# Reference-free logged energy-oracle recovery for neural approximations of symmetric coercive variational problems: conforming Riesz reconstruction and archive-level selection

arXiv: https://arxiv.org/abs/2608.16473 · HTML: https://mlsanity.com/paper/2608.16473/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 20 references that are in the map.
- First posted on arXiv 17 Aug 2026.

## Builds on (most cited first, 3 of 3)

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control](https://mlsanity.com/paper/2105.14094/) (2021)

## Abstract (arXiv)

Neural PDE training yields a finite checkpoint archive, yet its logged energy errors are inaccessible without the exact solution, while loss-based selection does not necessarily recover the logged energy oracle. For admissible neural approximations of symmetric coercive variational problems, we introduce a reference-free selection rule based on minimizing a computable conforming Riesz monitor. The exact residual-energy identity and conforming projection make the monitor an unconditional lower bound converging monotonically to each logged energy error under nested conforming refinement; under saturation, hierarchical enrichment yields a computable upper estimate and hence a lower-upper bracket. A key finding is that archive selection is order-sensitive: unresolved checkpoint-dependent components can reverse the oracle-non-oracle ranking at finite resolution, so checkpointwise recovery alone is insufficient. For finite archives, we prove uniform recovery, yielding convergence to the logged-oracle error and, without saturation, logged-oracle selection at sufficiently fine auxiliary resolution. Under saturation, the bracket gives a computable near-oracle bound and certifies unique logged-oracle selection upon interval separation. We also bound logging-resolution loss and certify oracle inclusion over prescribed comparison trajectories. The resulting criterion replaces inaccessible exact-error minimization by computable, training-independent post-training selection on the intrinsic energy-error scale, requiring only the computed candidates and the variational problem. Experiments on diffusion and elasticity, including a non-manufactured perforated plate, demonstrate energy-scale calibration, oracle-level selection, and modest post-processing cost.