用群结构张量代数强制物理选择规则，无需重训即可修正 Orb-v3 对称性缺陷

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Abstract:Symmetry determines which physical responses of molecules and crystals are allowed and underlies many models used throughout the physical sciences. Yet several widely used machine-learned models learn symmetry only approximately from data, raising the question of whether approximate symmetry is sufficient for scientific prediction. Using the Orb-v3 force field, we show that it is often not: the relaxed space group of a crystal depends on the orientation of the input for $47\%$ of a $150$-crystal subsample, and relaxation lowers the space group for half of all $399$ crystals studied. To characterize, quantify, and correct such failures, we introduce a tensor algebra, $\star_G$, defined by convolution over a finite symmetry group and applicable to tensors with noncommuting entries. We prove that the symmetry group determines the algebra's block-diagonal transform and use this structure to analyze model errors on symmetric targets. The resulting projections decompose the average squared error into orthogonal components arising from selection-rule violations, inconsistencies across symmetry-related inputs, and residual error. The removable components reveal how much error arises from broken symmetry, provide reference-free lower bounds on prediction error, and can be eliminated without sacrificing accuracy. For operators, projection followed by blockwise truncation yields the optimal symmetric low-rank approximation. Applying these projections to force predictions reduces both rates of space-group change to at most $2\%$. Exact selection rules, reference-free error bounds, and symmetry-preserving corrections are therefore available for any trained model without retraining.

Submission history

From: Shashanka Ubaru [view email]
[v1] Tue, 19 May 2026 19:47:40 UTC (12,390 KB)
[v2] Wed, 29 Jul 2026 04:24:05 UTC (12,507 KB)
[v3] Mon, 12 Oct 2026 18:55:10 UTC (12,181 KB)