High-precision stress prediction using deep energy network enhanced by stress equilibrium with Delaunay integration on refined grid

We present the deep energy method enhanced by stress equilibrium (DEM-SE), a physics-informed neural network (PINN) architecture for high-precision prediction of complex stress fields in elastic plates. The method calculates total potential energy via Delaunay integration on a locally refined triangular grid and enforces stress-equilibrium equations as physical constraints to construct a hybrid loss function. Its accuracy in stress concentration prediction and ability to characterize complex stresses in nonhomogeneous materials are validated through four typical elastic deformation cases. Using sparse displacement measurements from just 200 local nodes, combined with equilibrium-based constraints, the PINN accurately reconstructs full-field displacements, outperforms purely data-driven methods, and enables simultaneous inversion of nonhomogeneous elastic modulus distributions with constant Poisson's ratio. Ablation studies clarify the contributions of grid refinement, Delaunay integration, and the equilibrium constraint. This work establishes DEM-SE as a robust tool for stress analysis and offers insights for designing PINNs in elasticity applications.