Physics-informed multi-task neural networks for joint mapping of fracture network and hydraulic conductivity in fractured aquifers: PI-XNET

We develop a novel, physics-informed multi-task learning framework (PI-XNET) for steady-state hydraulic tomography in fractured aquifers. The model employs a SegNet-based encoder-decoder architecture with feature fusion to jointly reconstruct hydraulic conductivity (K) and the fracture network. The residuals of the governing partial differential equations (PDEs) are incorporated into the model to integrate the groundwater flow dynamics and enforce physical constraints. The unified loss combines data mismatch residuals, PDE constraints, and hard constraint loss, with each component weighted based on the task uncertainty. Through synthetic experiments, we evaluate the performance of PI-XNET and its robustness to data noise, reduced pumping datasets, and data resolution. In comparison to the standard multi-task learning network (RMSE_median= 1.27, median R²_k= 0.73), PI-XNET (RMSE_median= 1.11, median R²_k= 0.78) has improved the conductivity reconstruction and achieved higher fracture segmentation accuracy (ACC_median>99%). Moreover, PI-XNET consistently achieved higher accuracy in hydraulic head reproducibility (median R²_h= 0.61, median L₁ norm = 0.19 m, median RMSE_h = 0.14 m²). With fewer pumping test data and with data noise, the performance of PI-XNET declines modestly yet remains reliable. With coarser data resolution, head predictions remain robust (median R²_h = 0.82), whereas K and fracture mapping deteriorated with increased fracture complexity. Our results demonstrate that incorporating physics constraints within an uncertainty-weighted, multi-task framework improves the parameter estimation and fracture mapping and achieves high accuracy even with reduced pumping data. Further, we emphasize that the reliability of PI-XNET in realistic fractured geologic settings depends on data quality and resolution.