Discrete energy as an exact label-free training objective for finite-element surrogates

• URL: http://arxiv.org/abs/2608.05437v1
• Date: Wed, 05 Aug 2026 22:15:38 GMT
• Title: Discrete energy as an exact label-free training objective for finite-element surrogates
• Abstract summary: This note records, with proofs, the identities that make this signal exact for linear elastostatics.<n>Label-free discrete-energy minimisation and supervised regression in the stiffness norm have the same unique minimiser and identical gradients at every point.
• Score: 0.8125368885437911
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Supervised training of finite-element (FE) surrogate models requires reference solutions, and each reference solution is obtained by solving the system that the surrogate is intended to replace. The assembled discrete potential energy provides a training signal that requires no reference solution. This note records, with proofs, the identities that make this signal exact for linear elastostatics: the difference between the energy of a prediction and the energy of the reference solution equals one half of the squared stiffness-norm error, and the gradient of the energy equals the stiffness-weighted error. Label-free discrete-energy minimisation and supervised regression in the stiffness norm therefore have the same unique minimiser and identical gradients at every point. Around this central result, the note states a conditioning lemma that bounds the displacement error by the energy gap, a modewise contraction identity that explains why the Euclidean displacement error is an unsuitable primary metric, the Chebyshev bound that governs conjugate-gradient post-processing of surrogate predictions, and a conditional latent-separation proposition for joint-embedding predictive architecture (JEPA) pretraining on a shared stiffness operator, with an explicit numerical counterexample that delimits its scope. Every claim with numeric content is implemented as an executable falsification check; the checks were executed twice, on synthetic test problems and on a probe set of 16 instances from the validation split of a pre-registered experimental run, and every inequality holds, with the measured tightness reported. A closing section explains why the construction does not extend to elastodynamics through direct minimisation of the action functional, and which time-discrete formulation restores exactness.

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