# Resolution-aware operator data budget

- ID: 469
- Canonical URL: https://synthcore.org/idea/469/resolution-aware-operator-data-budget
- API JSON: https://synthcore.org/api/idea/469.json
- API Markdown: https://synthcore.org/api/idea/469.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2607.06287](https://arxiv.org/abs/2607.06287)
- Category: training
- Solves: scalability, sample-efficiency
- ML areas: neural-operator, training, sample-efficiency, scalability
- Math tags: approximation-theory, metric-geometry, numerical-analysis, statistics
- Ratings: usefulness 6/10; difficulty 4/10; novelty 7/10

## Idea description

Couple the number of operator training pairs to the output resolution instead of increasing the output grid independently. Refine the output discretization only while the oracle reconstruction improves, and increase the training set when the learned predictor remains substantially worse than the oracle decoder.

## Mathematical statement

For a finite set \(\Xi=\{\boldsymbol\xi_i\}_{i=1}^M\) in a domain \(D\), the paper defines the fill distance \(h_{\Xi,D}=\sup_{\boldsymbol\xi\in D}\min_i\|\boldsymbol\xi-\boldsymbol\xi_i\|_V\). For a compact input class \(\mathcal M\), the discretized training inputs \(\boldsymbol u_i\) should cover \(S_\Phi(\mathcal M)\), with \(h_{\{\boldsymbol u_i\},S_\Phi(\mathcal M)}\le h_0\). The information loss from input observations is \(\iota_\Phi(u)=\inf_{w\in\ker S_\Phi}\|u-w\|_{\mathcal U}\). The surrogate is \(\mathcal A=A_{\mathrm{on}}\circ A_{\mathrm{off}}\circ S_X\), with exact output observations \(\boldsymbol v^\dagger=S_Y(\mathcal G(u))\) and predicted observations \(\widehat{\boldsymbol v}=A_{\mathrm{off}}(S_X(u))\). Its total error separates reconstruction error of \(A_{\mathrm{on}}\) from learning error of \(A_{\mathrm{off}}\). The reported experiment used \(N=m^\kappa\): \(\kappa=0.5\) stagnated, while \(\kappa\ge1\) reached the oracle rate. The engineering rule is therefore to test \(N(m)=\lceil C_Nm^\kappa\rceil\), initially with \(\kappa\in\{0.5,1,1.5\}\), and stop increasing \(m\) when the learned-oracle gap dominates.

## Key formulas

- $$h_{\Xi,D}:=\sup_{\boldsymbol{\xi}\in D}\min_{1\leq i\leq M}\|\boldsymbol{\xi}-\boldsymbol{\xi}_{i}\|_{V}.$$
- $$h_{\{\boldsymbol{u}_{i}\},S_{\Phi}(\mathcal{M})}\leq h_{0},\qquad \iota_{\Phi}(u):=\inf_{w\in\ker S_{\Phi}}\|u-w\|_{\mathcal{U}}.$$
- $$\mathcal{A}=A_{\mathrm{on}}\circ A_{\mathrm{off}}\circ S_X,\qquad \boldsymbol v^\dagger=S_Y(\mathcal G(u)),\qquad \widehat{\boldsymbol v}=A_{\mathrm{off}}(S_X(u)).$$
- $$N(m)=\left\lceil C_Nm^\kappa\right\rceil,\qquad \kappa\in\{0.5,1.0,1.5\}.$$

## Implementation notes

Apply this to an FNO, DeepONet, or operator transformer whose output has an explicit number \(m\) of spatial points or output tokens. For each candidate resolution \(m\), train an oracle reconstruction model using exact output observations and record \(E_{\mathrm{oracle}}(m)\). Then train the actual neural operator using \(N=\lceil C_Nm^\kappa\rceil\) input/output pairs, testing \(\kappa=0.5,1.0,1.5\). Estimate input coverage on a held-out validation set \(V\) through the sampled-feature proxy \(\widehat h=\max_{u\in V}\min_i\|S_X(u)-S_X(u_i)\|_2\). At each resolution, measure learned error \(E_{\mathrm{learned}}(m,N)\), oracle error, total training FLOPs, and wall-clock cost. If increasing \(m\) lowers oracle error but leaves learned error unchanged, increase \(N\); if both errors stagnate, stop refining the grid. If the learned/oracle gap is small, additional training pairs are unnecessary. The first cheap experiment is 2D Poisson or Darcy flow at grids 32, 64, and 128, using the same FNO architecture and optimizer for all conditions. Compare against a fixed-\(N\) baseline and plot error versus training FLOPs. Success is lower error at equal data or compute, with the learned curve approaching the oracle curve near linear scaling \(N\propto m\).

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: no_effect
- Verdict: Built a compact kernel-based two-stage operator-learning MVP with exact-output cubic reconstruction, learned offline kernel regression, fill-distance verification, a fixed-N baseline, and N=ceil(m^kappa) schedules. The math check reproduced uniform fill distance h=1/(2(m-1)) to numerical precision with log-log slope -1.09. In the toy operator, fixed N=16 stagnated at errors 0.720, 0.727, 0.728 for m=16,32,64, while kappa=1.0 improved to 0.720, 0.632, 0.470 and kappa=1.5 reached 0.471, 0.310, 0.311; the oracle error fell from 0.204 to 0.0142 to 0.000385. This is a clear qualitative signal supporting resolution-aware data growth, but not evidence of superiority at equal compute because larger N costs more.

### Mechanism check

- Verdict: Built a compact kernel-based two-stage operator-learning MVP with exact-output cubic reconstruction, learned offline kernel regression, fill-distance verification, a fixed-N baseline, and N=ceil(m^kappa) schedules. The math check reproduced uniform fill distance h=1/(2(m-1)) to numerical precision with log-log slope -1.09. In the toy operator, fixed N=16 stagnated at errors 0.720, 0.727, 0.728 for m=16,32,64, while kappa=1.0 improved to 0.720, 0.632, 0.470 and kappa=1.5 reached 0.471, 0.310, 0.311; the oracle error fell from 0.204 to 0.0142 to 0.000385. This is a clear qualitative signal supporting resolution-aware data growth, but not evidence of superiority at equal compute because larger N costs more.
- Confidence: 8/10
- Limitations: This is a small synthetic smooth operator and kernel regressor rather than an FNO, DeepONet, or PDE benchmark. The kappa=1.5 schedule is capped at N=128 at m=64, only one random train/test split was used, wall-clock measurements are noisy, and no equal-FLOP comparison or GPU run was performed.

### Practical benchmark

- Paired seeds: 8
- Baseline mean: 0
- Idea mean: 0
- p-value: 1
- Paired wins: 0/0
- Benchmark verdict: no measurable effect

## Artifacts

- [bench_report.json](https://synthcore.org/code/1314/bench_report.json)
- [registered_resolution_bench.py](https://synthcore.org/code/1314/registered_resolution_bench.py)
- [report.md](https://synthcore.org/code/1314/report.md)
- [report_bench_2026-09-04T150248.md](https://synthcore.org/code/1314/report_bench_2026-09-04T150248.md)
- [resolution_aware_bench.py](https://synthcore.org/code/1314/resolution_aware_bench.py)
- [resolution_budget_experiment.py](https://synthcore.org/code/1314/resolution_budget_experiment.py)
- [results.json](https://synthcore.org/code/1314/results.json)
- [stage2_bench.py](https://synthcore.org/code/1314/stage2_bench.py)
- [Download all files as ZIP](https://synthcore.org/download/1314)

## Disclaimer

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