Organizations: School of Computer Science, Jiangsu University of Science and Technology, Zhenjiang 212100, People′s Republic of China · State Key Laboratory of Space Weather, National Space Science Center, Chinese Academy of Sciences, Beijing 100190, People′s Republic of China · Yunnan Astronomical Observatory, Chinese Academy of Sciences, Kunming 650216, China · Radio Cosmology Lab, Centre for Astronomy and Astrophysics Research, Department of Physics, Faculty of Science, Universiti Malaya, 50603 Kuala Lumpur, Malaysia · National Centre for Particle Physics, Universiti Malaya, 50603 Kuala Lumpur, Malaysia · Department of Electrical Engineering, Faculty of Engineering, Universiti Malaya, 50603 Kuala Lumpur, Malaysia
In this study, we construct Dataset A for training, validation, and testing, and Dataset B to evaluate generalization. We propose a novel F10.7 index forecasting method using wavelet decomposition, which feeds F10.7 together with its decomposed approximate and detail signals into the iTransformer model. We also incorporate the International Sunspot Number (ISN) and its wavelet-decomposed signals to assess their influence on prediction performance. Our optimal method is then compared with the latest method from S. Yan et al. (2025) and three operational models (SWPC, BGS, CLS). Additionally, we transfer our method to the PatchTST model used in H. Ye et al. (2024) and compare our method with theirs on Dataset B. Key findings include: (1) The wavelet-based combination methods overall outperform the baseline using only F10.7 index. The prediction performance improves as higher-level approximate and detail signals are incrementally added. The Combination 6 method integrating F10.7 with its first to fifth level approximate and detail signals outperforms methods using only approximate or detail signals. (2) Incorporating ISN and its wavelet-decomposed signals does not enhance prediction performance. (3) The Combination 6 method significantly surpasses S. Yan et al. (2025) and three operational models, with RMSE, MAE, and MAPE reduced by 18.22%, 15.09%, and 8.57%, respectively, against the former method. It also excels across four different conditions of solar activity. (4) Our method demonstrates superior generalization and prediction capability over the method of H. Ye et al. (2024) across all forecast horizons. To our knowledge, this is the first application of wavelet decomposition in F10.7 prediction, substantially improving forecast performance.
Figures & tables
Figure 1: Daily F10.7 index of DRAO data from 1957 to 2020. The training set, validation set and testing set are divided according to solar cycles. The blue part in the data is used for model training, the green part for model validation, and the yellow part for model testing.
Figure 2: Comparison of Langfang data and DRAO data for the time period 2024-2025.
Figure 3: Wavelet multi-level decomposition. X AL and X DL respectively represent the approximate signals and the detail signals of the L-th level.
Figure 4: The prediction scheme of the F10.7 index for future periods based on the iTransformer model.
Figure 5: The schematic diagram for F10.7 prediction method based on the iTransformer model and wavelet decomposition. The lower section of the diagram displays the integrated inputs to the model, which include the F10.7 index, ISN, their respective approximate signals and detail signals derived from wavelet decomposition, along with the original data. The upper section presents the iTransformer model architecture, which primarily consists of an embedding layer, a projection layer, and multiple inverted Transformer Blocks.
Hyperparameter
Concrete content
Model dimension
256
Training epochs
50
Number of heads
4
Number of encoder layers
2
Loss function
MSE
Feed-forward network dimension
512
Table 1: Hyperparameter settings of the iTransformer model that integrates wavelet-decomposed signals.
Evaluation indicators
Formulas
Descriptions
RMSE
RMSE=T×S1∑t=1T∑s=1S(Y^t,s−Yt,s)2
Better closer to zero
MAE
MAE=T×S1∑t=1T∑s=1S∣Y^t,s−Yt,s∣
Better closer to zero
MAPE
MAPE=T×S1∑t=1T∑s=1SYt,sY^t,s−Yt,s×100%
Better closer to zero
Table 2: Formulas and descriptions of evaluation indicators.
Figure 6: The wavelet-decomposed approximate signals of F10.7 index.
Figure 7: The wavelet-decomposed detail signals of F10.7 index.
Figure 8: The wavelet-decomposed approximate signals of ISN.
Figure 9: The wavelet-decomposed detail signals of ISN.
Combination
Number of
The content of the combination
number
variables
1
1
Original F10.7 index
2
3
The original F10.7 index with its first level approximate signals and detail signals (F10.7, F10.7 A1 , F10.7 D1 )
3
5
The original F10.7 index with its first to second level approximate signals and detail signals (F10.7, F10.7 A1 , F10.7 D1 , F10.7 A2 , F10.7 D2 )
4
7
The original F10.7 index with its first to third level approximate signals and detail signals (F10.7, F10.7 A1 , F10.7 D1 , ⋯ , F10.7 A3 , F10.7 D3 )
5
9
The original F10.7 index with its first to fourth level approximate signals and detail signals (F10.7, F10.7 A1 , F10.7 D1 , ⋯ , F10.7 A4 , F10.7 D4 )
Table 3: Different combinations that integrates the approximate signals and detail signals of the F10.7 index.
Evaluation indicators
Combination
Prediction length (days)
1
2
3
5
10
15
20
25
27
30
35
40
45
RMSE
Combination 1
5.236
6.089
6.919
8.397
11.875
11.768
12.349
13.633
13.143
13.298
13.775
14.320
14.495
Combination 2
4.547
5.416
6.618
8.758
11.189
12.063
13.197
13.298
13.639
13.947
13.307
14.550
13.868
Combination 3
4.204
4.695
5.700
7.348
10.469
11.565
12.987
11.663
14.076
12.004
13.497
12.666
13.708
Combination 4
4.165
4.445
5.290
5.604
8.083
10.090
10.889
11.305
12.519
12.689
12.442
13.781
14.573
Combination 5
4.232
4.484
5.189
6.027
7.974
10.056
10.352
11.606
11.975
12.421
12.438
13.126
13.658
Table 4: Performance results for each combination method on the corresponding testing datasets under different prediction lengths.
Figure 14Figure 15
Figure 10: Bar chart comparing the prediction performance results for Combinations 1 to 6 methods under different prediction lengths.
Figure 11: Bar chart comparing the prediction performance of methods that incorporate different level of wavelet-decomposed signals when the prediction length is 27 days.
Figure 18Figure 19
Figure 12: Bar chart comparing the prediction performance results for the methods of Combination 6, Combination A and Combination D under different prediction lengths.
Prediction length (days)
Combination 6
Combination A
Combination D
RMSE
MAE
MAPE
RMSE
MAE
MAPE
RMSE
MAE
MAPE
1
4.526
2.225
2.062
4.525
2.295
2.134
4.505
2.246
2.089
2
4.587
2.381
2.223
4.711
2.523
2.353
4.908
2.577
2.401
3
5.173
2.644
2.458
5.487
2.949
2.727
5.608
3.088
2.871
5
6.125
3.327
3.100
6.404
3.552
3.290
6.691
3.688
3.420
10
8.039
4.562
4.220
8.753
4.946
4.545
8.276
4.645
4.343
Table 5: Performance results for the methods of Combination 6, Combination A and Combination D on the corresponding testing datasets under different prediction lengths.
Combination
Number of
The content of the combination
number
variables
7
12
Combination 6, ISN
8
14
Combination 6, as well as the ISN and its first level approximate signals and detail signals (Combination 6, ISN, ISN A1 , ISN D1 )
9
16
Combination 6, as well as the ISN and its first to second level approximate signals and detail signals (Combination 6, ISN, ISN A1 , ISN D1 , ISN A2 , ISN D2 )
10
18
Combination 6, as well as the ISN and its first to third level approximate signals and detail signals (Combination 6, ISN, ISN A1 , ISN D1 , ⋯ , ISN A3 , ISN D3 )
11
20
Combination 6, as well as the ISN and its first to fourth level approximate signals and detail signals (Combination 6, ISN, ISN A1 , ISN D1 , ⋯ , ISN A4 , ISN D4 )
Table 6: Different combinations that integrates ISN along with its approximate signals and detail signals.
Figure 23Figure 24
Figure 13: Bar chart comparing the prediction performance results of the Combination 6 method and the Combination 7 to 12 methods under different prediction lengths.
Method
Evaluation Indicator
2018
2019
2020
2021
2022
2023
Total
Combination 6
RMSE
1.357
1.847
4.649
7.050
18.735
23.206
10.774
MAE
1.047
1.291
2.409
4.353
12.986
15.318
5.573
MAPE
1.448
1.827
2.952
5.288
9.632
8.771
5.057
S. Yan et al. [32]
RMSE
1.400
1.998
4.862
8.714
21.379
24.278
13.175
MAE
1.063
1.343
2.581
5.319
14.923
15.428
6.563
MAPE
1.526
1.891
3.176
5.883
11.063
9.131
5.531
Table 7: The annual and total prediction performance results of the Combination 6 method, S. Yan et al. [32] , and the other three operational models in the prediction length of 27 days.
Figure 14: Bar chart of the total prediction performance of the Combination 6 method, S. Yan et al. [32] , and the other three operational models.
Condition
Model
RMSE
MAE
MAPE
Low (F10.7 ≤ 75)
Combination 6
2.121
1.459
2.062
BGS
2.748
1.929
2.728
CLS
2.977
2.219
3.165
SWPC
2.470
1.726
2.437
Moderate (75 < F10.7 ≤ 150)
Combination 6
11.836
8.696
7.807
BGS
16.521
11.837
10.878
Table 8: Prediction performance of the Combination 6 method and the three operational models across four conditions in the prediction length of 27 days.
Figure 29Figure 30
Figure 15: On the DRAO data and Langfang data, the line graphs showing the prediction performance of the Combination 6 method and the method of H. Ye et al. [35] under different prediction lengths.
Prediction length (days)
The method of H. Ye et al. [35]
Combination 6
RMSE
MAE
MAPE
RMSE
MAE
MAPE
1
9.568
6.854
3.884
4.144
3.176
1.850
2
11.582
8.466
4.855
6.698
4.956
2.829
3
13.748
9.947
5.719
8.529
6.320
3.601
5
17.108
12.600
7.377
11.783
8.795
5.010
10
21.933
16.807
9.996
18.515
13.870
7.764
Table 9: Prediction performance results of the Combination 6 method and the method of H. Ye et al. [35] on the DRAO data under different forecast lengths.
Prediction length (days)
The method of H. Ye et al. [35]
Combination 6
RMSE
MAE
MAPE
RMSE
MAE
MAPE
1
11.645
8.978
5.167
7.851
6.065
3.457
2
14.202
10.488
6.046
9.322
7.037
3.997
3
16.535
12.367
7.116
10.824
8.16 4
4.635
5
19.551
14.766
8.646
13.693
10.343
5.867
10
24.138
18.712
11.113
20.058
15.225
8.479
Table 10: Prediction performance results of the Combination 6 method and the method of H. Ye et al. [35] on the Langfang data under different forecast lengths.
Time series forecasting predicts future values from ordered observations using statistical, neural, and probabilistic models to support decisions under uncertainty.
The F10.7 and F30 solar indices are the solar radio fluxes measured at wavelengths of 10.7 cm and 30 cm, respectively, which are key indicators of solar activity. F10.7 is valuable for explaining the impact of solar ultraviolet (UV) radiation on the upper atmosphere of Earth, while F30 is more sensitive and could improve the reaction of thermospheric density to solar stimulation. In this study, we present a new deep learning model, named the Solar Index Network, or SINet for short, to predict daily values of the F10.7 and F30 solar indices. The SINet model is designed to make medium-term predictions of the index values (1-60 days in advance). The observed data used for SINet training were taken from the National Oceanic and Atmospheric Administration (NOAA) as well as Toyokawa and Nobeyama facilities. Our experimental results show that SINet performs better than five closely related statistical and deep learning methods for the prediction of F10.7. Furthermore, to our knowledge, this is the first time deep learning has been used to predict the F30 solar index.
Zhenduo Wang, Yasser Abduallah, Jason T. L. Wang +6
Institute for Space Weather Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA · Department of Computer Science, New Jersey Institute of Technology, Newark, NJ 07102, USA · Center for Solar-Terrestrial Research, New Jersey Institute of Technology, Newark, NJ 07102, USA +3
Wavelet-based financial forecasters typically use the transform only to denoise, or reduce it to a single spectral snapshot at the forecast origin, and the convolution that produces the coefficients is usually bilateral, so it can read past the forecast origin. DSTNet instead retains the recent evolution of filter-bank magnitudes as a causal Dynamic Spectral Trajectory, built from seven trailing technical indicators over a twenty-day lookback with a one-sided Morlet-derived filter bank and an explicit burn-in for the left-boundary transient. A factorized Scale-Temporal Spectral Transformer attends along the time and filter-bank axes separately, a learned gate fuses the spectral branch with a CNN-BiLSTM, and horizon-specific gates emit one, three, five, and ten day forecasts in a single pass. We evaluate seven equity indices and gold under a common expanding-window protocol and an untouched one-year hold-out, against nine learned baselines and a random-walk persistence benchmark. Under MAE and MAPE, persistence is the strongest of the ten fixed competitors in 29 of the 32 series-horizon cells and DSTNet is the only model below it in every cell, by 0.7 to 0.9 percent at one day and 3.4 to 4.5 percent at ten days. At one day, paired testing favours DSTNet against the weaker learned baselines but is inconclusive against persistence and the strongest learned forecasters. A downstream allocation diagnostic does not support an equity-timing advantage on any of the seven indices.
Aashish Bohra, Lokendra Vishwakarm
Department of Computer Science and Engineering, Indian Institute of Technology Jodhpur, Jodhpur, Rajasthan 342030, India · School of Computing and Data Sciences, FLAME University, Pune Maharashtra, India
Long-horizon autoregressive forecasting of oscillatory physical signals, such as seismograms, gravitational-wave strain, and similar wavefields is limited by error accumulation: as a causal model is fed its own outputs over hundreds of steps, small per-step errors compound into phase drift that pointwise metrics fail to detect. We ask when such rollout stays stable, using synthetic three-component seismograms as a physically structured testbed and the SeismoGPT autoregressive forecaster as the model under study. Through controlled, intra-architecture ablations evaluated on free-running rollout with paired significance tests, we isolate the contribution of each design choice. Multi-token prediction is the dominant stabilizer, accounting for almost the entire improvement over a single-token baseline (+0.040 median NCC); a horizon-embedding hybrid prediction head and a cross-horizon STFT-magnitude coherence loss each add a small but consistent further gain. Performance depends sharply on a context-ratio threshold near one, roughly the full P-S interval of observed signal, below which rollout generalization collapses. The dominant residual failure is a polarity inversion that a magnitude-based spectral loss cannot, by construction, penalize, identifying phase-aware objectives as the natural next step. We frame this as a controlled study of rollout stability on oscillatory wavefields, not a benchmark of forecasting architectures.
Waleed Esmail, Stuart Russell, Jana Klinge +2
Institut für Kernphysik, Universität Münster Wilhelm-Klemm-Straße 9, 48149 Münster, Germany · Institut für Geophysik, Universität Münster Corrensstraße 24, 48149 Münster, Germany · James Cook University, 1 James Cook Drive, Douglas, QLD 4814, Australia +1