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Machine Learning-Based Characterization of Solar p-Mode Frequency Shifts during Solar Cycle 25

This paper develops and applies standard time-series analysis and machine learning methods to characterize and forecast solar p-mode frequency shifts throughout the remainder of Solar Cycle 25, aiming to establish a robust link between the Sun's interior dynamics and space weather while providing an early indicator of solar activity phases.

Original authors: Rekha Jain (School of Mathematical and Physical Sciences, University of Sheffield), Akash Kumar (School of Mechanical, Aerospace and Civil Engineering, University of Sheffield), Sushanta C. Tripathy (

Published 2026-04-23

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CC BY 4.0

Original authors: Rekha Jain (School of Mathematical and Physical Sciences, University of Sheffield), Akash Kumar (School of Mechanical, Aerospace and Civil Engineering, University of Sheffield), Sushanta C. Tripathy (National Solar Observatory)

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 Sun's Heartbeat: Predicting the Next Solar Quiet Period

Imagine the Sun not just as a blazing ball of fire, but as a giant, living drum. Just like a drum vibrates when hit, the Sun vibrates with sound waves traveling through its interior. These vibrations are called p-modes.

For decades, scientists have been listening to this "heartbeat." They've noticed something fascinating: the pitch of the Sun's heartbeat changes in a rhythm that matches its mood swings. When the Sun is "angry" (full of sunspots and solar storms), the pitch goes up. When the Sun is "calm," the pitch goes down. This mood swing happens roughly every 11 years, a cycle known as the Solar Cycle.

The Problem:
We are currently in the middle of the 11th cycle (Solar Cycle 25). Scientists want to know: When will this cycle end, and when will the Sun go quiet again? Knowing this is crucial for "Space Weather"—the solar storms that can knock out satellites, disrupt GPS, and cause power grid failures on Earth.

The Solution:
This paper is like a team of detectives (Rekha Jain, Akash Kumar, and Sushanta Tripathy) using two different tools to solve the mystery of the Sun's future:

The Direct Listener: They looked directly at the Sun's heartbeat data from the last 30 years.
The Proxy Detective: They noticed that the heartbeat changes in sync with two other famous solar "mood rings":
- Sunspots: Dark spots on the Sun's surface (like freckles).
- Radio Flux: The amount of radio waves the Sun blasts out (like a radio station's volume knob).

They used Machine Learning (smart computer programs) to analyze these patterns and predict the future.

The Detective Work: How They Did It

Think of the Sun's data as a very long, noisy song. To find the melody, the scientists had to separate the music from the static. They used three different "filters" (algorithms) to do this:

The Wavelet Filter: Imagine breaking a song down into its individual notes and timing them perfectly to see the rhythm.
The Smoothing Filter (LOESS): Imagine running your hand over a bumpy road to smooth out the potholes and see the general shape of the path.
The Deep Learning Filter (N-BEATS): This is like a super-smart AI that has listened to millions of songs and learned to guess the next note based on complex, hidden patterns.

They applied these filters to the Sun's heartbeat, the sunspots, and the radio waves to see when the "volume" would drop to its lowest point (the minimum).

The Verdict: What Did They Find?

The results are a bit like looking at a weather forecast that says, "It's going to rain, but the exact time depends on which model you trust."

Current Status: The Sun has already reached its peak "storminess" for this cycle (around early 2025) and is now slowly calming down.
The Prediction: All the models agree that the Sun will reach its quietest point (the bottom of the cycle) around 2030 or 2031.
The Duration: This quiet period will last for about seven years, similar to the last big cycle.

The Analogy of the "Proxy" vs. "Direct" Method:

Direct Method: Trying to predict the future by looking only at the Sun's heartbeat. The problem? We don't have enough history (only about 30 years of data) to be 100% sure about the exact month. It's like trying to predict the end of a movie after only watching the first 30 minutes.
Proxy Method: Using the Sun's "freckles" (sunspots) and "radio volume" (F10) as clues. Since we have records of sunspots going back to the 1800s, this gives the computer a much longer history to learn from. This method gave a slightly more confident prediction of 2030.

Why Should You Care?

You might think, "I don't live on the Sun, why do I care about its heartbeat?"

Space Weather: When the Sun is quiet (around 2030), the risk of massive solar storms hitting Earth drops significantly. This is good news for our satellites, astronauts, and power grids.
Connecting the Dots: This study proves that what happens deep inside the Sun (where the sound waves travel) is directly linked to what happens on the surface (sunspots) and in the atmosphere (radio waves). It's like realizing that a person's heartbeat, their skin temperature, and their voice tone are all connected.
Better Forecasts: By using Machine Learning, scientists are building a better "weather forecast" for space. Just as we use computers to predict rain on Earth, we are learning to predict solar storms.

The Bottom Line

The Sun is currently winding down its current 11-year storm cycle. Thanks to smart computers and clever math, we can now say with reasonable confidence that the Sun will hit its "nap time" around 2030-2031.

While the exact month might still be a little fuzzy (like a weather forecast saying "rain sometime next week"), the big picture is clear: the Sun is getting ready for a long, quiet rest. This gives us time to prepare our technology for the next cycle of activity, which will eventually start building up again.

1. Problem Statement

Solar p-mode oscillations (acoustic waves) exhibit frequency shifts that correlate with the Sun's 11-year magnetic activity cycle. While these shifts are well-documented for Solar Cycles (SC) 23 and 24, forecasting their behavior for the remainder of Solar Cycle 25 and the transition to Cycle 26 remains challenging.

The Gap: Existing models often rely on standard proxies (sunspot numbers, radio flux) to infer solar activity. There is a need for an independent, physics-based indicator (p-mode frequency shifts) that links the solar interior to space weather conditions.
The Challenge: The dataset for p-mode frequencies (GONG data) is relatively short (starting 1995), covering only two full cycles and a partial third. This scarcity makes traditional time-series forecasting difficult, necessitating advanced machine learning (ML) techniques to extrapolate trends and identify the timing of the next solar minimum.

2. Methodology

The authors employed a multi-faceted approach combining standard time-series analysis with advanced machine learning and deep learning frameworks.

Data Sources

Primary Data: Global Oscillation Network Group (GONG) p-mode frequency shifts ( $\delta\nu$ ) from May 7, 1995, to July 31, 2025. Data was averaged over 9-day intervals.
Proxy Data:
- 10.7 cm Radio Flux (F10): From Jan 9, 1954, to July 31, 2025.
- Sunspot Numbers (SSN): From Jan 9, 1954, to July 31, 2025.
Preprocessing: The authors established a quadratic relationship between p-mode shifts and the proxies (F10 and SSN) to account for saturation effects at high activity levels, rather than a simple linear fit.

Machine Learning Frameworks

Three distinct forecasting strategies were applied:

Direct Prediction (Wavelet-LGBM):
- Decomposition: Used Discrete Wavelet Transform (DWT) with 'db4' wavelets to separate the signal into trend and residual components.
- Modeling: Trained Light Gradient-Boosting Machine (LGBM) models separately on the trend and residuals.
- Features: Included temporal lags and seasonality (derived from wavelet power spectrum peaks).
Ensemble Prediction (LOESS-FFT + LGBM):
- Decomposition:
  - LOESS: Locally Estimated Scatterplot Smoothing to extract trends.
  - FFT: Fast Fourier Transform used as a low-pass filter (cutoff > 0.01 cycles/9 days) to isolate trends longer than 2.5 years.
- Ensembling: Predictions from LOESS-LGBM and FFT-LGBM models were averaged to generate the final forecast.
Deep Learning (N-BEATS):
- Architecture: Neural Basis Expansion Analysis for Time Series (N-BEATS), a deep learning model with specific stacks for trend and seasonality.
- Application: Applied only to the proxy datasets (F10 and SSN) to forecast the proxies first, which were then scaled to $\delta\nu$ using the established quadratic relationship. This was done because the raw $\delta\nu$ dataset was too short for effective deep learning training.
Uncertainty Quantification:
- Used LGBM-based quantile regression to generate 90% prediction intervals (5th and 95th percentiles).

3. Key Results

Forecasting the Solar Cycle 25 Minimum

Current Status: All methods confirm that Solar Cycle 25 has passed its maximum (early 2025) and is in the descending phase.
Timing of the Minimum (SC25/SC26 Transition):
- Direct $\delta\nu$ Forecast: Suggests a minimum around 2031.
- Scaled Forecast (via F10 and SSN): Consistently predicts a minimum around 2030.
- Comparison: The direct forecast (Figure 3) shows a longer descending phase compared to the scaled forecasts (Figures 5 & 7), likely due to the limited data points in the direct $\delta\nu$ series.

Methodological Performance

Wavelet-LGBM: Produced the smoothest descent curve.
LOESS-FFT-LGBM: Showed a steeper descent.
N-BEATS: Generally qualitatively similar but struggled to reproduce Quasi-Biennial Oscillations (QBOs) seen in the other methods, underestimating the ascending phase in some instances.
Correlation: The quadratic fit between p-mode shifts and proxies (F10/SSN) successfully reproduced high-magnitude data points better than linear fits.

Historical Context (Table 1)

The study tabulated the minimum and maximum periods for cycles 23–26:

SC23 Min: March 1996
SC24 Min: September 2009
SC25 Min (Observed): February 2020
SC26 Min (Predicted): March 2030 (scaled) / January 2031 (direct)

4. Key Contributions

First ML-Based Characterization of SC25: This is one of the first studies to apply modern ML (LGBM, N-BEATS) specifically to forecast p-mode frequency shifts for the remainder of Solar Cycle 25.
Hybrid Approach: Successfully combined signal decomposition (Wavelet, LOESS, FFT) with gradient boosting and deep learning to handle non-stationary solar data.
Independent Indicator Validation: Demonstrated that p-mode frequency shifts can serve as an independent indicator of solar activity, corroborating predictions made by sunspot numbers and radio flux.
Link to Space Weather: Established a quantitative framework to link interior solar dynamics (probed by p-modes) with exterior conditions, potentially improving space weather forecasting models.

5. Significance and Limitations

Significance: The results suggest that the descending phase of SC25 will last approximately seven years, similar to SC23, indicating a stronger cycle than SC24. The ability to predict the solar minimum (approx. 2030–2031) is crucial for long-term space weather planning.
Limitations:
- Data Scarcity: The direct $\delta\nu$ dataset (1995–2025) is too short for robust deep learning, forcing reliance on proxy scaling.
- Smoothing: The 9-day averaging smooths out small-scale fluctuations, meaning the models capture long-term trends rather than short-term variability.
- QBOs: The N-BEATS model failed to capture Quasi-Biennial Oscillations (2–3 year variations), highlighting a need for improved deep learning architectures for solar data.
Future Work: The authors emphasize the need for continued helioseismic monitoring and the integration of additional parameters (line widths, amplitudes) to refine these models.

In conclusion, the paper provides a promising, nascent step toward using oscillation frequencies as a robust, independent tool for forecasting solar activity evolution, bridging the gap between the solar interior and geospace.