△ Mechanism confirmed, baseline not beaten 2026

Energy-Riesz checkpoint selector

Implementation & benchmark of arXiv:2608.16473 — Reference-free logged energy-oracle recovery for neural approximations of symmetric coercive variational problems: conforming Riesz reconstruction and archive-level selection

Usefulness7/10

Difficulty4/10

Novelty7/10

Training Accuracy Stability Generalization Training Loss Functional analysis Geometry Numerical analysis

Source paper: Reference-free logged energy-oracle recovery for neural approximations of symmetric coercive variational problems: conforming Riesz reconstruction and archive-level selection arXiv:2608.16473 ⓘ · analyzed Sep 1, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Replace raw neural PDE training-loss checkpoint selection with a residual monitor measured in the variational energy geometry. For every archived network, solve an auxiliary conforming Riesz problem and select the checkpoint with the smallest reconstructed residual norm; nested auxiliary spaces make this score converge monotonically to the inaccessible energy error.

Formulas

$$r_k(v)=\ell(v)-a(u_k,v)=a(u^*-u_k,v),\qquad a(z_k,v)=r_k(v)\quad\forall v\in V_0,$$

$$\|z_k\|_a^2=a(z_k,z_k)=r_k(z_k)=a(u^*-u_k,u^*-u_k)=\|u_k-u^*\|_a^2,$$

$$M_m(k):=\|z_{k,m}\|_a=\sqrt{a(z_{k,m},z_{k,m})}\leq\|u_k-u^*\|_a,$$

$$M_m(k)\leq M_{m+1}(k)\longrightarrow\|u_k-u^*\|_a.$$

Mathematical statement

Let $\mathcal A=u_b+V_0$ be the affine admissible space, with homogeneous direction space $V_0$, and let $a:V_0\times V_0\to\mathbb R$ be symmetric and coercive. The exact solution $u^*\in\mathcal A$ satisfies $a(u^*,v)=\ell(v)$ for every $v\in V_0$. For candidate $u_k\in\mathcal A$, define the residual $r_k(v)=\ell(v)-a(u_k,v)=a(u^*-u_k,v)$. The full-space Riesz representative $z_k\in V_0$ satisfies $a(z_k,v)=r_k(v)$ for all $v\in V_0$. Symmetry gives $\|z_k\|_a^2=\|u_k-u^*\|_a^2$, where $\|v\|_a=\sqrt{a(v,v)}$. In an auxiliary conforming space $W_m\subset V_0$, define $M_m(k)=\|z_{k,m}\|_a$. Galerkin projection implies $M_m(k)\leq\|u_k-u^*\|_a$, and nested spaces yield monotone convergence.

Implementation notes

Integrate this as a post-training archive selector for a PINN or neural Galerkin solver. Store network parameters $\theta_k$, reconstruct $u_k=u_{\theta_k}$, and retain the PDE bilinear form $a$ and load functional $\ell$. Construct a conforming auxiliary basis $\{\phi_i\}_{i=1}^n$ satisfying homogeneous boundary conditions. For diffusion, use $a(u,v)=\int_\Omega\kappa\nabla u\cdot\nabla v\,dx$. Assemble $A_{ij}=a(\phi_j,\phi_i)$ once. For each checkpoint, compute $b_i^{(k)}=\ell(\phi_i)-a(u_k,\phi_i)$, solve $Az^{(k)}=b^{(k)}$, and score $M(k)=\sqrt{(z^{(k)})^TAz^{(k)}}$. Return the checkpoint with minimum score rather than minimum training loss. Repeat with nested auxiliary bases to verify monotonic score growth and ranking stability. The residual identity and lower-bound guarantee come from the mathematics; quadrature, basis resolution, and solver tolerances are numerical choices. First test on a small 2D Poisson PINN with a manufactured solution and 100 archived checkpoints. Compare training-loss selection, pointwise residual selection, and Riesz selection against the exact energy-error oracle. Measure selected energy error, rank correlation with the oracle, and post-processing time.

Verification

Mechanism confirmed, baseline not beaten

Stage 1 · Toy mechanism gate: Passed ✓

Stage 2 · Mechanism transferred to benchmark: Confirmed ✓

Stage 2 · Practical benchmark result: Significantly worse ✗

Methodology: 8 paired seeds, permutation test (p<0.05) against a learning-rate-tuned baseline. How verification works

Stage 1 — Mechanism check agent confidence 9/10

Built a readable conforming Poisson Energy-Riesz checkpoint selector in energy_riesz_experiment.py. The mechanism manifested strongly: full-space identity relative error was 1.49e-15, every auxiliary score was a valid lower bound, nested-space monotonicity had zero violations, and mean oracle gaps decreased from 0.8675 to 0.0741 as auxiliary size increased. However, this archive did not demonstrate a practical selection win over raw loss: raw loss selected error 4.3881, while Riesz selections ranged from 4.3881 to 4.5623.

Agent confidence: 9/10

Baseline: Raw residual-loss checkpoint: oracle energy error 4.3881; selected checkpoint 99.
Idea: Riesz selected oracle errors by auxiliary size 3,7,15,31,63: 4.5623, 4.3881, 4.3881, 4.5623, 4.5623. Spearman rank correlation improved 0.9904 -> 0.9994; mean relative gap to oracle improved 0.8675 -> 0.0741; full-space identity error 1.49e-15; 500 solves took 0.263 s.

Limitations: This is a 1D finite-dimensional manufactured Poisson test rather than a 2D neural PINN. The archive is synthetic and mostly monotonically improving, so it does not stress checkpoint mis-selection under noisy or non-monotone training. No GPU, wall-clock training comparison, quadrature error study, or truly independently assembled neural network checkpoints was tested.

How to run: python3 energy_riesz_experiment.py

Stage 2 — Benchmark vs. tuned baseline (arXiv:2608.16473) Sep 2, 2026

Significantly WORSE than the tuned baseline (+44.1%, p=0.0149, wins 0 / 8 paired seeds).

Benchmark: Poisson_boundary
Model: mlp_tiny
Paired seeds: 8
Baseline mean: 0.0439
Idea mean: 0.0632
Effect (Δ): 0.0193 (+44.1%; negative = idea better)
Wins: 0 / 8 paired seeds
p-value: 0.0149 (permutation test, 20 000 shuffles)
Smallest detectable effect: n/a
Mechanism: Confirmed ✓
Practical effect: Hurts
Baseline tuning: swept over 3 configs

Limitations:

Tested one built-in PDE track, one mlp_tiny architecture, 24 epochs, 400 training and 400 test samples, three learning rates, and three auxiliary levels. No larger models, longer schedules, PINN collocation loss, or quadrature-resolution study was run.

How to run: python3 bench_energy_riesz.py

Verdict computed by deterministic test code from paired-seed statistics — not by the language model.

Stage 2 — Benchmark Sep 1, 2026 latest Failed ✗

Agent confidence: 9/10

Implemented Energy-Riesz checkpoint selection on the structurally matched built-in poisson_boundary PDE track using shared mlp_tiny and Adam training. The trained-model signature confirmed monotone nested auxiliary scores, but held-out MSE was significantly worse: delta_mean=+0.019328 and permutation p=0.01485. The mechanism transferred mathematically, but not as a practical benchmark improvement.

Baseline: poisson_boundary/mlp_tiny, Adam lr=0.01, 24 epochs: mean test MSE 0.0438708, std 0.0028396
Idea: Best selector setting lr=0.01, auxiliary level=2: mean test MSE 0.0631988, std 0.0111461; paired delta +0.019328, p=0.01485; 0/8 wins
Mechanism signature: map[confirmed:true nested_scores:map[level_2:3.402205 level_3:3.536085 level_4:3.538054]]

Benchmark result