Evaluating Water Retention and Hydraulic Conductivity Properties Based on the Richardson–Richards Equation Using Progressive Training and Trainable Weights

ABSTRACT The Richardson–Richards equation serves as a fundamental model for understanding water movement in soil. The water retention curve (WRC) and the Hydraulic Conductivity Function (HCF) are key equations involved in defining the Richardson–Richards equation. The intrinsic nonlinearity of the Richardson–Richards equation presents major challenges in modeling soil–water processes, including the estimation of variables governed by WRCs and HCFs, especially under nonlinear and complex boundary conditions. To address this challenge, this paper innovatively proposes a physics–informed neural network (PINN) architecture for solving the Richardson–Richards equation, based on the concept of progressive training and trainable weights. Experimental results indicate that the proposed PINN architecture can effectively capture the highly nonlinear relationships WRC and HCF for the whole suction range. The proposed approach achieves high prediction accuracy for key soil–water variables, with R 2 consistently exceeding 0.88, and maintains good prediction accuracy even under significant environmental changes, demonstrating excellent generalization capabilities. This work provides a robust and adaptable framework for modeling soil moisture dynamics across a wide range of complex environmental and physical scenarios.