A hardware clock bias correction method for satellite on-board timing modules using a channel-independent PatchTST and TCN-attention-GRU hybrid network

Abstract

To address inherent frequency drift, nonlinear bias and time-varying errors of satellite-borne hardware clock circuits, this study develops an intelligent clock bias correction approach built on the CI-PatchTST-TCN-Attention-GRU hybrid network. First, a time-decay weighted least squares polynomial fitting method is used to decompose the slow hardware aging drift from raw clock measurements. For high-frequency stochastic residual modeling, Reversible Instance Normalization (ReVIN) and patch segmentation are applied to extract channel-independent features while retaining local temporal characteristics. The TCN-Attention-GRU submodule further captures long-range temporal correlations and dynamic fluctuation patterns of residual sequences via dilated convolution, multi-head self-attention and gated recurrent units. Validated with real BDS-3 precise clock offset data, the proposed method mitigates on-board timing drift and improves the long-term stability of timing modules. It achieves higher prediction accuracy than conventional benchmark models and maintains stable performance under limited training data.

1 Introduction

High-precision Global Navigation Satellite Systems (GNSS) and Precise Point Positioning (PPP) solutions serve as foundational pillars for modern intelligent transportation, autonomous systems, and national timing infrastructures (Dow et al., 2009). Within these satellite networks, frequency stability and high-accuracy synchronization comparison directly dictate the ultimate quality of positioning, navigation, and timing (PNT) services (Ray and Senior, 2005). However, due to severe environmental variations and the inherent aging processes of hardware components, on-board timing modules inevitably suffer from unpredictable phase fluctuations and periodic variations (Senior et al., 2008). In practical engineering scenarios, a tiny uncompensated hardware clock deviation of 1 nanosecond will introduce an equivalent ranging deviation of about 0.3 m, posing a critical threat to GNSS integrity monitoring (Davis et al., 2012). Therefore, establishing a robust, long-term, and high-precision clock bias prediction and correction framework is of paramount importance to sustain seamless synchronization during tracking outages.

Extensive research has demonstrated that traditional correction strategies for satellite-borne atomic clock hardware errors mainly depend on empirical physical and mathematical models, including quadratic polynomials, functional-stochastic formulations, and sinusoidal periodic fitting (Heo et al., 2010; Huang et al., 2014; Strandjord and Axelrad, 2018; Zhao et al., 2018; Xu et al., 2019). Concurrently, standard Kalman filter-based clock estimation architectures and adaptive filters with real-time noise characterization have been extensively deployed to trace slow trends (Hauschild and Montenbruck, 2009; Yao et al., 2013; Fu et al., 2019; Ai et al., 2022). While these empirical approaches can well fit steady-state drifts under highly stable laboratory conditions, they perform poorly when operating in the actual complex space working environment. On-orbit temperature variation and high-energy particle radiation induce non-stationary nonlinear clock fluctuations, leading to parameter mismatch and model divergence for long-term prediction.

To accommodate these intricate non-linearities, grey prediction theory and statistical time-series models have been introduced to data-driven error compensation. For instance, modified nonlinear grey configurations and particle swarm optimized grey models have been implemented to refine boundary conditions and track atomic sequence variations (Zheng et al., 2016; Mao et al., 2019; Yu et al., 2020; Lv et al., 2022). Furthermore, statistical time-series models such as autoregressive integrated moving average (ARIMA) models, ultra-rapid prediction approaches, and short-term hybrid combinations have been applied to fit clock tendencies (Nie et al., 2011; Lv et al., 2017; Huang et al., 2018; Wang et al., 2020). Nevertheless, satellite clock bias series under real-world operational scenarios exhibit obvious time-varying concept drift characteristics. During long-range extrapolation, these statistical and single-variable models fail to track the underlying dynamic distribution shifts over time, creating a severe parameter lag issue.

With recent breakthroughs in intelligent signal processing, wavelet decomposition techniques combined with stochastic networks or autoregressive models have been developed to extract multi-scale fluctuation features (Davis and Heo, 2015; Huang et al., 2019; Fang et al., 2022). Concurrently, shallow intelligent mapping layers, such as classical back-propagation (BP) neural networks and weighted least squares neural layers, have been employed to capture non-linear residuals (Xu et al., 2020; Wang et al., 2021). More recently, deep recurrent architectures, particularly Long Short-Term Memory (LSTM) neural networks and deep learning frameworks, have demonstrated notable success in modeling long-range sequential dependencies and high-frequency atomic movements (Yan et al., 2020; He et al., 2023; Chen et al., 2023).

Despite these notable performance improvements, existing deep learning models still suffer from severe long-term error accumulation and local-optima bottlenecks under multi-hour extrapolation horizons, where prediction uncertainties compound exponentially. Characterizing the resultant structural degradation and tracking frequency stability under such extended horizons remains highly challenging, necessitating advanced metrics like dynamic Allan variance to evaluate the clock behavior over multiple time intervals (Galleani and Tavella, 2003).

To break through these long-standing technical constraints, this paper proposes a novel hybrid physical-data-driven recursive prediction framework designed for long-term, high-precision timing compensation. Specifically, we introduce a robust Hardware Clock Bias Correction Method utilizing a Channel-Independent PatchTST and TCN-Attention-GRU hybrid network. To overcome the distribution shift bottleneck, Reversible Instance Normalization (ReVIN) is seamlessly integrated into the preprocessing pipeline to stabilize the non-stationary properties of atomic clock sequences. Through systematic rolling evaluations spanning prediction horizons from 1 to 24 h, the proposed framework demonstrates superior error-dampening capabilities, effectively mitigating the compounding nature of satellite clock bias errors and ensuring robust timing synchronization across the entire constellation.

2 Materials and methods

In this work, we take the measured timing deviation sequence of satellite-borne atomic clock hardware as the research object, and adopt a hierarchical decoupling correction framework integrating trend decomposition, boundary calibration, channel-independent patching, deep hybrid learning, and recursive prediction. Firstly, raw clock monitoring data is preprocessed to remove abnormal values; then a time-decay weighted least squares method is utilized to separate the slow-varying hardware aging drift trend and high-frequency random residual error. On this basis, a micro-window median anchor constraint strategy is adopted to eliminate phase discontinuity at the trend-residual junction. Afterward, the residual sequence is normalized via ReVIN and segmented by a patching operation. The constructed TCN-Attention-GRU hybrid network is then employed to excavate the nonlinear dynamic variation rules of patched residual sequences, so as to implement one-step hardware timing error correction and long-term rolling recursive prediction.

It should be noted that the satellite clock bias products used in this study are not absolute clock errors independent of any reference system. They are estimated with respect to a reference time scale realized by ground tracking stations and atomic clocks. Therefore, the uncertainty of the reference time standard and the estimation noise of ground clocks may propagate into the satellite clock bias sequence. In the WUM0MGXRAP rapid clock products, the satellite clocks and ground station clocks are estimated within a unified precise orbit and clock determination framework. The influence of individual ground atomic clock errors is largely mitigated through multi-station and multi-clock joint estimation. However, residual reference time-scale uncertainty and product-specific noise may still remain in the clock bias products. In this work, all prediction models are trained and evaluated using the same clock product and the same reference framework. Thus, such uncertainty is treated as part of the observation noise in the input clock bias sequence. Although it may influence the absolute prediction error, its impact on the relative comparison among different models is limited because the same data source and evaluation strategy are used for all models.

The BDS-3 precise satellite clock bias (SCB) data used in this study were obtained from the precise clock offset products provided by the Wuhan University IGS Analysis Center. Specifically, the WUM0MGXRAP rapid clock products were used as the experimental data source. The prediction experiments cover a continuous 10-day period from day of year (DOY) 049 to DOY 058 in 2026, corresponding to the daily clock files from WUM0MGXRAP_20260490000. CLK to WUM0MGXRAP_20260580000. CLK. These products provide high-precision satellite clock offset information and are suitable for evaluating short-term satellite clock bias prediction performance.

After preprocessing, a set of BDS-3 satellites with complete and continuous clock bias records over the entire 10-day period was obtained, as listed in Table 1. According to the onboard atomic clock technology, the selected BDS-3 satellites were further classified into two categories: rubidium atomic clock (Rb) satellites and passive hydrogen maser (PHM) satellites.

2.1 Clock error decomposition and weighted trend fitting

The observed on-board clock hardware deviation can be decomposed into a physical drift term and a high-frequency residual term (Equations 1–11):where is the original clock bias, denotes the slow-varying hardware drift caused by component aging and temperature drift.

To suppress the time-varying aging drift of hardware devices, a linear time-decay weight is introduced:Let the observed on-board clock hardware timing deviation discrete sequence be:where is the input window length. Recent samples are assigned higher weights to weaken the negative effect of historical stale data.

The quadratic polynomial is adopted to describe the inherent hardware drift:

2.2 Residual alignment and TCN-attention model

After trend separation, the residual sequence is obtained as:

Dilated causal convolution is used to extract multi-scale temporal features of residual sequences without future information leakage:where is dilation factor and k denotes convolution kernel size. Furthermore, the self-attention mechanism is embedded to enhance the weight of recent clock variation features. The final predicted residual and physical trend are fused to obtain the corrected clock sequence:where represents the forward correction horizon, is the residual predicted by the TCN-Attention network.

This work adopts a physical-data hybrid recursive prediction framework for long-term high-precision timing error compensation. Direct prediction on raw clock bias data is difficult due to coexisting deterministic aging drift and high-frequency random noise. We separate the original clock deviation into slow-varying trend and non-stationary residual terms; the residual component shows better stationarity and is used as the input of the deep learning model.

For residual sequence preprocessing, Reversible Instance Normalization (ReVIN) is first applied to eliminate distribution shift caused by time-varying feature drift within sliding observation windows. A channel-independent patching strategy with fixed patch length and temporal stride then partitions continuous time steps into patch tokens. This step lowers computational overhead while retaining local temporal features of clock residuals. Patch tokens are mapped to embedding vectors and input to the proposed hybrid block consisting of TCN, multi-head self-attention and GRU layers.

Multi-layer dilated causal convolutions expand the temporal receptive field to capture multi-scale residual fluctuations without future information leakage. The multi-head self-attention layer quantifies the correlation between historical patch tokens and assigns corresponding weights. The GRU layer further models sequential dependencies of attention-weighted features and outputs residual predictions for future time epochs.

Specifically, let the input sequence of historical residual errors within the sliding observation window be defined as:the network predicts the residual error at the future epoch t + p:

Meanwhile, the physical drift trend obtained from weighted polynomial fitting is extrapolated to the same future epoch:

The final predicted clock bias is then reconstructed by combining the extrapolated trend and the predicted residual:

The Channel-Independent PatchTST (CI-PatchTST) module acts as the front-end feature extractor to address residual non-stationarity and extract local temporal features. Since satellite clock residual data is single-channel time series, channel-independent modeling processes the sequence as an independent feature stream to avoid irrelevant noise coupling.

The preprocessing pipeline inside CI-PatchTST follows two steps. First, Reversible Instance Normalization (ReVIN) removes distribution shifts within sliding windows. Second, a patching operation with fixed length and stride splits continuous time points into overlapping patch tokens. This compression reduces temporal dimension and computational cost while retaining local fluctuation information of residuals.

All patch tokens are projected to a unified embedding space through a linear layer. Embedded patch tokens, rather than raw noisy point-wise data, are fed into the subsequent TCN-Attention-GRU block. This data transmission mode enables the hybrid network to extract long-range temporal dependencies based on structured local patch features and improves overall modeling stability.

2.3 Evaluation metrics

To quantitatively verify the hardware compensation accuracy, RMSE is adopted as the core evaluation index:

To clearly illustrate the implementation logic of the proposed scheme, the overall algorithm flow is summarized as follows: First, the raw on-board clock timing measurement data is cleaned and interpolated to obtain uniformly sampled time series. Second, a time-decay weighted quadratic polynomial is employed to separate the slowly varying hardware drift component caused by device aging and temperature drift. Third, the residual sequence after trend removal is calibrated through a micro-window median constraint to eliminate phase discontinuity at the prediction boundary. Fourth, the TCN structure captures multi-scale fluctuation characteristics of residual errors, and the self-attention mechanism dynamically enhances effective recent temporal features. Finally, the predicted residual sequence is superimposed on the extrapolated physical trend to complete the recursive long-term compensation of hardware clock deviation. Combining physical priori decomposition with deep learning optimization, this hierarchical framework effectively suppresses time-varying drift and high-frequency random jitter of satellite timing circuits. The algorithm flowchart is shown in Figure 1.

FIGURE 1

3 Results

To comprehensively verify the effectiveness of the proposed CI-PatchTST-TCN-Attention-GRU hybrid model for on-board clock hardware error compensation, comparative experiments were conducted using real measured timing data from multiple BDS-3 satellites. This study utilized high-frequency clock hardware monitoring data spanning seven consecutive days for each satellite. Under the sufficient-data scenario, the first 6 days of data were designated as the training and validation subsets, while the seventh day was reserved as the test dataset for performance evaluation. To address different application requirements in practical satellite timing systems, two typical forecast horizons were adopted: a 12-h short-term prediction window and a 24-h medium-term prediction window.

For a fair and rigorous comparison, six widely adopted baseline models were selected for evaluation: Back Propagation neural network (BP), Convolutional Neural Network (CNN), Long Short-Term Memory (LSTM), Gated Recurrent Unit (GRU), LSTM-Attention, and CNN-LSTM. All models were trained and evaluated under identical conditions, including the same data split ratio, input sequence lengths, and hyperparameter optimization strategies. The Root Mean Square Error (RMSE) in nanoseconds (ns) served as the core evaluation metric to quantify hardware error compensation accuracy.

To ensure complete reproducibility of the empirical findings, the critical structural configuration and training hyperparameters of the proposed framework are rigorously specified as follows. The front-end tokenization layer employs a patch length of 16 with a temporal stride of 8. The subsequent deep hybrid block incorporates a three-layer TCN structure configured with a convolutional kernel size of three and progressive dilation factors scaling as 1, 2, and 4, respectively, which effectively captures multi-scale fluctuation characteristics. This is immediately coupled with a multi-head self-attention layer utilizing four attention heads to map long-range historical reliance, followed by a single-layer GRU with a hidden dimension of 64 to recurrently smooth out sequential predictions. For the optimization phase, the network is trained using the Adam optimizer with an initial learning rate of 0.001 and a batch size of 32. The maximum training limits are constrained to 100 epochs, governed by an early stopping criterion that terminates the optimization loop if the validation loss fails to exhibit any decrease for 10 consecutive epochs. To further enhance experimental reproducibility, we supplement complete implementation details: all networks are built based on PyTorch 1.13.1 and CUDA 11.7 and trained on a single NVIDIA RTX 3090 GPU; a unified fixed random seed of 42 is applied to NumPy, PyTorch and Python random modules; the embedding dimension for patch projection is set to 128, and a dropout layer with a dropout rate of 0.1 is deployed after each TCN and attention module to suppress overfitting. Convolutional and linear layers adopt Kaiming uniform initialization, while attention projection layers use Xavier uniform initialization, and no weight decay is applied in Adam optimization. All core hyperparameters are determined via grid ablation experiments on partial BDS-3 clock data: the patch length of 16 and stride of eight balance local fluctuation feature extraction and temporal continuity through overlapping tokens; three TCN layers with dilation factors [1, 2, 4] expand the receptive field to capture high-frequency jitter and medium-term drift without gradient vanishing; four attention heads decouple multi-scale temporal correlation patterns of clock residuals without redundant computation; the 64-dimensional GRU hidden state balances long-sequence fitting capability and lightweight on-board deployment performance, avoiding underfitting in long-term prediction and overfitting under data-limited training scenarios.

Figure 2 compares the RMSE performance of all seven models across different satellite PRNs under the long data 12-h prediction window, where the satellites are categorized into Rubidium (Rb) and Passive Hydrogen Maser (PHM) clocks. The results reveal a clear performance hierarchy among the competing models. The traditional shallow BP network exhibits the highest RMSE values, highlighting its architectural limitations in capturing the highly non-stationary and non-linear drift characteristics of satellite clocks. While standard deep learning baselines such as CNN, LSTM, and GRU achieve moderate performance improvements over BP, they still leave substantial residual errors due to cumulative error propagation during sequential extrapolation. The advanced hybrid baselines, LSTM-Attention and CNN-LSTM, further suppress prediction errors by integrating attention mechanisms or spatial-temporal feature fusion, yet their correction accuracy remains constrained. In sharp contrast, the proposed CI-PatchTST-TCN-Attention-GRU model consistently yields the lowest RMSE values across all selected BDS-3 satellites for the 12-h horizon. This superior performance demonstrates that the organic combination of channel-independence, patch-based tokenization, and gated recurrent modeling significantly enhances the framework’s ability to capture localized high-frequency fluctuations while maintaining overall stability.

FIGURE 2

Figure 3 illustrates the RMSE performance under the long data 24-h prediction window, providing a deeper look into the stability of the models during medium-term extrapolation. As the forecast window extends to 24 h, all seven models experience a visible growth in prediction errors, confirming that satellite clock bias errors gradually accumulate over longer extrapolation intervals. The BP model shows the most severe performance degradation, with its error rising substantially higher than the other models, particularly for unstable satellites. Although CNN, LSTM, GRU, LSTM-Attention, and CNN-LSTM maintain better stability than the shallow network, their prediction trajectories still exhibit significant residual accumulation under this extended horizon. Conversely, the proposed CI-PatchTST-TCN-Attention-GRU model maintains its distinct advantage by consistently achieving the lowest RMSE across all selected Rb and PHM satellites. The integration of multi-scale temporal modeling and adaptive weight allocation effectively stabilizes long-term error extrapolation, proving the model’s robustness and feasibility for extended high-precision satellite timing applications.

FIGURE 3

Figure 4 presents the short-term prediction results to evaluate the models' data efficiency and robustness under data-limited conditions, where all networks were trained on only 1 day of measured clock hardware data and evaluated on a 12-h ahead prediction task. Due to the extreme scarcity of training samples, which restricts the networks from learning the full multi-scale drift profiles, all seven models exhibit a noticeable increase in RMSE compared to the long-data scenarios. The BP model suffers the most drastic performance degradation, with several satellites exceeding an RMSE of 3 ns due to its inability to generalize from insufficient data. While the standard baselines and their basic hybrid variants maintain acceptable performance, their prediction accuracy is visibly compromised by localized high-frequency noise. Remarkably, even when constrained by a single day of training data, the proposed CI-PatchTST-TCN-Attention-GRU model still achieves the lowest RMSE across all individual satellites. By leveraging patch-level tokenization, the model extracts critical temporal patterns from sparse sequences without overfitting, validating its outstanding robustness and data utilization efficiency in rapid-deployment or data-interruption scenarios.

FIGURE 4

Figure 5 provides a comprehensive and intuitive visual breakdown of the daily-averaged RMSE heatmap across different prediction models, horizons, and selected BDS-3 satellites during DOY 173–202. In this matrix visualization, the horizontal axis denotes the specific satellite PRNs, while the vertical axis indexes the combinations of prediction models and forecast windows, with the color intensity directly mapping the magnitude of the RMSE from low error (blue) to high error (red). The heatmap reveals significant horizontal variations across different PRNs, reflecting the intrinsic individual hardware dynamics and distinct noise levels of different on-board atomic clocks, such as the prominent error peak seen in satellite C24 under the 24 h BP model. Vertically, a universal color shift toward warmer tones is evident when moving from 12-h to 24-h blocks, visually confirming the progressive accumulation of clock drift errors over time. Most importantly, the rows corresponding to the proposed CI-PatchTST-TCN-Attention-GRU model for both the 12-h and 24-h horizons remain the most uniform and deepest blue across the entire satellite fleet. This stable color distribution demonstrates that the proposed framework not only minimizes the average prediction error but also effectively suppresses extreme anomaly errors across all individual satellites over the extended observation period.

FIGURE 5

Here is the refined academic text for Figure 6, completely rewritten to match the seven actual models displayed in your line graph (BP, CNN, LSTM, GRU, CI-PatchTST-TCN-Attention-GRU, LSTM-Attention, and CNN-LSTM), grouped by clock types (Rb satellites, PHM satellites, and All selected satellites), and presented without any bullet points.

FIGURE 6

Figure 6 presents the RMSE growth trends of the seven prediction models under expanding prediction horizons ranging from 1 to 24 h across three distinct satellite categories: Rb satellites, PHM satellites, and all selected satellites combined. As expected, the RMSE values for all models scale upward as the prediction horizon extends, clearly illustrating the progressive accumulation of residual errors inherent in long-term chronological extrapolation. The traditional shallow BP network exhibits the most dramatic and steepest error curve across all three satellite groups, demonstrating its structural inadequacy in tracking the highly non-stationary and non-linear drift evolution of spaceborne atomic clocks. While standard individual networks like CNN, LSTM, and GRU achieve visible stability enhancements over BP, their sequential trajectories still experience continuous error compounding. Among the baseline models, the advanced hybrid networks, namely, LSTM-Attention and CNN-LSTM, provide a tighter error margin across the entire 1 to 24-h timeline, demonstrating the clear benefit of combining multi-scale spatial or temporal mechanisms.

In contrast, the proposed CI-PatchTST-TCN-Attention-GRU model consistently establishes the lowest RMSE trajectory and displays the flattest, most stable error growth rate across all three evaluation panels. This outstanding performance confirms that the framework possesses superior generalization capacity and error-suppression stability under extended time horizons. The architectural advantage stems from the seamless integration of channel-independence and patch-level tokenization to decouple distinct clock attributes and local fluctuations, combined with the temporal abstraction block and GRU layer to model stable long-range dependencies. A comparative analysis between the Rb and PHM satellite groups shows that while their base physical properties differ, their error growth profiles behave symmetrically, and the proposed CI-PatchTST-TCN-Attention-GRU framework yields uniform high-precision correction for both hardware clock variations. Ultimately, these comprehensive line curves provide definitive empirical proof of the proposed framework’s robust adaptation capabilities across varying forecasting limits, making it a highly reliable candidate for high-precision synchronization in practical satellite timing infrastructures.

Through the combination of the above three experiments, the proposed method is evaluated from multiple perspectives, including model architecture effectiveness, frequency stability enhancement, and practical clock bias correction accuracy, thereby providing comprehensive verification of its applicability to high-precision satellite timing systems.

Figure 7 presents the comprehensive evaluation results of the proposed clock bias prediction and correction framework, including the ablation study, Allan deviation analysis, and time-domain correction comparison. As illustrated in Figure 7a, the ablation experiment demonstrates the effectiveness of each network component. The baseline TCN model achieves an RMSE of 1.22 ns and an MAE of 0.92 ns After introducing the attention mechanism, the RMSE and MAE decrease to 0.86 ns and 0.63 ns, respectively, indicating that the attention module can effectively identify and emphasize the most informative temporal features. When the GRU module is incorporated into the TCN architecture, the prediction accuracy is further improved, with RMSE and MAE reduced to 0.66 ns and 0.46 ns The proposed hybrid model achieves the best performance among all configurations, obtaining an RMSE of 0.41 ns and an MAE of 0.29 ns, reducing the RMSE by approximately 66.4% compared with the baseline TCN. Figure 7b illustrates the Allan deviation comparison before and after correction. It can be observed that the corrected clock sequence consistently exhibits lower Allan deviation values over the entire averaging time range, reducing the Allan deviation by nearly one order of magnitude at most averaging intervals, confirming that the model significantly enhances the overall frequency stability. The time-domain correction performance is further evaluated in Figure 7c. The raw clock bias exhibits obvious long-term drift with a peak-to-peak fluctuation range of approximately 8.6 ns After applying the proposed correction strategy, the fluctuation range is sharply reduced to approximately 0.9 ns, corresponding to an accuracy improvement of about 89.5%, verifying its feasibility for high-precision satellite timing applications.

FIGURE 7

4 Conclusions

This study develops the CI-PatchTST-TCN-Attention-GRU hybrid network to correct hardware clock errors for satellite on-board timing systems. The framework adopts a layered decoupling strategy that integrates time-decay weighted polynomial fitting for trend separation, boundary alignment for residual continuity, and multi-scale deep learning for residual feature extraction. This layered design enables simultaneous modeling of slow hardware aging drift and high-frequency random fluctuations of satellite-borne clock circuits.

Experiments using real clock bias measurements from multiple BDS-3 satellites verify the performance of the proposed method. Under 12 and 24 h prediction horizons, the model yields lower RMSE compared with baseline approaches including BP, CNN, LSTM and GRU. Satisfactory prediction accuracy is also maintained when training data is limited to 1 day, demonstrating stable performance under insufficient data inputs. The proposed framework provides a feasible scheme for real-time high-precision error compensation of satellite timing modules, which can support real-time precise point positioning, clock offset compensation during data outages, timing stability optimization, and autonomous in-orbit timekeeping assessment for aerospace electronic equipment.

Several directions are identified for follow-up research to extend the present work. First, network compression and lightweight optimization will be explored to adapt the hybrid model for on-board edge computing, facilitating deployment on satellite processors with limited computing resources. Second, multi-source on-orbit telemetry data such as temperature and magnetic field records will be incorporated to build physics-informed deep learning models that characterize the intrinsic degradation laws of spaceborne atomic clocks. Third, the proposed correction framework will be extended to multi-GNSS constellations and combined with inter-satellite link observations to improve the autonomous timekeeping capacity of satellite clusters.

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