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

2026-08-05

AI Summary

This work addresses the challenge of scaling traditional finite element surrogate models, which rely on costly reference solutions for supervised training. The authors propose a novel unsupervised training approach that eliminates the need for reference solutions by rigorously establishing, for the first time, an exact equivalence between discrete potential energy and stiffness-norm error. Leveraging this relationship, they design a gradient-consistent training mechanism that integrates discrete energy functionals, stiffness-weighted error analysis, and a JEPA (Joint-Embedding Predictive Architecture) framework. Validation across synthetic benchmarks and 16 experimental cases demonstrates that the resulting energy gap effectively controls displacement error. Furthermore, the study reveals that Euclidean error is ill-suited as a primary evaluation metric and delineates the method’s applicability boundaries.

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.

Problem

• finite-element surrogate
• label-free training
• discrete energy
• linear elastostatics
• training objective

Innovation

• discrete energy
• label-free training
• finite-element surrogate
• stiffness-norm error
• elastostatics