The deep operator finite element method: A discrete variational neural operator for elasticity under variable loading

Learning solution operators for parametric partial differential equations is essential for real-time simulation and engineering analysis. However, existing physics-informed neural operators often suffer from unstable training due to numerical quadrature errors, sampling variability, and optimization difficulties associated with continuously defined physics losses. To address this, we propose the deep operator finite element method (DOFEM), a discrete variational neural operator for elasticity under variable loading. DOFEM extends the deep finite element method to operator learning by combining DeepONet with a physics loss constructed directly from the assembled FEM stiffness matrix and equivalent nodal load vectors. For linear elasticity, this loss is exactly quadratic with respect to the predicted nodal displacement vector, and its stationarity condition in the nodal-output space coincides with the discrete FEM equilibrium equation. The stiffness-matrix-based loss introduces a deterministic algebraic curvature structure that can improve optimization robustness. Numerical experiments demonstrate that DOFEM is more stable and accurate than the energy-based PI-DeepONet baseline. In particular, DOFEM remains robust in a three-dimensional problem in which the baseline becomes numerically unstable and fails to converge. Once trained, DOFEM serves as an efficient differentiable surrogate for rapid prediction under unseen loads, effectively bridging discrete variational mechanics and neural operator learning.