Conformal Uncertainty Quantification Guarantees for Neural Operators
2026-08-28 • Artificial Intelligence
Artificial Intelligence
AI summaryⓘ
The authors developed a method to add reliable uncertainty measures to neural operators, which are tools that quickly predict solutions for complex mathematical problems. Their approach uses a statistical technique called split conformal prediction to create bands around the neural operator outputs that contain the true solution most of the time. They provide theoretical guarantees and validate their method with experiments on physics problems, showing it produces tighter and trustworthy uncertainty estimates compared to previous methods. This helps users know how confident they can be in the neural operator's predictions.
Neural operatorsUncertainty quantificationSplit conformal predictionConfidence bandsResidual fieldDarcy flowNavier–Stokes equationsCalibration datasetMarginal coverage
Authors
Tom Stent, Nicolas Boullé
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.