Discrete energy as an exact label-free training objective for finite-element surrogates

What the math gives to ML

The paper provides an exact label-free objective for neural surrogates of linear finite-element elasticity. Its transferable asset is the identity equating the discrete potential-energy gap with one half of the squared stiffness-norm displacement error, plus a gradient computable using only a sparse matrix-vector product. A neural operator or graph network can therefore train on mesh, material, boundary, and load data without generating reference displacement labels. The method is most useful when reference FEM solves dominate dataset construction and the stiffness matrix is already available or cheap to assemble.