Abstract
This work develops and systematically evaluates a physics-informed neural network (PINN) solver for the fully coupled, time-dependent Muskat–Leverett system with capillarity modeled in the pressure equation. A single shallow–wide multilayer perceptron jointly predicts wetting pressure and water saturation; physical capillary pressure regularizes the saturation front, while a small numerical diffusion term in the saturation residual acts as a training stabilizer rather than a shock-capturing device. To guarantee admissible states in stiff regimes, we introduce a saturation soft-clamping head enforcing 0