Physics-Informed Neural Networks with Uncertainty Quantification for High-Precision Tropospheric Zenith Wet Delay Prediction

High-precision modeling of tropospheric wet delay (ZWD) is one of the core elements for enhancing GNSS rapid precise positioning and GNSS water vapor detection. However, traditional analytical models suffer from insufficient accuracy and robustness under extreme weather conditions and complex terrain. Pure data-driven approaches lack physical consistency constraints, making it difficult to ensure spatial-temporal generalization performance. This study aims to develop a deep learning framework integrating physical constraints to achieve accurate ZWD prediction while maintaining high robustness in time migration and cross-station scenarios, and simultaneously quantifying prediction uncertainty to enhance application credibility. We construct a Physics-Informed Neural Network (PINN) framework by explicitly embedding atmospheric refraction physical constraints into the loss function to enable data-physics collaborative training. A hierarchical feature strategy is established based on feature importance analysis, with water vapor pressure, wet delay coefficients, and temperature as core layers, and tropospheric static delay parameters as auxiliary layers. To provide reliable predictions, we incorporate a Bayesian Neural Network (BNN) and use Monte Carlo Dropout techniques to quantify prediction uncertainty. Cross-station validation experiments show that the PINN model significantly outperforms traditional Saastamoinen models and deep learning algorithms such as CNN and LSTM, particularly in time generalization and high-humidity regions, achieving centimeter-level robust accuracy. The combination of PINN and BNN framework delivers high-precision predictions (R²=0.97, RMSE = 1.2 cm, MAE = 1 cm), with prediction uncertainty showing a significant positive correlation with actual error (R = 0.42, p < 0.001). The 95% confidence interval coverage reaches 97.0%, close to the theoretical value of 95.4%. This study successfully bridges the gap from "point prediction" to "credible probabilistic prediction," offering a new paradigm of physical constraints and uncertainty quantification for high-precision GNSS geodetic data processing and application technologies. The framework demonstrates excellent robustness and adaptability under extreme weather conditions, providing a more comprehensive and reliable technical solution for tropospheric delay correction in practical applications.