Large language models (LLMs) are usually evaluated in chess by how closely their position evaluations agree with a strong engine, but such agreement does not show that a model evaluates equivalent positions consistently. We apply metamorphic relations to chess evaluation: a 180-degree rotation of pawnless positions, left-right mirroring, color swapping, position pairs one move apart, conversion between Forsyth-Edwards Notation and a natural-language description, and illegal positions. Six LLMs (Claude Sonnet 4.5, DeepSeek Chat, Gemini 2.5 Flash, GPT-5.1, Grok 4.1 Fast and Kimi K2 Turbo) evaluated seven test sets of 500 items each, with Stockfish 16.1 as the reference. All six models failed on rotation: at most 10.6% of the 500 rotated pairs received evaluations within 300 centipawns (cp) of each other, and the models often declared the legal rotated positions illegal. On the other four transformations, stability ranked the models as accuracy did except for the first two places: GPT-5.1 had the lowest mean stability error (186.17 cp) and Claude Sonnet 4.5 the lowest accuracy error (330.41 cp). Rejection rates for illegal positions ranged from 44.2% to 96.0%. Stability should be reported with the share of consistent pairs, because means over answered pairs concealed most rotation failures.
Finite-element surrogates predict the stresses of a new design in one evaluation, but are usually trained on solutions with a displacement loss and compared by it, although engineering decisions rest on stresses. Application in engineering: stress analysis of perforated plates and solids with cavities, parametric families in which holes concentrate stress. Contribution to artificial intelligence: training and evaluating such surrogates in the energy norm, with or without solutions, and screening their predictions without solutions. For linear elastostatics, minimising the assembled discrete potential energy is exactly supervised regression in this norm: the energy gap of a prediction is the strain energy of its stress error. A displacement loss under-weights the stiff modes that carry the strains. In pre-registered comparisons, changing only the norm of the supervised loss cut a three-dimensional transformer's median energy-norm error 7.5-fold and its median von Mises error 6.1-fold; in two dimensions it more than halved the von Mises error, also on held-out instances. In three dimensions, the transformer trained on the energy alone came within a factor of 1.3 of the energy-norm-supervised one in median energy-norm and von Mises error. It was more accurate than the displacement-supervised one on all five metrics in every seed. Given the stiffness matrix, a Ritz projection was more accurate than the label-free transformer, in the seed means, on every metric but peak stress. No network transferred to a substantially finer mesh. The energy ranks predictions of a load case, and its sign flags those worse than the zero field.