A Mixed-Variable Physics-Informed Neural Network for Direct and Converse Piezoelectricity in a Bimorph Cantilever

In design applications, simulations enable rapid iterations and adjustments to device architecture, reducing the need for physical prototyping. The simulation of piezoelectric devices has traditionally relied on the Finite Element Method (FEM). Physics-Informed Neural Networks (PINNs) offer an alternative based on governing equations without requiring labeled solution data. This study develops a mixed PINN architecture for the static direct and converse piezoelectric responses of a polyvinylidene fluoride (PVDF) bimorph cantilever. The network predicts two mechanical displacements, electric potential, three stress components, and two electric-displacement components. The methodology integrates the piezoelectric governing equations into a first-order loss formulation. The models are evaluated against coupled FEM solutions. For the converse effect, the relative L2 errors in horizontal displacement, vertical displacement, and electric potential are 0.107, 0.150, and 0.045, respectively. For the direct effect, they are 0.117, 0.153, and 0.067. The mixed configurations give lower displacement errors than the tested networks predicting only displacement and electric potential. However, they also use hard traction enforcement in the direct problem and mechanical gradient routing in the converse problem, so the improvement cannot be attributed to the additional outputs alone. Discrepancies between independently predicted stresses and those reconstructed from displacement and potential derivatives from the PINN reveal incomplete physical consistency. These results support approximate displacement and potential prediction while identifying constitutive consistency as a remaining limitation.