h. The SEP forecasts performed by the KSWC could have significant impacts, particularly regarding the potential damage to geostationary satellites from SEP events. In this regard, an alert system has also been developed alongside the SEP prediction models.

While we have achieved promising accuracy in SEP prediction using a simple ML model based on multi-linear regression and an analytic approach based on shock acceleration theory at a stationary shock structure, we anticipate that more detailed models could further enhance the accuracy of SEP predictions. For instance, pre-existing turbulence influences the acceleration efficiency following shocks, e.g., [28,29], or can enhance particle acceleration on its own, e.g., [30]. Additionally, numerical simulations, such as those using the Monte Carlo method, e.g., [31], or combining magnetohydrodynamics with particle-in-cell simulations, e.g., [32], could provide a better understanding of the nonlinear acceleration process. We leave these aspects as future work.