Thermodynamically admissible neural solvers for stiff electro-elastodynamics via exact geometric constraints

Abstract

Physics-Informed Neural Networks (PINNs) serve as continuous, mesh-free solvers for partial differential equations, but they frequently encounter optimization failures when applied to strongly coupled, stiff multiphysics systems. In piezoelectricity, the disparity in energetic scales between mechanical stress and electric displacement creates severely ill-conditioned loss landscapes, resulting in gradient pathologies. Standard PINN formulations rely on penalty-based soft constraints for boundary and initial conditions, which exacerbate this stiffness, leak unphysical energy, and corrupt boundary stress calculations. In this work, we present a constrained PINN architecture for 1D coupled electro-elastodynamics that structurally bypasses this gradient competition. By applying the quasi-static approximation and utilizing analytical distance functions, we enforce exact Dirichlet boundaries and second-order time initial conditions directly within the neural network’s topology. This geometric constraint restricts the optimizer to a physically consistent energy manifold and enables recovery of maximal boundary stresses without internal gradient noise, a capability that is critical for predicting mechanical fatigue and failure in high-frequency transducer architectures. Stabilized by a deterministic quasi-Newton optimization stage, our fully constrained model resolves the mechanical displacement and electric potential fields with a global relative L2 error of O(10−5). These results demonstrate that the gradient pathologies typically observed in stiff multiphysics PINNs can be systematically neutralized through the exact algebraic imposition of boundary and initial constraints.