Estimation of soil hydraulic properties based on physics-informed neural networks

Soil hydraulic properties are key parameters for characterizing water infiltration, storage, and transport in soils, as well as their interactions with surface runoff and groundwater recharge. They play an essential role in hydrological simulation, soil and water conservation assessment, and agricultural water management. However, conventional methods for estimating soil hydraulic properties are often labor-intensive, time-consuming, and limited in their applicability. To address these limitations, this study proposes a method for estimating soil hydraulic properties based on physics-informed neural networks (PINN) and the Richardson–Richards equation (RRE). By incorporating the governing equation of soil water movement as the physical constraint, the proposed method is trained only with soil moisture observations and does not require initial or boundary conditions as training constraints, thereby reducing modeling complexity while maintaining physical consistency.Three representative soil types, namely cork oak forest soil, Pinus tabuliformis forest soil, and farmland soil, were selected as the study objects. Time series of volumetric water content measured at four soil depths during infiltration experiments were used as the input data for the PINN model. Meanwhile, the inverse modeling module of HYDRUS was employed to estimate soil hydraulic parameters and construct reference soil water characteristic curves (SWCC) and unsaturated hydraulic conductivity (K_u) data for validating the PINN estimation results. The root mean square error (RMSE), relative root mean square error (RRMSE), and relative error were used to evaluate the estimation accuracy of the proposed model.The results showed that the RMSEs between the volumetric water contents estimated from the PINN-derived SWCCs and the measured volumetric water contents were 0.144, 0.040, and 0.015 cm³/cm³ for the cork oak forest soil, Pinus tabuliformis forest soil, and farmland soil, respectively, with corresponding RRMSEs of 39.26%, 9.45%, and 4.85%. In the unsaturated range where the volumetric water content was lower than 0.4 cm³/cm³, the RMSEs for the two forest soils further decreased to 0.072 and 0.008 cm³/cm³, respectively, while the corresponding RRMSEs decreased to 37.16% and 2.37%, indicating that the proposed model achieved higher estimation accuracy in the unsaturated range. The relative errors between the PINN-estimated saturated hydraulic conductivity (K_s) and the measured values were 2.56%, 2.50%, and 7.89%, respectively. Owing to the lack of measured unsaturated hydraulic conductivity data, the PINN-estimated K_u values were further compared with the reference hydraulic conductivity curves constructed from independent observations. The corresponding RMSEs were 3.0×10⁻⁴, 7.3×10⁻⁵, and 7.1×10⁻³ cm/min, with corresponding RRMSEs of 8.39%, 1.19%, and 39.96%, respectively. Overall, the proposed PINN model effectively captured the hydraulic behavior of different soil types and provided stable estimates of soil hydraulic properties. Moreover, the method does not require the explicit specification of initial or boundary conditions, thereby reducing modeling complexity and demonstrating strong potential under limited data availability. This study provides a reliable technical approach for soil hydrological analysis and offers promising applications in soil moisture dynamics research, agricultural water management, and ecohydrological modeling.