Deep neural network homogenization of multiphysics behavior for periodic piezoelectric composites

We present a physically informed deep neural network homogenization theory for identifying homogenized moduli and local electromechanical fields of periodic piezoelectric composites. The theory employs a multi-network model to represent solutions to stress equilibrium and electrostatic partial differential equations in the inclusion and the matrix phases, leading to a more accurate depiction of field variables. Satisfaction of periodicity conditions for hexagonal and cubic symmetries are efficiently tackled using a series of sinusoidal functions with known periods and adjustable parameters. Additionally, the theory applies fully trainable weights to the collocation points. These trainable weights, trained concurrently with the neural network weights, compel the neural networks to enhance their performance when faced with large local stress/deformation gradients. The predictive capability of the proposed theory is illustrated by comparison with finite-element solutions for composites reinforced with unidirectional fiber, spherical particle, or weakened by an ellipsoidal cavity over a wide range of volume fractions.