# Probabilistic neural operators for functional uncertainty quantification

arXiv: https://arxiv.org/abs/2502.12902 · HTML: https://mlsanity.com/paper/2502.12902/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 46 references that are in the map.
- First posted on arXiv 18 Feb 2025.

## Builds on (most cited first, 29 of 29)

- [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 Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Variational Dropout and the Local Reparameterization Trick](https://mlsanity.com/paper/1506.02557/) (2015)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Laplace Redux -- Effortless Bayesian Deep Learning](https://mlsanity.com/paper/2106.14806/) (2021)
- [WeatherBench 2: A benchmark for the next generation of data-driven global weather models](https://mlsanity.com/paper/2308.15560/) (2023)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [Practical Gauss-Newton Optimisation for Deep Learning](https://mlsanity.com/paper/1706.03662/) (2017)
- [Neural networks for post-processing ensemble weather forecasts](https://mlsanity.com/paper/1805.09091/) (2018)
- [Do Bayesian Neural Networks Need To Be Fully Stochastic?](https://mlsanity.com/paper/2211.06291/) (2022)
- [Real-time high-resolution CO2 geological storage prediction using nested Fourier neural operators](https://mlsanity.com/paper/2210.17051/) (2022)
- [Probabilistic Forecasting with Generative Networks via Scoring Rule Minimization](https://mlsanity.com/paper/2112.08217/) (2021)
- [Plasma Surrogate Modelling using Fourier Neural Operators](https://mlsanity.com/paper/2311.05967/) (2023)
- [Generative machine learning methods for multivariate ensemble post-processing](https://mlsanity.com/paper/2211.01345/) (2022)
- [Calibrated Uncertainty Quantification for Operator Learning via Conformal Prediction](https://mlsanity.com/paper/2402.01960/) (2024)
- [Uncertainty quantification for data-driven weather models](https://mlsanity.com/paper/2403.13458/) (2024)
- [Variational Bayesian dropout: pitfalls and fixes](https://mlsanity.com/paper/1807.01969/) (2018)
- [Solving Seismic Wave Equations on Variable Velocity Models with Fourier Neural Operator](https://mlsanity.com/paper/2209.12340/) (2022)
- [IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learning](https://mlsanity.com/paper/2302.03271/) (2023)
- [Strictly proper kernel scores and characteristic kernels on compact spaces](https://mlsanity.com/paper/1712.05279/) (2017)
- [Linearization Turns Neural Operators into Function-Valued Gaussian Processes](https://mlsanity.com/paper/2406.05072/) (2024)
- [Easy Uncertainty Quantification (EasyUQ): Generating Predictive Distributions from Single-valued Model Output](https://mlsanity.com/paper/2212.08376/) (2022)
- [Forecasting subcritical cylinder wakes with Fourier Neural Operators](https://mlsanity.com/paper/2301.08290/) (2023)
- [Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures](https://mlsanity.com/paper/2206.07588/) (2022)

## Built on by (most cited first, 18 of 18)

- [Probabilistic Retrofitting of Learned Simulators](https://mlsanity.com/paper/2603.01949/) (2026)
- [Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators](https://mlsanity.com/paper/2508.00643/) (2025)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [Method of Manufactured Learning for Solver-free Training of Neural Operators](https://mlsanity.com/paper/2511.12890/) (2025)
- [Locally Adaptive Conformal Inference for Operator Models](https://mlsanity.com/paper/2507.20975/) (2025)
- [Improved probabilistic regression using diffusion models](https://mlsanity.com/paper/2510.04583/) (2025)
- [Uncertainty Quantification for Regression: A Unified Framework based on kernel scores](https://mlsanity.com/paper/2510.25599/) (2025)
- [Nonparametric estimation of conditional probability distributions using a generative approach based on conditional push-forward neural networks](https://mlsanity.com/paper/2511.14455/) (2025)
- [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)
- [Geometry-Aware Post-Hoc Uncertainty Quantification in Operator Learning](https://mlsanity.com/paper/2606.17513/) (2026)
- [Neural Operator Processes for Probabilistic Operator Learning under Partial Observations](https://mlsanity.com/paper/2606.22946/) (2026)
- [On the sensitivity of machine-learned probabilistic weather forecast models to scale-aware scoring rules](https://mlsanity.com/paper/2607.19161/) (2026)
- [GeoQ: Geometry-Aware Conditional Quantile Error Estimation for Scientific Surrogate Models](https://mlsanity.com/paper/2608.21652/) (2026)
- [When Does AI for PDEs Yield Scientific Evidence?](https://mlsanity.com/paper/2608.22504/) (2026)
- [Generative Neural Operators through Diffusion Last Layer](https://mlsanity.com/paper/2602.04139/) (2026)
- [Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorization](https://mlsanity.com/paper/2605.15806/) (2026)
- [Conditional Distribution Estimation for Functional Responses with Random Forests](https://mlsanity.com/paper/2608.08247/) (2026)
- [Conformal Uncertainty Quantification Guarantees for Neural Operators](https://mlsanity.com/paper/2608.28515/) (2026)

## Abstract (arXiv)

Neural operators aim to approximate the solution operator of a system of differential equations purely from data. They have shown immense success in modeling complex dynamical systems across various domains. However, the occurrence of uncertainties inherent in both model and data has so far rarely been taken into accounttextemdasha critical limitation in complex, chaotic systems such as weather forecasting. In this paper, we introduce the probabilistic neural operator (PNO), a framework for learning probability distributions over the output function space of neural operators. PNO extends neural operators with generative modeling based on strictly proper scoring rules, integrating uncertainty information directly into the training process. We provide a theoretical justification for the approach and demonstrate improved performance in quantifying uncertainty across different domains and with respect to different baselines. Furthermore, PNO requires minimal adjustment to existing architectures, shows improved performance for most probabilistic prediction tasks, and leads to well-calibrated predictive distributions and adequate uncertainty representations even for long dynamical trajectories. Implementing our approach into large-scale models for physical applications can lead to improvements in corresponding uncertainty quantification and extreme event identification, ultimately leading to a deeper understanding of the prediction of such surrogate models.