Ruifeng Cao

Finite-element surrogates are usually trained and compared by displacement error, although engineering decisions rest on stresses. This study replaces the norm of the supervised loss with the energy norm of the assembled stiffness matrix, leaving everything else fixed; in pre-registered comparisons a three-dimensional transformer's median von Mises error fell 6.1-fold and the two-dimensional mean fell by more than half, also on held-out instances, and training on the energy alone without solutions came close to that accuracy.

Finite-element surrogates are usually trained and compared by displacement error, although engineering decisions rest on stresses. This study replaces the norm of the supervised loss with the energy norm of the assembled stiffness matrix, leaving everything else fixed; in pre-registered comparisons a three-dimensional transformer's median von Mises error fell 6.1-fold and the two-dimensional mean fell by more than half, also on held-out instances, and training on the energy alone without solutions came close to that accuracy.