Adversarial deep energy method for solving saddle point problems involving dielectric elastomers

In this work, we develop an adversarial deep energy method (adversarial DEM) for solving saddle point problems with electromechanical coupling. Our approach uses physics-informed neural networks (PINN) to find the saddle point equilibrium of the underlying potential energy. Specifically, our deep energy method employs multiple distinct neural networks to model the mechanical and electrostatic fields in an adversarial relationship. The optimal configuration of an electromechanical system is calculated by using saddle point optimization algorithms. This involves minimizing the maximum value of the cost function which is related to the potential energy of the system. We demonstrate the performance of the proposed framework by solving various benchmark problems for dielectric elastomers and comparing the results with the analytical or finite element solutions. The proposed method significantly extends the applicability of DEM and has versatility for various material models in dielectric elastomer problems. In particular, the buckling phenomenon in electromechanical coupling problems, with which conventional finite element method (FEM) has struggled, can be successfully solved without difficulties. The proposed method is easy to implement using open-source machine-learning environments and is expected to pave the way for solving multiphysics problems effectively.