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GCR Spectra Reconstructed with Neutron Monitor Yield Function and Artificial Neural Networks: Comparison of Two Methods

This paper presents and compares two methods—a calibrated yield function with a force-field scheme and artificial neural networks—for reconstructing time-resolved galactic cosmic-ray proton and helium spectra from global neutron monitor data, demonstrating that the neural network approach offers superior accuracy and robustness in extending spectral records to periods lacking direct satellite observations.

Original authors: Stepan Siruk, Vladislav Alekseev, Victor Kuzminov, Rustam Yulbarisov, Andrey Mayorov

Published 2026-07-27

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Stepan Siruk, Vladislav Alekseev, Victor Kuzminov, Rustam Yulbarisov, Andrey Mayorov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Invisible Rain and the Cosmic Weather Station

Imagine the Earth is constantly being pelted by an invisible rain of tiny, super-fast particles called Galactic Cosmic Rays (GCRs). These aren't water droplets, but high-energy atoms from deep space, zooming through the galaxy at nearly the speed of light. However, they don't just hit us directly. Before they reach our atmosphere, they have to navigate a giant, swirling magnetic bubble surrounding our Sun called the heliosphere. Think of this bubble like a cosmic wind tunnel; as the Sun blows its own "solar wind" (a stream of charged particles) outward, it creates magnetic turbulence that pushes back against the incoming cosmic rain, slowing some down and deflecting others. This process is called "solar modulation."

Why do we care about this invisible rain? Because it's a double-edged sword. On one hand, it's a natural radiation hazard for astronauts and airplane crews. On the other, it acts like a giant, invisible brush that paints chemical changes in our atmosphere, potentially influencing weather patterns and the Earth's electric circuits. To understand this, scientists need to know exactly how many of these particles are hitting us and how energetic they are at any given moment. Usually, we rely on satellites to catch these particles directly, but satellites have limits: they can break, they can't be fixed in space, and they often take a long time to process their data. This leaves us with a gap in our knowledge, especially for the past or for times when satellites are quiet. We need a way to "see" this cosmic rain using tools we already have on the ground.

The Paper: Turning Ground Sensors into a Cosmic Telescope

This paper is a story about two different ways to solve a puzzle: how to reconstruct the full energy spectrum of these cosmic rays using only data from a global network of ground-based detectors called Neutron Monitors (NMs). These monitors are like giant, sensitive ears on the ground; they don't hear the cosmic rays directly (which are blocked by the atmosphere), but they hear the "echo" or "splash" created when cosmic rays crash into the air and create a shower of secondary particles, including neutrons.

The researchers compared two very different methods to turn these ground-level "splashes" back into a picture of the original cosmic rain.

Method 1: The Physics Detective (The Old Way)
The first method is like a detective trying to solve a crime using a rulebook and a few clues. This approach uses a "yield function" (a mathematical recipe that tells you how many splashes a specific energy of cosmic ray should create) combined with a "force-field" model. The force-field model is a simplified way of describing how the Sun's magnetic bubble pushes the cosmic rays away. The researchers tried to make this old-school method better by adding a few extra knobs and dials to the math, such as accounting for the different types of heavy particles in the cosmic rain and the specific mix of helium isotopes. They calibrated this model using data from a satellite called AMS-02 (which was flying from 2011 to 2019) to make sure the "recipe" was accurate.

Method 2: The AI Learner (The New Way)
The second method is like training a super-smart student (an Artificial Neural Network) to look at the same clues and guess the answer. Instead of following a strict rulebook, this AI was fed a massive amount of data: the daily counts from 17 different neutron monitors around the world, plus information about the Sun's activity (like sunspot numbers) and the Earth's magnetic mood (geomagnetic indices). The AI learned the patterns by itself, figuring out how the ground counts relate to the actual cosmic ray energy without being told the specific physics equations.

The Showdown: What They Found
When the researchers compared the two methods against real satellite data, the results were clear.

The "Physics Detective" method (Method 1) was okay at seeing the big picture. It could track the long-term 11-year cycle of solar activity, where cosmic rays get stronger when the Sun is quiet and weaker when the Sun is stormy. However, it struggled with the details. It tended to overestimate how much the cosmic rays changed during short-term storms and underestimated how much the Sun's magnetic bubble actually slowed them down at high energies. It was like a weather forecast that got the season right but missed the daily rain showers.

The "AI Learner" (Method 2) was the clear winner. It reproduced the data with much higher precision.

Accuracy: The AI reduced the average error by about four times at low energies and two times at high energies compared to the physics method.
Reliability: While the physics method sometimes guessed wrong by a lot, the AI's predictions were usually within a tiny margin of error (a "chi-squared per degree of freedom" value near 1, which is the gold standard for a good fit).
Time Travel: Because the AI learned the patterns so well, the researchers could use it to "fill in the blanks" for times when we didn't have satellite data. They successfully reconstructed cosmic ray data for 2006–2011 (matching older PAMELA satellite data) and 2019–2022 (matching newer AMS-02 data), effectively turning the ground network into a time machine for cosmic rays.

The Twist: The Magnetic Flip-Flop
One of the most interesting discoveries was a "hysteresis" effect. When the researchers plotted the results, they saw that the physics method behaved differently depending on the direction of the Sun's magnetic field. Before 2014, when the Sun's magnetic polarity was negative, the physics method consistently overestimated the cosmic ray flux. After 2014, when the polarity flipped to positive, the method got much closer to the truth. The AI, however, didn't get confused by this flip; it handled both periods equally well. This suggests that the old physics rules need a serious update to account for how the Sun's magnetic field changes over time.

The Bottom Line
The paper concludes that while the old physics-based method is useful for understanding the general trends, the new AI approach is far superior for getting the details right. By training on data from a global network of neutron monitors, we can now effectively use the ground as a giant, real-time spectrometer. This means we can monitor the cosmic radiation environment with high precision, even without a satellite in the sky, and we can look back into the past to see how the cosmic rain behaved during times when we didn't have direct eyes on the sky. The global network of neutron monitors, once just a collection of ground sensors, has been upgraded into a unified, high-tech telescope for the cosmos.

Technical Summary: GCR Spectra Reconstructed with Neutron Monitor Yield Function and Artificial Neural Networks

Problem Statement
Galactic cosmic rays (GCRs) are modulated by the heliosphere, causing their energy spectra near Earth to differ from the local interstellar spectra (LIS). Understanding these variations is critical for studying solar activity, space weather, and atmospheric processes. While space-based instruments like PAMELA and AMS-02 provide direct measurements, they suffer from data latency, potential radiation degradation, and limited temporal resolution due to statistical constraints. Conversely, ground-based neutron monitors (NMs) offer continuous, high-temporal-resolution data but only measure the integral flux of secondary atmospheric particles, not the primary GCR spectra directly. Reconstructing the primary energy spectra from NM data is challenging due to uncertainties in yield functions (YF), coupling functions, and the need for empirical calibration that often fails to capture energy-dependent detector non-idealities.

Methodology
The authors present and compare two distinct frameworks for reconstructing time-resolved GCR proton and helium energy spectra using global NM network data:

Physics-Based Approach (YF + Force-Field Approximation):
- Yield Function Calibration: Theoretical count rates are calculated using NM yield functions. The authors refine the standard approach by incorporating a scaling factor $S(T)$ to account for the contribution of heavier nuclei (Z > 2) and the isotopic composition of helium ( $^3$ He/ $^4$ He ratio), using AMS-02 time-averaged data.
- Calibration: To address detector non-idealities, the authors introduce an empirical calibration scheme involving a scaling factor $\kappa$ and an effective atmospheric depth parameter $h^*$ . However, fitting this to AMS-02 data (2011–2019) revealed that $h^*$ was nearly identical to the actual atmospheric depth, suggesting this specific parameterization was unnecessary.
- Force-Field Model (FFA): The GCR spectra are parameterized using a modified two-parameter force-field approximation ( $\phi_0, \phi_1$ ) to describe the solar modulation potential $\phi(T)$ . This model is fitted daily to the count rates of five low-cutoff NMs to reconstruct the spectra.
Data-Driven Approach (Artificial Neural Networks):
- Inputs: The models utilize daily count rates from 17 NMs, the sunspot number (SSN), the Ap geomagnetic index, and the heliospheric magnetic field polarity ( $A$ ). Crucially, the SSN and polarity inputs are time-delayed and averaged to account for the physical propagation time of solar wind effects to the outer heliosphere.
- Architecture: Two neural network architectures were trained: a Convolutional Neural Network (CNN) and a Multilayer Perceptron (MLP).
- Training: Models were trained on daily AMS-02 proton and helium spectra (2011–2019). The training minimized mean squared error across energy bins.
- Interpretability: SHAP (SHapley Additive exPlanations) values were used to verify that the models learned physically consistent relationships, such as low-rigidity NMs contributing to low-energy predictions and high-rigidity NMs to high-energy predictions.

Key Contributions

Refined Physics Model: The study introduces a more rigorous calculation of the heavy-nuclei contribution to NM count rates and explicitly accounts for the $^3$ He isotopic fraction, which previous studies often neglected.
Modified FFA: A two-parameter force-field model is applied to NM data, attempting to better capture the energy dependence of solar modulation, though the authors note limitations in the low-energy range.
Comprehensive ML Framework: Unlike previous machine learning studies that relied on single inputs or limited datasets, this work integrates multi-station NM data with physically motivated heliospheric indices (including time delays) to reconstruct both proton and helium fluxes.
Extended Time Series: The authors successfully extend reconstructed spectra into periods lacking daily spacecraft data: 2006–2011 (validated against PAMELA) and 2019–2022 (validated against AMS-02 Bartels rotation averages).

Results

Performance Comparison: The neural network-based methods significantly outperformed the YF+FFA approach.
- Error Metrics: The neural networks reduced the Mean Absolute Percentage Error (MAPE) by a factor of ~4 at low energies and ~2 at high energies compared to the physics-based method.
- Goodness of Fit: The $\chi^2$ /dof for neural networks typically ranged between 0.1 and 2, whereas the FFA method showed higher deviations, particularly during solar minima and maxima.
- Hysteresis: A scatter plot analysis revealed a pronounced hysteresis in the FFA-based reconstructions relative to the neural networks, particularly during periods of negative heliospheric polarity (pre-2014). The FFA method systematically overestimated fluxes during these periods, while the neural networks maintained consistency with observations across polarity cycles.
Validation:
- 2011–2019: Neural networks reproduced both long-term solar cycle modulation and short-term disturbances (e.g., 27-day variations) with high fidelity.
- 2006–2011 & 2019–2022: The reconstructed spectra agreed well with PAMELA and AMS-02 data, respectively. Discrepancies observed (e.g., ~10% deviation in low-energy protons in 2009) were attributed to known instrumental differences between PAMELA and AMS-02 and the extrapolation limits of the models during the deep solar minimum of cycle 23/24.

Significance and Claims
The paper claims that the global neutron monitor network, when processed through trained artificial neural networks, functions effectively as a unified, real-time GCR spectrometer. The study demonstrates that data-driven models can overcome the limitations of traditional yield-function approaches, particularly regarding the energy dependence of modulation and detector calibration uncertainties.

The authors conclude that while the physics-guided FFA approach recovers long-term trends, it requires refinement, especially in the low-energy range and during negative polarity epochs. In contrast, the neural network approach provides a robust, high-fidelity reconstruction of GCR spectra, capable of filling gaps in satellite data and providing continuous spectral information for both protons and helium. The work establishes a new benchmark for utilizing ground-based detectors for real-time space weather monitoring and GCR spectral analysis.

The Invisible Rain and the Cosmic Weather Station

The Paper: Turning Ground Sensors into a Cosmic Telescope

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