Physics–Informed Neural Network With Enhanced Parameter–State Coupling for Inverse Modeling of Unsaturated Flow in Heterogeneous Soils

Modeling unsaturated flow remains challenging due to the interplay of uncertain atmospheric forcing, parameter heterogeneity, and sparse observations. This study presents the application of physics–informed neural networks (PINNs) with Karhunen–Loève Expansion (KLE) to unsaturated flow, specifically designed to handle both soil heterogeneity and boundary uncertainty. We propose KLE–PINN–EC (Enhanced Coupling), a novel architecture that explicitly couples parameter and state representations through a branch–trunk design to enhance learning from sparse data. Through numerical experiments, we compare KLE–PINN–EC against (a) standard KLE–PINN, previously successful in groundwater modeling but untested for highly nonlinear unsaturated flow, and (b) ensemble smoother with multiple data assimilation (ES–MDA), a well–established data assimilation method. Our findings reveal that: (a) KLE–PINN successfully handles combined uncertainties in parameters and boundary conditions; (b) KLE–PINN–EC achieves superior performance over standard KLE–PINN in sparse data scenarios; and (c) while ES–MDA performs competitively when boundary timing is known, its performance degrades significantly under uncertainty in boundary timing, whereas KLE–PINN–EC maintains robust performance. These results suggest that the KLE–PINN–EC framework provides a flexible and robust alternative for characterizing unsaturated zone processes in environments where both boundary conditions and subsurface properties are poorly constrained.