PaperlayerView exact revision on arXiv

Source-linked AI summary

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

Karim Bounja, Lahcen Laayouni, Boujemaa Achchab, Abdeljalil Sakat

arXiv:2608.16473v1cs.LGmath.NA

TL;DR

Selecting the best neural PDE checkpoint is difficult because logged energy errors require the exact solution, and loss-based selection may misorder finite archives. The paper introduces a computable conforming Riesz monitor and proves that refinement eventually recovers the logged energy oracle, with finite-resolution certificates under saturation.

  • Problem

    Logged energy errors are inaccessible without the exact solution, while checkpointwise recovery or loss minimization may not preserve the archive’s energy-error ordering at finite resolution.

  • Method

    The paper uses a computable, training-independent conforming Riesz monitor based on residual–energy identities and conforming reconstruction to select archived candidates.

  • Results

    Monitor refinement eventually recovers the logged energy oracle; under saturation, nested reconstructions provide near-oracle bounds and interval-separation certificates.

  • Takeaways & Limitations

    The framework enables reference-free archive selection on the intrinsic energy-error scale, with experiments demonstrating calibrated monitoring, oracle-level selection, and low overhead.

  • Takeaways & Limitations

    Logged-oracle guarantees assume the prescribed archive contains a minimizer of the energy error over the computed training trajectory; retaining such a minimizer is a separate archive-coverage question.

Abstract

from arXiv · show

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.

1 Introduction

The paper develops a reference-free, conforming Riesz-monitor rule for selecting energy-optimal checkpoints from finite neural PDE archives without the exact solution. Uniform recovery resolves finite-resolution ordering errors, while refinement, saturation, and logging-resolution results provide oracle selection, near-oracle bounds, and trajectory certificates.

  • Motivation: The inaccessible exact energy criterion motivates computable diagnostics aligned with the problem-intrinsic energy geometry.The exact solution is unknown, so direct evaluation of checkpoint energy errors is unavailable.
  • Reference-free monitor: The conforming Riesz monitor is exactly calibrated to each logged checkpoint’s energy error and converges monotonically under nested conforming refinement.The monitor is built from the energy-dual residual and conforming projection, providing an intrinsic reference-free quantity.
  • Ordering obstruction: Checkpoint-dependent unresolved components can reverse oracle–non-oracle rankings at finite auxiliary resolution, so checkpointwise recovery alone is insufficient for archive selection.The paper addresses this obstruction by proving uniform recovery over finite archives.
  • Oracle recovery: For finite archives, uniform recovery eventually restores oracle separation and yields logged-oracle selection without any saturation assumption.Monitor minimization is therefore consistent in value and selection as auxiliary resolution becomes sufficiently fine.
  • Certification: Under saturation, nested reconstructions provide finite-resolution near-oracle bounds and interval-separation certificates, while logging-resolution control quantifies archive-subsampling loss and certifies trajectory-oracle inclusion.These guarantees distinguish auxiliary refinement from the resolution at which checkpoints were logged.
  • Numerical validation: Experiments on diffusion, elasticity, and a non-manufactured perforated plate demonstrate energy-error calibration, oracle-level selection, controlled logging loss, and low monitoring overhead.The perforated-plate results use an independent refined FEM reference.

2 Problem setting

The problem is posed on an affine admissible class whose differences lie in a homogeneous constraint space, with a symmetric bilinear form coercive there. Admissible neural approximations enable residual–energy identification, whereas penalty-only boundary enforcement may not.

  • Affine admissible class: Admissible functions lie in A := ub + V0, while differences of admissible functions lie in V0; homogeneous data give A = V0.Here V0 is the closed linear space of homogeneous admissible directions and ub lifts prescribed essential data.
  • Coercive variational structure: A continuous symmetric bilinear form is coercive on V0, inducing the energy inner product and norm used for the variational problem.Subtracting the lifting reduces the affine problem to a coercive problem on V0, where Lax–Milgram applies.
  • Residual–energy identity: For admissible uθ, the error uθ − u∗ belongs to V0, so the energy-dual residual norm equals the energy error.Residual–energy identities are formulated on V0, while Riesz reconstructions and hierarchical estimators use conforming subspaces of V0.
  • Neural admissibility: Hard boundary ansätze enforce admissibility by construction, but finite-weight boundary penalties do not guarantee uθ ∈ A or direct applicability of the residual–energy identity.Natural boundary conditions enter through the variational forms rather than imposing additional affine constraints on uθ.

3 Residual–energy structure and conforming Riesz error assessment

This section establishes that the energy-dual residual exactly equals the energy error and develops a computable conforming Riesz monitor for admissible neural approximations. The monitor increases monotonically from below under nested refinement, while saturation supplies a conditional lower–upper bracket for archive-level selection.

  • Exact residual–energy structure: For every admissible approximation, the energy-dual residual norm coincides exactly with the energy error, enabling reference-free minimization of the unknown error.Unlike Ritz values, the residual norm does not require the unknown reference level J(u∗).
  • Conforming Riesz assessment: The continuous Riesz representative of the residual is the signed energy error, so its conforming projection yields a computable lower monitor.The unresolved component is the energy-norm difference between the continuous representative and its auxiliary-space projection.
  • Conforming Riesz assessment: Under nested conforming refinement, the monitor increases monotonically to the exact energy error from below.Nestedness makes the unresolved projection error nonincreasing, while density drives it to zero.
  • Hierarchical enrichment: Nested auxiliary spaces decompose the error into resolved, enrichment-revealed, and unresolved components independently of network parametrization or training.This makes the construction applicable post-training to finite checkpoint archives.
  • Hierarchical enrichment: Under saturation, the lower monitor and enrichment increment form a conditional lower–upper bracket whose width bounds possible underestimation.A wide bracket indicates that further auxiliary refinement may be required.

4 Reference-free checkpoint selection and logged-oracle guarantees

This section establishes reference-free selection by minimizing a conforming Riesz monitor over a fixed finite checkpoint archive. Conforming refinement removes finite-resolution rank reversals and yields eventual logged-oracle recovery, while saturation enables computable near-oracle control and exact certification by interval separation.

  • Finite-resolution obstruction: At fixed auxiliary resolution, checkpoint-dependent unresolved projection defects can reverse the monitor ordering even when the monitor remains a valid lower bound.The unresolved component depends on each checkpoint’s error direction relative to the auxiliary space, not only its energy-error norm.
  • Conforming refinement: Common nested conforming refinement makes checkpointwise recovery uniform over the finite archive, eliminating rank reversals and yielding value consistency.The result follows because finitely many projection defects converge to zero uniformly.
  • Reference-free selection: The conforming Riesz monitor is minimized over the prescribed finite archive using only the unconditional lower monitor, independently of saturation.Saturation enters only the later finite-resolution guarantees.
  • Logged-oracle recovery: Sufficiently fine auxiliary resolution eventually selects a logged energy-oracle checkpoint, but this asymptotic guarantee is not itself a computable finite-resolution certificate.Stable monitor minimizers across tested levels provide empirical robustness indications, not certification without reference information.
  • Finite-resolution certification: Under saturation, hierarchical lower–upper intervals provide finite-resolution near-oracle control and can certify a unique logged energy oracle when intervals are separated.The lower-monitor selection remains independent of the prescribed factor q, which enters only through conditional upper estimates.
  • Scope and contribution: The guarantees recover only the minimum-energy-error candidate retained in the prescribed archive and cannot compensate for omitted better iterates from the training trajectory.Archive coverage is a distinct selection problem.

5 Logging resolution and trajectory-wide guarantees

The section quantifies how archive coarsening increases trajectory-wide selection loss through omitted iterates and develops competitive-coverage bounds and exact oracle-inclusion certificates. These guarantees support choosing the sparsest logging resolution meeting prescribed accuracy or certification criteria, with joint logging and auxiliary refinement recovering the trajectory-oracle level asymptotically.

  • Logging-resolution effects: Coarser logging cannot improve the best energy-error level in the retained archive because it may omit lower-energy iterates.The logged oracle therefore depends on the logging resolution and can worsen under archive subsampling.
  • Trajectory-wide control: Competitive coverage restricts trajectory-wide analysis to iterates that could improve on the selected checkpoint and measures their energy distance to retained checkpoints.These pairwise distances do not require access to the exact solution.
  • Trajectory-wide control: Trajectory-wide suboptimality decomposes into monitor-based selection loss within the archive and archive-subsampling loss relative to the comparison trajectory.Proposition 11 bounds these contributions through an auxiliary-resolution term and an archive-coverage term.
  • Logging-resolution selection: Candidate strides can be assessed by deterministic subsampling of one densely recorded comparison run, identifying the sparsest tested resolution satisfying quantitative accuracy or certification requirements.This avoids retraining and exposes the trade-off between logging sparsity and trajectory-wide accuracy.
  • Oracle certification: Exact certificates determine when a coarsened archive contains a trajectory oracle or when its selected checkpoint is the unique trajectory oracle.The criteria use lower bounds for omitted iterates and upper bounds for archive-attained errors; valid conservative upper bounds may replace direct coverage radii.
  • Asymptotic recovery: Joint decay of the auxiliary-resolution gap and archive-coverage loss yields asymptotic recovery of the trajectory-oracle level under joint logging and auxiliary refinement.The result holds relative to the same finite prescribed comparison trajectory.

6 Numerical validation

Numerical experiments show that the conforming Riesz monitor calibrates energy errors, recovers logged or FEM-reference oracle checkpoints, and remains effective across scalar, elastic, singular, and perforated-plate problems. Auxiliary enrichment improves calibration and certification, while operational monitoring remains low-cost and reference-free.

  • Scalar benchmarks: Across all twenty scalar runs, η24 selects the same logged checkpoint as the Equad-oracle, with η24/Equad = 0.975–0.983 and η48/Equad = 0.994–0.996.The level-24 monitor incurs below-1% observed overhead, while conditional q = 0.9 upper estimates remain within about 5%–7% of reference error.
  • Logging resolution and trajectory coverage: Trajectory experiments distinguish archive certification from coverage: K400 certifies all five logged oracles but contains no K50-oracle, whereas K100 limits maximum deterioration to 0.91%.Coarser logging increases trajectory-wide bounds and observed deterioration.
  • Elasticity: In elasticity, the Riesz criterion is oracle-level in all three material cases, while training-loss selection has mean deterioration factors Dc = 6.60–12.29.Mean η24/Equad calibration ranges from 0.971 to 0.986 and rises to 0.993–0.996 at level 48.
  • Reentrant-corner singularity: For the reentrant-corner benchmark, η64 and η128 recover the logged oracle in all five runs, although finer auxiliary resolution is required than for smooth square-domain tests.At η64, η64/Equad = 0.911 ± 0.022 already recovers the oracle, while η128 raises the captured fraction to 0.965 ± 0.009.
  • Computational cost and diagnostic scope: Operational monitoring costs less than 1% overhead in scalar and L-shaped tests and 8.01%–8.72% in elasticity, reserving enriched reconstructions for qualification and certification.The diagnostics are reference-free relative to the logged archive, but deterministic-quadrature quantities are not certified continuous-energy-norm bounds unless quadrature error is controlled.
  • Perforated plate: In the perforated-plate experiment, η48 and η96 select the FEM-reference oracle in all five runs, while ηm/EFEM increases from 0.977±0.002 to 0.998±0.000.The conditional q = 0.9 bracket has upper-to-lower factor 1.023 ± 0.002, and η48 adds 4.19% overhead.

7 Discussion and outlook

The framework enables reference-free, archive-level recovery of the logged energy oracle through conforming Riesz reconstruction, while finite-dimensional projection defects make ordering preservation essential. The discussion identifies extensions toward broader formulations, certification, and optimization-integrated selection.

  • Contributions: Conforming Riesz reconstruction provides a computable, training-independent monitor that recovers the logged energy oracle without the exact solution or a reference solve.Monitor refinement yields eventual logged-oracle recovery; nested reconstructions provide finite-resolution certification under saturation.
  • Ordering preservation: Finite-dimensional projection defects can reverse oracle–non-oracle ordering, so checkpointwise recovery alone cannot guarantee archive-level selection.Finite-archive uniform recovery eventually restores the correct ordering.
  • Ordering preservation: In the symmetric coercive setting, continuous residual and energy-error orderings coincide, leaving finite-dimensional reconstruction as the source of the ordering obstruction.Exact norm equivalence alone does not generally preserve the ordering relevant to archive selection beyond this setting.
  • Outlook: Natural extensions include saturation-free finite-resolution certification, archive selection beyond symmetric coercive formulations, and explicit quadrature-error control for fully certified continuous-energy-norm bounds.The framework currently selects among optimizer-generated candidates; carrying order preservation upstream into optimization remains future work.

A Sensitivity to the prescribed saturation parameter

The prescribed saturation parameter q affects only conditional upper estimates, not monitor-based checkpoint selection. Across the tested range, certification patterns and perforated-plate upper estimates remain qualitatively stable, with increasing conservatism as q grows.

  • A Sensitivity to the prescribed saturation parameter: q enters only the conditional upper estimate and does not affect monitor-based checkpoint selection.The tested values are q ∈ {0.70, 0.80, 0.90, 0.96}.
  • A Sensitivity to the prescribed saturation parameter: Certification remains stable over a broad q range, especially for V48 ⊂ V96, while upper-bound conservatism gradually reduces certificate counts.Higher counts on coarser subarchives reflect greater checkpoint separation rather than improved trajectory coverage.
  • A Sensitivity to the prescribed saturation parameter: 1.048 ± 0.005 is the perforated-plate upper-estimate ratio at q = 0.96 relative to the independent FEM-reference error.The estimate remains close to the FEM-reference error across the tested q range.
  • A Sensitivity to the prescribed saturation parameter: The qualitative conclusions remain stable across the tested q range, with the expected increase in conservatism as q grows.This sensitivity concerns the finite-resolution certificates and conditional upper estimates.

B Spatial diagnostics of the conforming Riesz reconstruction

Spatial diagnostics show that conforming Riesz-projected energy densities recover the dominant energetic regions of reference error in both manufactured stress tests. The L-shaped test also retains nonzero contributions away from the reentrant corner, motivating global auxiliary refinement.

  • Spatial diagnostics: Across two manufactured stress tests, conforming Riesz diagnostics qualitatively identify the dominant energetic regions captured by the reference energy-error density.The comparisons provide a spatial view of where the energy error is concentrated.
  • Smoothed high-contrast inclusion: In the smoothed high-contrast inclusion, the Riesz-projected density identifies the same dominant energetic region associated with the localized material transition as the reference density.Figure 6 compares the reference and conforming Riesz-projected densities on a log10(· + ε) scale.
  • L-shaped-domain stress test: In the L-shaped-domain stress test, both densities concentrate primarily near the reentrant corner while retaining nonzero contributions away from it.The spatial pattern is consistent with the observed need for global auxiliary refinement; Figure 7 uses a 128 × 128 auxiliary space.

C Discrete-hierarchy and quadrature-refinement audit

The discrete-hierarchy and quadrature-refinement audits were conducted post-training using the same trained trajectories and saved checkpoints as the main experiments.

  • Audit setup: All audits were performed post-training.The audit procedure was applied after training had completed.
  • Audit setup: The audits reused the same trained trajectories as the main experiments.No separate training trajectories were used for these audits.
  • Audit setup: The audits reused the saved checkpoints from the main experiments.Checkpoint data were carried over from the primary experimental runs.

C.1 Monitor-hierarchy audit

The monitor-hierarchy audit found no monotonicity violations when recomputing diagnostics at saved selected checkpoints. Increasing quadrature order produced only small monitor-value changes in both benchmarks.

  • C.1 Monitor-hierarchy audit: Monitor diagnostics were recomputed at saved operationally selected checkpoints on the same auxiliary levels using common finest-level Galerkin restrictions.The audit covered the scalar-diffusion and L-shaped benchmarks.
  • C.1 Monitor-hierarchy audit: No monotonicity violation was observed in either the scalar-diffusion or L-shaped benchmark.
  • C.1 Monitor-hierarchy audit: 1.314×10−6 was the maximum scalar-diffusion monitor change when quadrature order increased from 3 to 5.
  • C.1 Monitor-hierarchy audit: 6.486 × 10−4 was the maximum L-shaped monitor change when quadrature order increased from 3 to 5.

C.2 Validation-oracle stability

Refinement audits show that the reported validation-oracle and logging-resolution conclusions are numerically stable under tested quadrature and FEM refinements. A sufficient gap-versus-error condition would certify the numerical oracle as the unique continuous-energy oracle, but the required integration-error bounds remain future work.

  • Quadrature-refinement audit: The manufactured-reference audit reevaluates energies at tensor-product Gauss orders Q = 3, 5, 7 without retraining to test ordering stability.The largest sensitivity occurs for the singular L-shaped problem and decreases from Q = 3 →5 to Q = 5 →7.
  • Quadrature-refinement audit: Quadrature refinement preserves the complete κ4 archive’s oracle and runner-up across all five trajectories, despite a smallest relative gap of 1.986 × 10−3.The trajectory-oracle inclusion counts remain unchanged: K100 : 3/5, K200 : 1/5, and K400 : 0/5.
  • FEM-reference refinement: For the perforated plate, FEM-reference refinement from level 192 to 256 preserves the FEM-reference oracle in all five runs.The maximum relative checkpoint-error change is 6.26 × 10−4, with rank correlation at least 0.999826.
  • Continuous-energy oracle certification: If every numerical energy error differs from its exact continuous counterpart by at most ε and the numerical oracle gap exceeds 2ε, the numerical minimizer is the unique continuous-energy oracle.Applying this certification to an Equad-based oracle requires a suitable uniform numerical-integration error bound, which is left to future work.
Loading 2608.16473v1…