Conformal Uncertainty Quantification Guarantees for Neural Operators

Tom Stent; Nicolas Boullé

math.NAAug 28, 2026

Conformal Uncertainty Quantification Guarantees for Neural Operators

Authors: Tom Stent, Nicolas Boullé

Organizations: Department of Mathematics Imperial College London London, SW7 2AZ, UK

Abstract

Neural operators provide fast surrogate models for approximating operators between function spaces, but their predictions often lack uncertainty quantification. We develop a split conformal framework to guarantee that a calibrated pointwise band around the neural operator output contains the true solution on at least a $1-γ$ fraction of the evaluation domain, with probability at least $1-α$ over test and calibration inputs, where $α,γ\in(0,1)$ . Our method reduces a normalized residual field to its spatial $(1-γ)$ -quantile and computes a scaling factor using a held-out calibration dataset. We prove marginal coverage guarantees for measurable residual fields defined on arbitrary probability spaces, covering both continuum domains and fixed discretizations. Under mild assumptions on the data distribution, we show that the coverage conditional on the calibration set follows a Beta distribution, which we verify with numerical experiments on Darcy flow and Navier--Stokes equations, where our calibration yields bands consistently tighter than existing corrections while retaining the target coverage.

Figures & tables

Figure 1: 2D Darcy flow dataset. Left: median calibrated scaling factor

\hat{\lambda}

with respect to the spatial resolution

N

over

3000

random disjoint re-splits (

n_{\mathrm{cal}}=500

n_{\mathrm{test}}=1000

) along with empirical

5

th–

95

th percentiles; smaller

\hat{\lambda}

indicates tighter bands. Right: empirical distribution of resampled coverage at resolution

N=60

compared with the exact Beta-Binomial law predicted by Theorem 2.5 at the target level

1-\alpha=0.9

. Our distribution aligns with the predicted law, while ( Ma et al., 2024 ) is shifted right with more conservative and wider bands.

Figure 2: Navier–Stokes at $N=64$ : input vorticity, solution at $t=21$ , FNO prediction, pointwise error, and estimated error field for a calibrated operator pair. The estimated error tracks the spatial error pattern, enabling tight pointwise bands while maintaining the target containment fraction $1-\gamma$ .

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

	Darcy flow	Navier–Stokes
Resolutions $N$	$12,20,30,42,60,84$	$16,32,64$
Fourier modes	$10$ at $N{=}12$ ; $16$ otherwise	$16,24,32$
Hidden channels	$32$	$32$
Fourier layers	$4$	$4$
Lifting / projection	$64$ / $64$	$64$ / $64$
Optimiser	AdamW	AdamW

Appendix

Table 1: Configuration for both problems. Fourier modes are given per axis and are capped by the grid size at each resolution.

	Ours		Ma et al.
$N$	$\hat{\lambda}$	$\overline{B}$	$\hat{\lambda}$	$\overline{B}$	ratio
12	1.84	$9.43\times 10^{-4}$	4.38	$2.24\times 10^{-3}$	2.37
20	1.59	$6.90\times 10^{-4}$	2.24	$9.71\times 10^{-4}$	1.41
30	1.48	$5.52\times 10^{-4}$	1.92	$7.13\times 10^{-4}$	1.29
42	1.64	$4.62\times 10^{-4}$	2.05	$5.77\times 10^{-4}$	1.25
60	1.42	$5.41\times 10^{-4}$	1.66	$6.34\times 10^{-4}$	1.17

Appendix

Table 2: Darcy flow: calibrated factor

\hat{\lambda}

(single calibration split), mean bandwidth

\overline{B}

, and width ratio between methods at each resolution. Mean bandwidth is

\overline{B}:=\frac{1}{n_{\mathrm{test}}}\sum_{i=1}^{n_{\mathrm{test}}}\frac{1}{M}\sum_{j=1}^{M}\hat{\lambda}\,\hat{\mathcal{E}}_{M}(A_{i})(x_{j})

Figure 3: Darcy flow: distribution of re-sampled coverage by resolution and method, with means and target

1-\alpha

. Our distributions are centred near the target at each

N

, consistent with Theorem 2.5 ; those of Ma et al. (2024) are shifted upward and collapse to coverage

1

at the coarsest resolution.

Figure 4: Darcy flow: coverage on the fine reference grid (

N_{\max}=420

) versus calibration resolution. Left: exact spectral interpolation of calibrated coarse-grid bands. Right: zero-shot super-resolution. This evaluates cross-grid transfer, which is outside the scope of Theorem 2.5 (a fixed-grid guarantee). The correction in Ma et al. (2024) increases width (up to

2.4\times

) yet reaches the target only at

N=84

, and zero-shot coverage is

0

at all resolutions.

	Ours		Ma et al.
$N$	$\hat{\lambda}$	$\overline{B}$	$\hat{\lambda}$	$\overline{B}$	ratio
16	1.92	0.419	3.42	0.747	1.78
32	1.76	0.354	2.34	0.472	1.33
64	1.72	0.354	2.11	0.436	1.23

Appendix

Table 3: Navier-Stokes: calibrated factor

\hat{\lambda}

, mean bandwidth

\overline{B}

, and the width ratio between the methods, per resolution.

Figure 5: Navier–Stokes. Left: resampled median calibrated scaling factor

\hat{\lambda}

versus resolution

N

, with empirical

5

th–

95

th percentile bars. Right: resampled coverage at

N=64

against the Beta-Binomial reference implied by Theorem 2.5 and the target

1-\alpha

Figure 6: Navier–Stokes: distribution of re-sampled coverage by resolution and method, with means and target

1-\alpha

. Our distributions centre near the target, while those of Ma et al. (2024) are shifted upward with smaller spread, indicating conservative validity.

Fair Conformal Prediction

See all 9 papers

Conformal Prediction for Neural Operators: Distribution-Free Uncertainty Quantification in Physics Simulation
Jun 7, 2026Michael ChinUncertainty Quantification PDE Surrogate Modeling
Independent Researcher
Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime
Jun 7, 2026Weinan Wang, Bowen Gang, Hao DengUncertainty Quantification Conformal Prediction
Department of Mathematics, University of Oklahoma, Norman, OK, USA · Department of Statistics and Data Science, School of Management, Fudan University, Shanghai, China
Physics-Informed Conformal Prediction: Embedding PDE Consistency into Distribution-Free Uncertainty Quantification for Neural Operators
Sep 14, 2026Michael ChinFourier Neural Operator Uncertainty Quantification
Independent Researcher

Jun 7, 2026J/K move · Enter open · S save

Conformal Prediction for Neural Operators: Distribution-Free Uncertainty Quantification in Physics Simulation

Michael Chin

Independent Researcher

Uncertainty Quantification PDE Surrogate Modeling Conformal Prediction Neural Operators Regression Prediction Intervals

Open paper AskPDF arXiv

Neural operators such as the Fourier Neural Operator (FNO) have emerged as powerful surrogates for solving partial differential equations (PDEs), achieving speedups of several orders of magnitude over traditional numerical solvers. However, deploying these models in safety-critical engineering applications -- such as thermal management of electronic components and battery systems -- requires not only accurate point predictions but also rigorous uncertainty guarantees. Existing uncertainty quantification (UQ) methods for neural operators, including Monte Carlo Dropout and Deep Ensembles, provide only relative uncertainty estimates without formal coverage guarantees. In this work, we propose the first application of split conformal prediction to neural operator-based physics simulation, providing distribution-free prediction intervals with finite-sample coverage guarantees. We further introduce a normalized conformal prediction scheme that leverages MC Dropout uncertainty to produce adaptive-width intervals, yielding tighter intervals in regions of low uncertainty and wider intervals where the model is less certain. Full-scale experiments (33.7M parameters, 800 training samples, 5 ensemble members, NVIDIA V100) on steady-state heat conduction benchmarks demonstrate that our method achieves 89.1% empirical coverage at the target level of alpha=0.1, while producing spatially adaptive prediction intervals that reflect the underlying physical uncertainty structure. We also provide an uncertainty decomposition framework that separates epistemic uncertainty (68% of total) from aleatoric uncertainty (32% of total), offering actionable guidance for data collection and model improvement. Our method is implemented in an open-source platform with REST API endpoints and interactive 3D visualization.

Abstract

Fair Conformal Prediction

Explore similar work

Conformal Prediction for Neural Operators: Distribution-Free Uncertainty Quantification in Physics Simulation

Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime

Physics-Informed Conformal Prediction: Embedding PDE Consistency into Distribution-Free Uncertainty Quantification for Neural Operators