General scheme of physics-augmented constitutive learning. From displacement fields and global reaction forces, the deformation gradient F is used to build invariant-based inputs of a constrained neural architecture. The network predicts a strain-energy density W(F), from which the first Piola–Kirchhoff stress $P(F)=\partial W/\partial F$ is derived, ensuring thermodynamic consistency. Additional architectural and normalization constraints enforce material symmetries and polyconvexity, while training is driven in an unsupervised manner by equilibrium residuals. Figure composed from [147].