FB-PINN: A Fischer-Burmeister-enhanced physics-informed neural network for phreatic line prediction in saturated-unsaturated seepage flow

Accurate phreatic line prediction is essential for seepage analysis in earth dams, yet traditional mesh-based methods require costly iterative procedures. Existing physics-informed neural network (PINN) approaches typically model only the saturated zone, neglecting the influence of the unsaturated zone. This study develops the FB-PINN, a novel mesh-free neural solver integrating the smoothed Fischer-Burmeister (FB) function into the PINN for steady-state saturated–unsaturated seepage with an unknown seepage face. The seepage face gives rise to a Signorini-type nonlinear complementarity problem (NCP), which the smoothed FB function reformulates as a single smooth equality constraint, eliminating the ill-conditioned loss landscape and gradient oscillation inherent in conventional separate-penalty formulations. Built upon a mixed-form Richards equation and a trunk-branch architecture, FB-PINN simultaneously predicts pressure head, water content, and phreatic line location with high accuracy. The framework is validated across varying water levels, diverse dam geometries, stratified anisotropic materials, and zoned dams with clay cores. A localized numerical cut-off is also introduced to alleviate the strong near-saturation VG nonlinearity in the low-n regime. Following validation of the core formulation, FB-PINN is further extended to support rapid multi-condition seepage prediction through a parameterized surrogate and sparse-observation-based identification of van Genuchten parameters.