Derivative-Enhanced Constraint Weights for PINNs in Groundwater Flow Modeling Under Unknown Aquifer Properties

Physics-Informed Neural Networks (PINNs) offer a promising framework for groundwater modeling in regions where hydrogeological data are limited. However, their performance significantly depends on the choice of constraint weights associated with governing equations and derivative-based regularizations. In this study, we develop a constraint-weight selection strategy for PINNs to simulate groundwater head dynamics in data-sparse environments where aquifer properties such as hydraulic conductivity (K) and specific yield/storativity (S) are unavailable. The proposed formulation incorporates first-, second-, and third-order spatial and temporal derivatives of hydraulic head and aquifer properties into the PINN loss function, enabling the model to capture fine-scale spatiotemporal variations without explicit knowledge of subsurface parameters. The approach is applied to a small section of the Varuna River Basin, using groundwater-level observations collected from 37 monitoring stations between 2022 and 2024. The dataset contains several missing values that the PINN framework handles seamlessly, unlike conventional simulation models such as MODFLOW, which require complete and continuous input fields for stable execution. An iterative optimization scheme is employed to balance data fidelity, physical constraints, and derivative-based regularization during training. The proposed method achieves a training R² of 0.986 and a testing R² of 0.947, with corresponding RMSE values of 0.721 and 1.416 meters, respectively. These results demonstrate that adaptive constraint weighting significantly improves prediction accuracy, robustness, and convergence compared to fixed-weight PINN formulations. Overall, the study highlights the potential of derivative-enhanced PINNs for groundwater modeling in data-sparse aquifers and provides a generalized framework for physics-guided learning under missing or incomplete observations.e data scarcity.