# Porous-Media Flow

With only 20 to 50 coordinate–state observations, physics-informed neural networks (PINNs, which write the governing equations and initial and boundary conditions into the loss) reach the highest mean primary-target accuracy on a heat equation and three heterogeneous two-phase-flow benchmarks, and the water-saturation advantage holds through 1000 labels. Hierarchical loss weighting (H-PINN) and its gradient-direction-aware extension (HO-PINN) activate physical constraints as higher-priority losses approach prescribed scales, reducing run-to-run variability in selected settings. The cost is fitting time substantially higher than tuned gradient boosting, support-vector regression and Gaussian-process regression.

With only 20 to 50 coordinate–state observations, physics-informed neural networks (PINNs, which write the governing equations and initial and boundary conditions into the loss) reach the highest mean primary-target accuracy on a heat equation and three heterogeneous two-phase-flow benchmarks, and the water-saturation advantage holds through 1000 labels. Hierarchical loss weighting (H-PINN) and its gradient-direction-aware extension (HO-PINN) activate physical constraints as higher-priority losses approach prescribed scales, reducing run-to-run variability in selected settings. The cost is fitting time substantially higher than tuned gradient boosting, support-vector regression and Gaussian-process regression.