Physics-Constrained Deep Learning with Adaptive Z-R Relationship for Accurate and Interpretable Quantitative Precipitation Estimation

Quantitative precipitation estimation (QPE) from radar reflectivity is fundamental for weather nowcasting and water resource management. Conventional Z-R relationship formulas, derived from Rayleigh scattering theory, rely heavily on empirical parameter fitting, which limits the estimation accuracy and generalization across different precipitation regimes. Recent deep learning (DL)-based QPE methods can capture the complex nonlinear relationships between radar reflectivity and rainfall. However, most of them overlook fundamental physical constraints, resulting in reduced robustness and interpretability. To address these issues, this paper proposes FusionQPE, a novel Physics-Constrained DL framework that integrates an adaptive Z-R formula. Specifically, FusionQPE employs a Dense convolutional neural network (DenseNet) backbone to extract multi-scale spatial features from radar echoes, while a modified squeeze-and-excitation (SE) network adaptively learns the parameters of the Z-R relationship. The final rainfall estimate is obtained through a linear combination of outputs from both the DenseNet backbone and the adaptive Z-R branch, where the trained linear weight and Z-R parameters provide interpretable insights into the model’s physical reasoning. Moreover, a physical-based constraint derived from the Z-R branch output is incorporated into the loss function to further strengthen physical consistency. Comprehensive experiments on real radar and rain gauge observations from Guangzhou, China, demonstrate that FusionQPE consistently outperforms both traditional and state-of-the-art DL-based QPE models across multiple evaluation metrics. The ablation and interpretability analysis further confirms that the adaptive Z-R branch improves both the physical consistency and credibility of the model’s precipitation estimation.