Finite-element surrogates swap the displacement loss for energy: median von Mises error falls 6.1-fold in 3D and more than halves in 2D

For linear-elastostatic surrogates of perforated plates and solids with spherical cavities, the norm of the training loss, not the use of finite-element solutions as labels, decided the stress accuracy. The usual practice trains on the Euclidean error of nodal displacements and compares by the same error, a norm that under-weights the stiff modes carrying the strains. In pre-registered comparisons that changed only the norm of the supervised loss, with labels and initialisation held fixed, a three-dimensional transformer's median energy-norm error fell 7.5-fold and its median von Mises error 6.1-fold, the two-dimensional mean von Mises error fell 54.9%, and the stress-error reduction replicated on held-out instances.

The assembled discrete potential energy exceeds its minimum by exactly half the squared energy-norm error of a prediction, which is the strain energy of its stress error. Minimising the energy is therefore supervised regression in the energy norm, with the same gradients, yet needs no finite-element solution. In three dimensions the transformer trained on the energy alone, without solutions, came within factors of 1.30 and 1.24 of the energy-norm-supervised network in median energy-norm and von Mises error, against factors of 7.5 and 6.1 for changing the norm.

The energy also ranks the predictions of one load case without a reference solution and, by its sign, flags those worse than the zero field. Ranked by displacement error, a three-dimensional graph network came first, yet its von Mises error was 5.3 times the label-free transformer's. On held-out instances the two networks trained on displacement error had 340 and 237 load cases with positive energy, exactly those worse than the zero field; after rescaling by the energy-optimal amplitude, no flagged prediction remained worse than the zero field.

The conclusions rest on linear elastostatics, where the energy identity is exact; the geometries are two-dimensional perforated plates and three-dimensional boxes with spherical cavities, with four load cases per dimension, a homogeneous isotropic material and three seeds per network. The demonstrated scope is prediction within the range of training mesh sizes, and no network transferred to a finer mesh at about half the element size.

In two dimensions the reduction of the mean relative energy gap by energy-norm supervision over displacement supervision did not replicate on held-out instances, because a few instances dominate the mean, while the reduction on the medians held. In three dimensions, whether the gap between the energy-only network and the energy-norm-supervised one comes from the labels, the weighting of the load cases or the learning rate was not separated.