Modeling Unsaturated Soil Water Transport Based on Loss-attentional Physics-informed Neural Networks

Physics-informed neural networks (PINNs) have recently attracted increasing attention as a data-efficient framework for solving partial differential equations governing complex subsurface flow processes. PINNs provide a promising alternative to conventional numerical methods for modeling unsaturated soil water flow, which is typically described by highly nonlinear governing equations. However, when applied to complex infiltration problems, conventional PINNs often suffer from imbalanced loss terms associated with initial conditions, boundary conditions, and governing equation residuals, leading to slow convergence and suboptimal accuracy.

In this study, a Loss-Attention Physics-Informed Neural Network (LAPINN) framework is employed to simulate unsaturated infiltration processes under both steady-state and transient conditions. The employed framework incorporates a loss-attention mechanism that adaptively reweights individual loss components during training, enabling the network to dynamically focus on regions and constraints that are more difficult to satisfy. This adaptive strategy effectively alleviates loss imbalance and enhances training stability without requiring manual tuning of loss weights.

The performance of LAPINN is systematically evaluated using three representative benchmark problems: (1) one-dimensional steady-state unsaturated infiltration, (2) one-dimensional transient unsaturated infiltration, including an inverse problem for hydraulic parameter identification, and (3) two-dimensional transient unsaturated infiltration with a prescribed Dirichlet boundary condition at the soil surface. Both forward and inverse modeling capabilities of the proposed framework are investigated.

The results demonstrate that LAPINN consistently outperforms standard PINNs in terms of prediction accuracy and convergence efficiency across all benchmark cases. In addition, the proposed method enables reliable inversion of hydraulic parameters using limited observational data. These results indicate that LAPINN provides a robust and efficient computational framework for modeling unsaturated soil water flow and offers strong potential for data-scarce hydrological and geotechnical applications.