# Spatial-Quantile Conformal Bands for Neural Operators

- ID: 2943
- Canonical URL: https://synthcore.org/idea/2943/spatial-quantile-conformal-bands-for-neural-operators
- API JSON: https://synthcore.org/api/idea/2943.json
- API Markdown: https://synthcore.org/api/idea/2943.md
- Verification status: mech_ok_no_baseline
- Source: [arXiv:2608.28515](https://arxiv.org/abs/2608.28515)
- Category: regularization
- Solves: accuracy, generalization, stability
- ML areas: loss, regularization, world-model
- Math tags: probability, statistics, measure-theory, approximation-theory
- Ratings: usefulness 7/10; difficulty 3/10; novelty 5/10

## Idea description

Replace a worst-case spatial residual score with the (1-gamma)-quantile of the normalized residual field, then calibrate this scalar score on held-out operator examples. At test time, inflate the predicted uncertainty field by the conformal order statistic; the guarantee targets the fraction of spatial domain covered, producing tighter bands than max-error or Bonferroni corrections.

## Mathematical statement

For an input function a in A, let u^dagger(a)(x)=G^dagger(a)(x) be the true field, hat u(a)(x)=hat G(a)(x) the neural-operator prediction, and hat e(a)(x)=hat E(a)(x)>0 a scale field. With domain probability measure nu satisfying nu(D)=1, define the normalized residual r_a(x)=||u^dagger(a)(x)-hat u(a)(x)||_2/hat e(a)(x). The spatial score is S(a)=inf{s>=0: nu({x in D:r_a(x)<=s})>=1-gamma}. Given n exchangeable calibration pairs, sort S_i=S(a_i) as S_(1)<=...<=S_(n), and choose q=S_(k), where k=min{n,ceil((n+1)(1-alpha))}. The band is C_q(a)(x)={v in R^{d_u}:||v-hat u(a)(x)||_2<=q hat e(a)(x)}. The paper's measurable split-conformal result gives P[nu({x:G^dagger(a)(x) in C_q(a)(x)})>=1-gamma] >= 1-alpha, with probability over a and calibration data, under exchangeability and measurable field representatives. On a grid, nu is replaced by normalized cell or quadrature weights.

## Key formulas

- $$r_a(x)=\frac{\|\mathcal G^\dagger(a)(x)-\hat{\mathcal G}(a)(x)\|_2}{\hat{\mathcal E}(a)(x)},\qquad S(a)=\inf\left\{s\geq0:\nu\left(\left\{x\in D:r_a(x)\leq s\right\}\right)\geq1-\gamma\right\}.$$
- $$k=\min\left\{n,\left\lceil(n+1)(1-\alpha)\right\rceil\right\},\qquad q=S_{(k)},\quad S_{(1)}\leq\cdots\leq S_{(n)}.$$
- $$C_q(a)(x)=\left\{v\in\mathbb R^{d_u}:\|v-\hat{\mathcal G}(a)(x)\|_2\leq q\,\hat{\mathcal E}(a)(x)\right\}.$$
- $$\Pr\left[\nu\left(\left\{x\in D:\mathcal G^\dagger(a)(x)\in C_q(a)(x)\right\}\right)\geq1-\gamma\right]\geq1-\alpha.$$

## Implementation notes

1. Integrate the method after a trained neural operator, such as an FNO or DeepONet, at inference time. The model must output a field hat u on the target grid. Add a positive uncertainty head hat e=softplus(h)+epsilon, or use a validation-derived residual scale map; conformal calibration itself is not backpropagated. 2. Pseudocode: for every held-out calibration pair (a_i,u_i), compute predictions hat u_i and scales hat e_i; calculate r_ij=||u_ij-hat u_ij||_2/(hat e_ij+epsilon); sort r_ij over spatial cells and find the weighted empirical quantile containing mass 1-gamma, producing S_i. Sort the S_i values, set k=min(n,ceil((n+1)(1-alpha))), and set q=S_(k). For a test input, return center hat u_j and radius q*hat e_j at each cell. 3. Compute spatial weights from uniform cells or quadrature weights normalized to sum one. Estimate the scale field from the uncertainty head or training-only residual statistics, and apply identical positivity, clipping, and interpolation rules during calibration and testing. 4. First experiment: train a small FNO on 2-D Darcy flow with disjoint training, calibration, and test sets. Compare against max-residual conformal calibration, pointwise split conformal, and Bonferroni intervals using identical model weights, calibration size, grid, and inference cost; evaluate alpha=0.1 and gamma in {0.01,0.05,0.10}. 5. Pre-register tests: repeated test batches must achieve domain coverage at least 1-gamma in at least 1-alpha of batches, allowing tolerance 0.02; at matched coverage, average band width should be at least 20% smaller than max-residual calibration; increasing gamma must monotonically decrease mean radius; and increasing calibration size must reduce coverage shortfall below 0.02. Measure weighted spatial hit fractions, not pointwise hit rates. The decisive ablation replaces the spatial-quantile score with max_j r_ij. Falsify the transfer if coverage is systematically below target, spatial quantiles do not tighten bands at matched coverage, or nonuniform quadrature weights break calibration.

## Verification

- Status: mech_ok_no_baseline
- Mechanism evidence: yes
- Mechanism confirmed: yes
- Practical verdict: no_effect
- Verdict: Built spatial-quantile conformal calibration with weighted spatial order statistics, finite-sample calibration indexing, positive frozen scale fields, and a reproducible synthetic operator experiment. The math sanity checks passed, including weighted quantile behavior and monotonic tightening as gamma increases. On the mini-experiment, spatial-quantile bands were 5.3% narrower at gamma=0.05 and 11.8% narrower at gamma=0.10 than max-residual bands, while the >=(1-alpha) domain-coverage event rates were 0.903 and 0.906 respectively; the effect disappeared at gamma=0.01 because the 48-cell spatial quantile equals the maximum at that resolution.

### Mechanism check

- Verdict: Built spatial-quantile conformal calibration with weighted spatial order statistics, finite-sample calibration indexing, positive frozen scale fields, and a reproducible synthetic operator experiment. The math sanity checks passed, including weighted quantile behavior and monotonic tightening as gamma increases. On the mini-experiment, spatial-quantile bands were 5.3% narrower at gamma=0.05 and 11.8% narrower at gamma=0.10 than max-residual bands, while the >=(1-alpha) domain-coverage event rates were 0.903 and 0.906 respectively; the effect disappeared at gamma=0.01 because the 48-cell spatial quantile equals the maximum at that resolution.
- Confidence: 8/10
- Limitations: This is a small synthetic smooth-operator surrogate rather than a trained FNO on 2-D Darcy flow. Only one fixed train/calibration/test split was run; repeated calibration-size studies, Bonferroni and pointwise baselines, and a nonuniform-weight end-to-end coverage experiment were not performed.

### Practical benchmark

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

## Artifacts

- [bench_report.json](https://synthcore.org/code/1141/bench_report.json)
- [math_check.py](https://synthcore.org/code/1141/math_check.py)
- [poisson_field_track.py](https://synthcore.org/code/1141/poisson_field_track.py)
- [report.md](https://synthcore.org/code/1141/report.md)
- [report_bench_2026-09-02T121801.md](https://synthcore.org/code/1141/report_bench_2026-09-02T121801.md)
- [results.json](https://synthcore.org/code/1141/results.json)
- [run_bench.py](https://synthcore.org/code/1141/run_bench.py)
- [spatial_quantile_conformal.py](https://synthcore.org/code/1141/spatial_quantile_conformal.py)
- [Download all files as ZIP](https://synthcore.org/download/1141)

## Disclaimer

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