# Calibrated Uncertainty Quantification for Operator Learning via Conformal Prediction

arXiv: https://arxiv.org/abs/2402.01960 · HTML: https://mlsanity.com/paper/2402.01960/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 36 references that are in the map.
- First posted on arXiv 2 Feb 2024.

## Builds on (most cited first, 23 of 23)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Conformalized Quantile Regression](https://mlsanity.com/paper/1905.03222/) (2019)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Distribution-Free, Risk-Controlling Prediction Sets](https://mlsanity.com/paper/2101.02703/) (2021)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [Image-to-Image Regression with Distribution-Free Uncertainty Quantification and Applications in Imaging](https://mlsanity.com/paper/2202.05265/) (2022)
- [A Conformal Prediction Approach to Explore Functional Data](https://mlsanity.com/paper/1302.6452/) (2013)
- [Beyond Pinball Loss: Quantile Methods for Calibrated Uncertainty Quantification](https://mlsanity.com/paper/2011.09588/) (2020)
- [Real-time high-resolution CO2 geological storage prediction using nested Fourier neural operators](https://mlsanity.com/paper/2210.17051/) (2022)
- [Learning Functional Priors and Posteriors from Data and Physics](https://mlsanity.com/paper/2106.05863/) (2021)
- [Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation](https://mlsanity.com/paper/2301.11509/) (2023)
- [Conformal Prediction Bands for Multivariate Functional Data](https://mlsanity.com/paper/2106.01792/) (2021)
- [Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators](https://mlsanity.com/paper/2311.11262/) (2023)
- [Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs](https://mlsanity.com/paper/2208.01565/) (2022)
- [Computationally efficient versions of conformal predictive distributions](https://mlsanity.com/paper/1911.00941/) (2019)
- [Uncertainty Quantification via Neural Posterior Principal Components](https://mlsanity.com/paper/2309.15533/) (2023)
- [Conformal Prediction Bands for Two-Dimensional Functional Time Series](https://mlsanity.com/paper/2207.13656/) (2022)

## Built on by (most cited first, 26 of 26)

- [A Library for Learning Neural Operators](https://mlsanity.com/paper/2412.10354/) (2024)
- [Probabilistic neural operators for functional uncertainty quantification](https://mlsanity.com/paper/2502.12902/) (2025)
- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEs](https://mlsanity.com/paper/2403.10642/) (2024)
- [MODNO: Multi Operator Learning With Distributed Neural Operators](https://mlsanity.com/paper/2404.02892/) (2024)
- [Uncertainty Quantification of Surrogate Models using Conformal Prediction](https://mlsanity.com/paper/2408.09881/) (2024)
- [Calibrated Physics-Informed Uncertainty Quantification](https://mlsanity.com/paper/2502.04406/) (2025)
- [Guaranteed prediction sets for functional surrogate models](https://mlsanity.com/paper/2501.18426/) (2025)
- [Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators](https://mlsanity.com/paper/2508.00643/) (2025)
- [Split Conformal Prediction in the Function Space with Neural Operators](https://mlsanity.com/paper/2509.04623/) (2025)
- [Valid Error Bars for Neural Weather Models using Conformal Prediction](https://mlsanity.com/paper/2406.14483/) (2024)
- [Locally Adaptive Conformal Inference for Operator Models](https://mlsanity.com/paper/2507.20975/) (2025)
- [Flow-Based Conformal Predictive Distributions](https://mlsanity.com/paper/2602.07633/) (2026)
- [Structure-Aware Epistemic Uncertainty Quantification for Neural Operator PDE Surrogates](https://mlsanity.com/paper/2603.11052/) (2026)
- [Distribution free uncertainty quantification in neuroscience-inspired deep operators](https://mlsanity.com/paper/2412.09369/) (2024)
- [Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects](https://mlsanity.com/paper/2605.16966/) (2026)
- [Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime](https://mlsanity.com/paper/2606.08654/) (2026)
- [GeoQ: Geometry-Aware Conditional Quantile Error Estimation for Scientific Surrogate Models](https://mlsanity.com/paper/2608.21652/) (2026)
- [Generative Neural Operators through Diffusion Last Layer](https://mlsanity.com/paper/2602.04139/) (2026)
- [Distribution-Free Robust Predict-Then-Optimize in Function Spaces](https://mlsanity.com/paper/2602.08215/) (2026)
- [Direct Learning of Calibration-Aware Uncertainty for Neural PDE Surrogates](https://mlsanity.com/paper/2602.11090/) (2026)
- [Reliability of Probabilistic Emulation of Physical Systems](https://mlsanity.com/paper/2606.12997/) (2026)
- [FAST-DeepONet: Factor-Augmented Branch Representations for High-Dimensional PDE Inputs in the Small-Sample Regime](https://mlsanity.com/paper/2608.15408/) (2026)
- [Conformal Uncertainty Quantification Guarantees for Neural Operators](https://mlsanity.com/paper/2608.28515/) (2026)
- [Adapting neural operators for mechanics decisions under changing operating conditions](https://mlsanity.com/paper/2609.33978/) (2026)
- [Threshold-Aware Conformal Routing](https://mlsanity.com/paper/2610.02487/) (2026)

## Abstract (arXiv)

Operator learning has been increasingly adopted in scientific and engineering applications, many of which require calibrated uncertainty quantification. Since the output of operator learning is a continuous function, quantifying uncertainty simultaneously at all points in the domain is challenging. Current methods consider calibration at a single point or over one scalar function or make strong assumptions such as Gaussianity. We propose a risk-controlling quantile neural operator, a distribution-free, finite-sample functional calibration conformal prediction method. We provide a theoretical calibration guarantee on the coverage rate, defined as the expected percentage of points on the function domain whose true value lies within the predicted uncertainty ball. Empirical results on a 2D Darcy flow and a 3D car surface pressure prediction task validate our theoretical results, demonstrating calibrated coverage and efficient uncertainty bands outperforming baseline methods. In particular, on the 3D problem, our method is the only one that meets the target calibration percentage (percentage of test samples for which the uncertainty estimates are calibrated) of 98%.