Conformal Uncertainty Quantification Guarantees for Neural Operators

Forum topic · 小凯 · 2026-09-01

Summary

Neural operators provide fast surrogate models for approximating operators between function spaces, but their predictions often lack uncertainty quantification. This paper by Tom Stent and Nicolas Boullé (arXiv:2608.28515) develops a split conformal framework that guarantees 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 ∈ (0,1). The method reduces a normalized residual field to its spatial (1-γ)-quantile and computes a scaling factor using a held-out calibration dataset. The authors prove marginal coverage guarantees for measurable residual fields on arbitrary probability spaces, covering both continuum domains and fixed discretizations, and show under mild assumptions that coverage for a given calibration set follows a Beta distribution. Numerical experiments on Darcy flow and Navier-Stokes equations demonstrate that the calibrated bands are consistently tighter than existing correction methods while maintaining the target coverage.

Paper Overview

Field: Machine Learning
Authors: Tom Stent, Nicolas Boullé
Published: 2026-08-28
arXiv: 2608.28515

Summary

Neural operators provide fast surrogate models for approximating operators between function spaces, but their predictions often lack uncertainty quantification. This work develops a split conformal framework to guarantee that a calibrated pointwise band around the neural operator output contains the true solution on at least a $1-\gamma$ fraction of the evaluation domain, with probability at least $1-\alpha$ over test and calibration inputs, where $\alpha,\gamma\in(0,1)$ .

Method

Reduce a normalized residual field to its spatial $(1-\gamma)$ -quantile.
Compute a scaling factor using a held-out calibration dataset.

Theoretical Guarantees

Marginal coverage guarantees are proven for measurable residual fields defined on arbitrary probability spaces, covering both continuum domains and fixed discretizations.
Under mild assumptions on the data distribution, the coverage for a given calibration set follows a Beta distribution.

Experiments

Numerical experiments on the Darcy flow and Navier-Stokes equations validate the theory: the proposed calibration produces bands that are consistently tighter than existing corrections while maintaining the target coverage.

Original Abstract (excerpt)

> 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 $1-\gamma$ fraction of the evaluation domain, with probability at least $1-\alpha$ over test and calibration inputs, where $\alpha,\gamma\in(0,1)$ . Our method reduces a normalized residual field to its spatial $(1-\gamma)$ -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 ...

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Conformal Uncertainty Quantification Guarantees for Neural Operators

Summary

Paper Overview

Summary

Method

Theoretical Guarantees

Experiments

Original Abstract (excerpt)

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