← 研踪 PaperTrailarXiv:2611.14574 · v1
The norm, not the labels: training and evaluating finite-element surrogates for stress analysis in the energy norm
首次提交 2026-11-10 07:35
版本更新 2026-11-10 07:35
雷达观测 2026-11-13 10:10
摘要
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.
作者备注：45 pages, 6 figures; supplementary material (37 pages, 4 figures) appended
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v1 · 2026-11-10 07:35 · 最新观测版本
The norm, not the labels: training and evaluating finite-element surrogates for stress analysis in the energy norm
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.
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