Physics-Informed Neural Networks for Flexoelectric Boundary Value Problems: A Reformulated Architecture for High- and Mixed-Order Electromechanically Coupled Systems

Abstract

Flexoelectricity denotes a universal electromechanical coupling in dielectric materials, whereby strain gradients break local inversion symmetry and induce electric polarization. The resulting boundary value problem (BVP) is governed by a coupled multi-physics system of high-order and mixed-order partial differential equations with nonstandard boundary conditions. Within this context, physics-informed neural networks (PINNs) offer an attractive mesh-free computational framework, providing $C^{\infty }$-continuous approximations of the solution fields and enabling the direct evaluation of high-order derivatives via automatic differentiation. However, we recognize that the distinctive differential structure of flexoelectric BVPs severely degrades the learning performance of naive PINN formulations. The direct use of primary field variables in strong-form residuals requires repeated evaluation of high-order derivatives, leading to unstable gradient propagation and degraded training efficiency. More critically, the problem exhibits a mixed-order structure across the coupled fields, which introduces conflicting gradients within a shared network and results in an ill-conditioned optimization landscape. In this work, we propose a reformulated PINN architecture for flexoelectric BVPs, retaining the strong-form and data-free manner. By incorporating pseudo outputs and auxiliary losses, the proposed architecture recasts the original high-order system into a lower-order reformulation; more importantly, it reduces the imbalance of differential orders across coupled fields within each loss component. This reformulation improves the compatibility between the differential structure of the governing equations and the PINN representation, thereby alleviating the associated optimization difficulty and enhancing training efficiency and accuracy. Numerical investigations on representative flexoelectric benchmark problems demonstrate strong performance as a promising numerical approach for flexoelectric BVPs and clear advantages over naive formulations. Beyond flexoelectricity, the present work provides new insights into the treatment of high- and/or mixed-order coupled partial differential systems within the PINN framework.