Physics-Guided Modeling of Water Flow in the Vadose Zone

Soil hydraulic properties, including saturated hydraulic conductivity and water retention parameters, play a central role in regulating infiltration, redistribution, and storage of water in the root zone, making them fundamental for understanding soil moisture dynamics and plant water availability. Their spatial variability must be characterized to enable accurate field‑scale predictions of soil water dynamics using process‑based models. The Richards equation, which serves as a core framework for modeling water movement in unsaturated soils, poses major difficulties for conventional numerical approaches because of its pronounced nonlinearity, intricate boundary conditions, and high computational demands. Physics‑informed neural networks (PINNs) have emerged as a promising tool that integrates governing physical laws into deep learning frameworks and provides a mesh‑free approach for inverse estimation of hydraulic parameters from limited and noisy datasets. While PINNs have proven effective for homogeneous soils, layered profiles remain challenging due to unknown interface depths and parameter heterogeneity. This study develops a novel PINN‑based framework with progressive physics training to estimate saturated and residual soil moisture contents and the α parameter of the van Genuchten model within layered soils by predicting volumetric water content variations from Time Domain Reflectometry (TDR) sensor data. The framework optimizes data fitting and physics regularization to predict soil moisture dynamics across multiple soil depths. Model performance is evaluated using multiple criteria, including Root Mean Square Error (RMSE), Kling–Gupta Efficiency (KGE), and the coefficient of determination (R²), at sensor‑aligned nodes. Incorporating hydraulic continuity constraints into the loss function enhances parameter identifiability and mitigates equifinality. The proposed approach advances vadose zone modeling by embedding hydrological principles within neural networks, thereby improving computational efficiency while preserving physical consistency. By coupling PINNs with field‑scale TDR observations, this framework bridges the gap between theoretical inverse modeling and practical soil monitoring.