Accurate coupled pressure-saturation prediction is essential in subsurface energy applications, where displacement often develops sharp saturation fronts. In the Buckley-Leverett model [1], the saturation equation is nonlinear and hyperbolic; its entropy solution contains moving shocks that standard physics-informed neural networks (PINNs) [2], a core tool in physics-informed machine learning [3], tend to oversmooth. We propose a unified, single-network PINN that outputs both pressure and water saturation. Training minimizes a coupled physics loss: (i) a saturation conservation residual with Darcy water flux depending on pressure gradients and saturation-dependent relative permeabilities, and (ii) an elliptic pressure equation from Darcy’s law and total-mobility conservation. Initial and boundary conditions are imposed with additional loss terms, avoiding operator splitting and multi-network architectures. Across 500+ ablation runs, the model consistently recovers pressure but exhibits front smearing and phase error in saturation; increasing capacity or sampling does not remove this pathology, consistent with analyses of PINN training dynamics [4]. To improve shock capture we add a vanishing-viscosity term to the saturation residual during training, following artificial-viscosity stabilization ideas for PINNs on hyperbolic PDEs. We then combine this with front-aware loss weighting and conservation-aware enforcement strategies inspired by conservative PINNs [5]. Relative to diffusion alone, the optimized configuration reduces mean relative L² error by 44.4% (saturation) and 27.86% (pressure), improving front localization without degrading the smooth pressure field [6].