Reference-free logged energy-oracle recovery for neural approximations of symmetric coercive variational problems: conforming Riesz reconstruction and archive-level selection
2026-08-17 • Machine Learning
Machine Learning
AI summaryⓘ
The authors address a problem in training neural networks to solve partial differential equations (PDEs), where it's hard to measure energy errors without knowing the exact solution. They propose a new method that uses a computable monitor based on mathematical projections, which can estimate the energy error reliably without needing the exact answer. Their approach helps pick the best neural network checkpoints from a finite archive by providing guaranteed bounds on errors and improving selection even when traditional methods fail. They tested their method on different PDE problems and showed it works well with reasonable computational effort.
neural PDE trainingenergy errorvariational problemsRiesz monitorconforming projectionresidual-energy identitycoercive operatororacle selectionenrichmentdiffusion and elasticity
Authors
Karim Bounja, Lahcen Laayouni, Boujemaa Achchab, Abdeljalil Sakat
Abstract
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.