Hybrid Physics-Informed Residual Learning for Robust BDS-3 Satellite Clock Bias Prediction

Real-time precise point positioning (RT-PPP) has enabled a wide range of high-precision positioning and navigation applications, while its reliability strongly depends on the availability and continuity of precise satellite clock products. In the third-generation BeiDou Navigation Satellite System (BDS-3), interruptions or gaps in real-time precise clock products can significantly degrade the continuity and performance of precise positioning services. Therefore, accurate and robust satellite clock bias (SCB) prediction is essential for supporting reliable RT-PPP applications under product outage conditions. To address this problem, this study proposes a hybrid physics-informed and data-driven framework for BDS-3 SCB prediction. The proposed method sequentially integrates a physics-informed neural network (PINN) and a long short-term memory (LSTM) network. Specifically, the PINN is used to model and extrapolate the physically consistent trend component of SCB increments by embedding clock dynamical constraints through automatic differentiation, while the LSTM is employed to learn and predict the residual sequence containing short-term stochastic variations that cannot be fully captured by the physical model. The final SCB prediction is obtained by reconstructing the trend and residual components and recovering the original clock bias series. The proposed framework is evaluated using BDS-3 precise clock products and compared with conventional models, including quadratic polynomial (QP), autoregressive integrated moving average (ARIMA), convolutional neural network-long short-term memory (CNN-LSTM), and attention-enhanced long short-term memory (LSTM-Attention). Experimental results show that the proposed PINN-LSTM framework consistently achieves superior prediction accuracy and stability at both 12 h and 24 h forecasting horizons. Specifically, compared with QP, ARIMA, CNN-LSTM, and LSTM-Attention, the proposed method improves prediction accuracy by 18.4%, 52.8%, 32.3%, and 33.8%, respectively, for the 12 h forecasting task, and by 34.8%, 58.5%, 41.8%, and 43.8%, respectively, for the 24 h forecasting task. The results further demonstrate reduced long-horizon error accumulation, improved robustness across satellites equipped with different atomic clock types, and stronger generalization across observation days. These findings indicate that the proposed framework can provide effective support for maintaining the continuity and reliability of BDS-3 precise clock products and has strong potential for improving real-time precise positioning applications.