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

arXiv: https://arxiv.org/abs/2608.05437 · HTML: https://mlsanity.com/paper/2608.05437/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 19 references that are in the map.
- First posted on arXiv 5 Aug 2026.

## Builds on (most cited first, 5 of 5)

- [Self-Supervised Learning from Images with a Joint-Embedding Predictive Architecture](https://mlsanity.com/paper/2301.08243/) (2023)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications](https://mlsanity.com/paper/1908.10407/) (2019)
- [An operator preconditioning perspective on training in physics-informed machine learning](https://mlsanity.com/paper/2310.05801/) (2023)
- [Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equations](https://mlsanity.com/paper/2304.04234/) (2023)

## Abstract (arXiv)

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.