# Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priors

arXiv: https://arxiv.org/abs/2203.03048 · HTML: https://mlsanity.com/paper/2203.03048/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 69 references that are in the map.
- First posted on arXiv 6 Mar 2022.

## Builds on (most cited first, 30 of 32)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [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)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Averaging Weights Leads to Wider Optima and Better Generalization](https://mlsanity.com/paper/1803.05407/) (2018)
- [Automatic chemical design using a data-driven continuous representation of molecules](https://mlsanity.com/paper/1610.02415/) (2016)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Deep Exploration via Bootstrapped DQN](https://mlsanity.com/paper/1602.04621/) (2016)
- [Deep Ensembles: A Loss Landscape Perspective](https://mlsanity.com/paper/1912.02757/) (2019)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Scalable Bayesian Optimization Using Deep Neural Networks](https://mlsanity.com/paper/1502.05700/) (2015)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Randomized Prior Functions for Deep Reinforcement Learning](https://mlsanity.com/paper/1806.03335/) (2018)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Deep Bayesian Bandits Showdown: An Empirical Comparison of Bayesian Deep Networks for Thompson Sampling](https://mlsanity.com/paper/1802.09127/) (2018)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Quantifying Uncertainties in Natural Language Processing Tasks](https://mlsanity.com/paper/1811.07253/) (2018)
- [DeepONet-Grid-UQ: A Trustworthy Deep Operator Framework for Predicting the Power Grid's Post-Fault Trajectories](https://mlsanity.com/paper/2202.07176/) (2022)
- [Accelerated replica exchange stochastic gradient Langevin diffusion enhanced Bayesian DeepONet for solving noisy parametric PDEs](https://mlsanity.com/paper/2111.02484/) (2021)
- [Physics-informed deep generative models](https://mlsanity.com/paper/1812.03511/) (2018)

## Built on by (most cited first, 30 of 39)

- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022)
- [NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators](https://mlsanity.com/paper/2208.11866/) (2022)
- [Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators](https://mlsanity.com/paper/2311.11262/) (2023)
- [RiemannONets: Interpretable Neural Operators for Riemann Problems](https://mlsanity.com/paper/2401.08886/) (2024)
- [Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networks](https://mlsanity.com/paper/2402.15406/) (2024)
- [Variationally Mimetic Operator Networks](https://mlsanity.com/paper/2209.12871/) (2022)
- [IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learning](https://mlsanity.com/paper/2302.03271/) (2023)
- [D2NO: Efficient Handling of Heterogeneous Input Function Spaces with Distributed Deep Neural Operators](https://mlsanity.com/paper/2310.18888/) (2023)
- [Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEs](https://mlsanity.com/paper/2403.10642/) (2024)
- [On Approximating the Dynamic Response of Synchronous Generators via Operator Learning: A Step Towards Building Deep Operator-based Power Grid Simulators](https://mlsanity.com/paper/2301.12538/) (2023)
- [Active operator learning with predictive uncertainty quantification for partial differential equations](https://mlsanity.com/paper/2503.03178/) (2025)
- [Variational Bayes Deep Operator Network: A data-driven Bayesian solver for parametric differential equations](https://mlsanity.com/paper/2206.05655/) (2022)
- [Composite Bayesian Optimization In Function Spaces Using NEON -- Neural Epistemic Operator Networks](https://mlsanity.com/paper/2404.03099/) (2024)
- [Federated scientific machine learning for approximating functions and solving differential equations with data heterogeneity](https://mlsanity.com/paper/2410.13141/) (2024)
- [Time-Series Forecasting, Knowledge Distillation, and Refinement within a Multimodal PDE Foundation Model](https://mlsanity.com/paper/2409.11609/) (2024)
- [Guaranteed prediction sets for functional surrogate models](https://mlsanity.com/paper/2501.18426/) (2025)
- [A hybrid Decoder-DeepONet operator regression framework for unaligned observation data](https://mlsanity.com/paper/2308.09274/) (2023)
- [Cauchy Random Features for Operator Learning in Sobolev Space](https://mlsanity.com/paper/2503.00300/) (2025)
- [Randomized prior wavelet neural operator for uncertainty quantification](https://mlsanity.com/paper/2302.01051/) (2023)
- [Multi-fidelity climate model parameterization for better generalization and extrapolation](https://mlsanity.com/paper/2309.10231/) (2023)
- [Uncertainty quantification for deeponets with ensemble kalman inversion](https://mlsanity.com/paper/2403.03444/) (2024)
- [Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantification](https://mlsanity.com/paper/2408.00681/) (2024)
- [E-PINNs: Epistemic Physics-Informed Neural Networks](https://mlsanity.com/paper/2503.19333/) (2025)
- [Conformalized-KANs: Uncertainty Quantification with Coverage Guarantees for Kolmogorov-Arnold Networks (KANs) in Scientific Machine Learning](https://mlsanity.com/paper/2504.15240/) (2025)
- [Probabilistic Predictions of Process-Induced Deformation in Carbon/Epoxy Composites Using a Deep Operator Network](https://mlsanity.com/paper/2512.13746/) (2025)
- [Scalable Bayesian optimization with high-dimensional outputs using randomized prior networks](https://mlsanity.com/paper/2302.07260/) (2023)
- [Learning Latent Space Dynamics with Model-Form Uncertainties: A Stochastic Reduced-Order Modeling Approach](https://mlsanity.com/paper/2409.00220/) (2024)
- [Empowering Bayesian Neural Networks with Functional Priors through Anchored Ensembling for Mechanics Surrogate Modeling Applications](https://mlsanity.com/paper/2409.05234/) (2024)
- [Towards Robust Surrogate Models: Benchmarking Machine Learning Approaches to Expediting Phase Field Simulations of Brittle Fracture](https://mlsanity.com/paper/2507.07237/) (2025)

## Abstract (arXiv)

We present a simple and effective approach for posterior uncertainty quantification in deep operator networks (DeepONets); an emerging paradigm for supervised learning in function spaces. We adopt a frequentist approach based on randomized prior ensembles, and put forth an efficient vectorized implementation for fast parallel inference on accelerated hardware. Through a collection of representative examples in computational mechanics and climate modeling, we show that the merits of the proposed approach are fourfold. (1) It can provide more robust and accurate predictions when compared against deterministic DeepONets. (2) It shows great capability in providing reliable uncertainty estimates on scarce data-sets with multi-scale function pairs. (3) It can effectively detect out-of-distribution and adversarial examples. (4) It can seamlessly quantify uncertainty due to model bias, as well as noise corruption in the data. Finally, we provide an optimized JAX library called em UQDeepONet that can accommodate large model architectures, large ensemble sizes, as well as large data-sets with excellent parallel performance on accelerated hardware, thereby enabling uncertainty quantification for DeepONets in realistic large-scale applications.