# Energy-Riesz checkpoint selector

- ID: 2403
- Canonical URL: https://synthcore.org/idea/2403/energy-riesz-checkpoint-selector
- API JSON: https://synthcore.org/api/idea/2403.json
- API Markdown: https://synthcore.org/api/idea/2403.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2608.16473](https://arxiv.org/abs/2608.16473)
- Category: training
- Solves: accuracy, stability, generalization
- ML areas: training, loss
- Math tags: functional-analysis, geometry, numerical-analysis
- Ratings: usefulness 7/10; difficulty 4/10; novelty 7/10

## 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.

## 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.

## Key formulas

- $$r_k(v)=\ell(v)-a(u_k,v)=a(u^*-u_k,v),\qquad a(z_k,v)=r_k(v)\quad\forall v\in V_0,$$
- $$\|z_k\|_a^2=a(z_k,z_k)=r_k(z_k)=a(u^*-u_k,u^*-u_k)=\|u_k-u^*\|_a^2,$$
- $$M_m(k):=\|z_{k,m}\|_a=\sqrt{a(z_{k,m},z_{k,m})}\leq\|u_k-u^*\|_a,$$
- $$M_m(k)\leq M_{m+1}(k)\longrightarrow\|u_k-u^*\|_a.$$

## 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

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: harms
- Verdict: 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.

### Mechanism check

- Verdict: 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.
- Confidence: 9/10
- 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.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 0.01485
- Paired wins: 0/8
- Benchmark verdict: idea worse (significant)

## Artifacts

- [bench_energy_riesz.py](https://synthcore.org/code/941/bench_energy_riesz.py)
- [bench_report.json](https://synthcore.org/code/941/bench_report.json)
- [energy_riesz_experiment.py](https://synthcore.org/code/941/energy_riesz_experiment.py)
- [poisson_track.py](https://synthcore.org/code/941/poisson_track.py)
- [report.md](https://synthcore.org/code/941/report.md)
- [report_bench_2026-09-01T161851.md](https://synthcore.org/code/941/report_bench_2026-09-01T161851.md)
- [results.json](https://synthcore.org/code/941/results.json)
- [Download all files as ZIP](https://synthcore.org/download/941)

## Disclaimer

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 Not peer-reviewed.