Physics-Informed Neural Networks For 3D Percolation Across Unseen Shapes

We investigate whether physics-informed neural networks (PINNs) can outperform conventional convolutional neural networks (CNNs) in predicting percolation behavior in three-dimensional voxelized shapes. Using seven shape families (cube, sphere, cylinder, ellipsoid, torus, elongated box, random porosity) and occupation probabilities p in [0.10, 0.60], we generate Monte Carlo ground truth labels for connectivity and train both CNN and PINN models under a leave-group-out protocol that withholds entire shapes for testing. The PINN augments a 3D CNN with auxiliary physics observables (largest-cluster fraction, second-moment of the cluster-size distribution, correlation length, and local connectivity ratio) and incorporates physics-based loss terms enforcing monotonicity in p, order-parameter consistency, and improved calibration. Across unseen geometries the PINN reduces RMSE by 8-15%, halves monotonicity violations, and improves calibration error by up to 35% while matching CNN accuracy on seen shapes. The results support the thesis that embedding coarse physical structure in learning systems improves robustness and generalization in discrete phase-transition problems.