Stress-Equilibrium-Enhanced Deep Energy Method for Electromagnetic-Mechanical Coupling in Superconductors

Abstract Electromagnetic-mechanical coupling in superconductors presents a nonlinear and tightly coupled challenge, where strong Lorentz forces during magnetization can induce severe stress concentrations and compromise structural reliability. Finite element methods provide a robust baseline, but their cost increases sharply in nonlinear, transient, or defect-sensitive regimes. Physics-informed neural networks (PINNs) offer a mesh-free alternative; however, the deep energy method (DEM), a widely used PINN formulation, often yields inaccurate stress predictions due to the lack of explicit equilibrium enforcement. Here we propose a Stress-Equilibrium-Enhanced Deep Energy Method (SeDEM) , which augments DEM by embedding strong-form equilibrium constraints together with Maxwell’s equations and nonlinear superconducting constitutive relations in a unified variational framework. Tailored strategies-including nondimensionalization, hard boundary enforcement, and dynamic loss weighting-further improve training stability and convergence. Benchmark studies on bulk superconductors confirm that SeDEM eliminates spurious oscillations, accurately resolves stress localization near defects, and achieves close agreement with finite element solutions. This work establishes SeDEM as a physically consistent and computationally efficient approach for coupled-field modeling in superconductors, with broader potential for advancing data-driven multi-physics analysis.