Physics-Consistent Neural Networks for Learning Deformation and Director Fields in Microstructured Media with Loss-Based Validation Criteria

In this work, we study the mechanical behavior of solids with microstructure using the framework of Cosserat elasticity with a single unit director. This formulation captures the coupling between deformation and orientational fields that arises in many structured materials. To compute equilibrium configurations of such media, we develop two complementary computational approaches: a finite element formulation based on variational principles and a neural network-based solver that directly minimizes the total potential energy. The neural architecture is constructed to respect the fundamental kinematic structure of the theory. In particular, it enforces frame invariance of the energy, satisfies the unit-length constraint on the director field, and represents deformation and director fields through separate networks to preserve their kinematic independence in the variational setting. Beyond satisfying balance laws, however, physically admissible solutions must also correspond to stable energy minimizers. To assess this requirement, we derive the quasiconvexity condition, rank-one convexity condition, and the Legendre-Hadamard inequalities for the Cosserat model and formulate them in a manner suitable for evaluating neural network predictions. These necessary stability conditions provide a physics-based validation framework: network outputs that violate these necessary conditions cannot correspond to stable energy minimizers and can therefore be rejected. In this way, we integrate classical variational stability theory with modern machine-learning solvers, establishing a computational workflow in which equilibrium solutions are not only learned but also assessed for energetic consistency.