# Surrogate Modelling

Finite-element surrogates are usually trained and compared by displacement error, although engineering decisions rest on stresses. This study replaces the norm of the supervised loss with the energy norm of the assembled stiffness matrix, leaving everything else fixed; in pre-registered comparisons a three-dimensional transformer's median von Mises error fell 6.1-fold and the two-dimensional mean fell by more than half, also on held-out instances, and training on the energy alone without solutions came close to that accuracy.

Finite-element surrogates are usually trained and compared by displacement error, although engineering decisions rest on stresses. This study replaces the norm of the supervised loss with the energy norm of the assembled stiffness matrix, leaving everything else fixed; in pre-registered comparisons a three-dimensional transformer's median von Mises error fell 6.1-fold and the two-dimensional mean fell by more than half, also on held-out instances, and training on the energy alone without solutions came close to that accuracy.

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.