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Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networks

arXiv 2402.15406 ↗First posted 23 Feb 2024cs.LG cs.NA math.NAcited by 28

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Abstract (arXiv)

In this paper, we adopt conformal prediction, a distribution-free uncertainty quantification (UQ) framework, to obtain confidence prediction intervals with coverage guarantees for Deep Operator Network (DeepONet) regression. Initially, we enhance the uncertainty quantification frameworks (B-DeepONet and Prob-DeepONet) previously proposed by the authors by using split conformal prediction. By combining conformal prediction with our Prob- and B-DeepONets, we effectively quantify uncertainty by generating rigorous confidence intervals for DeepONet prediction. Additionally, we design a novel Quantile-DeepONet that allows for a more natural use of split conformal prediction. We refer to this distribution-free effective uncertainty quantification framework as split conformal Quantile-DeepONet regression. Finally, we demonstrate the effectiveness of the proposed methods using various ordinary, partial differential equation numerical examples, and multi-fidelity learning.

Key facts. Cited by 28 papers in the map (data through 7 Oct 2026). Builds on 35 of its 71 references that are in the map. First posted on arXiv 23 Feb 2024.

Builds on

35 of 71 references.

References · 35

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2021A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantificationcited by 777
  4. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 748
  5. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  6. 2019Conformalized Quantile Regressioncited by 620
  7. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  8. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  9. 2014Stochastic Gradient Hamiltonian Monte Carlocited by 341
  10. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 292
  11. 2017Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysiscited by 242
  12. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
23 more
  1. 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 169
  2. 2023Conformal Language Modelingcited by 163
  3. 2017User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradientcited by 116
  4. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 109
  5. 2022Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transportcited by 92
  6. 2023Conformal PID Control for Time Series Predictioncited by 85
  7. 2023Uncertainty Quantification over Graph with Conformalized Graph Neural Networkscited by 79
  8. 2022Reliable extrapolation of deep neural operators informed by physics or sparse observationscited by 62
  9. 2023Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustnesscited by 60
  10. 2023Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestrationcited by 46
  11. 2022Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priorscited by 39
  12. 2020Non-convex Learning via Replica Exchange Stochastic Gradient MCMCcited by 33
  13. 2022DeepONet-Grid-UQ: A Trustworthy Deep Operator Framework for Predicting the Power Grid's Post-Fault Trajectoriescited by 30
  14. 2023PPDONet: Deep Operator Networks for Fast Prediction of Steady-State Solutions in Disk-Planet Systemscited by 19
  15. 2023IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learningcited by 18
  16. 2023D2NO: Efficient Handling of Heterogeneous Input Function Spaces with Distributed Deep Neural Operatorscited by 18
  17. 2022DeepGraphONet: A Deep Graph Operator Network to Learn and Zero-shot Transfer the Dynamic Response of Networked Systemscited by 14
  18. 2024DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domainscited by 14
  19. 2023On Approximating the Dynamic Response of Synchronous Generators via Operator Learning: A Step Towards Building Deep Operator-based Power Grid Simulatorscited by 13
  20. 2023Deep Operator Learning-based Surrogate Models with Uncertainty Quantification for Optimizing Internal Cooling Channel Rib Profilescited by 10
  21. 2024Accelerating Approximate Thompson Sampling with Underdamped Langevin Monte Carlocited by 10
  22. 2024Speeding up and reducing memory usage for scientific machine learning via mixed precisioncited by 8
  23. 2023A Physics-Guided Bi-Fidelity Fourier-Featured Operator Learning Framework for Predicting Time Evolution of Drag and Lift Coefficientscited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 10; 2025: 9; 2026: 9.

28 papers.

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  1. 2024Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolationcited by 45
  2. 2024VICON: Vision In-Context Operator Networks for Multi-Physics Fluid Dynamics Predictioncited by 22
  3. 2024MODNO: Multi Operator Learning With Distributed Neural Operatorscited by 14
  4. 2024LeMON: Learning to Learn Multi-Operator Networkscited by 13
  5. 2025Active operator learning with predictive uncertainty quantification for partial differential equationscited by 12
  6. 2024DeepONet as a Multi-Operator Extrapolation Model: Distributed Pretraining with Physics-Informed Fine-Tuningcited by 11
  7. 2025PI-MFM: Physics-informed multimodal foundation model for solving partial differential equationscited by 6
  8. 2024Conformalized Prediction of Post-Fault Voltage Trajectories Using Pre-trained and Finetuned Attention-Driven Neural Operatorscited by 5
  9. 2025A Conformal Prediction Framework for Uncertainty Quantification in Physics-Informed Neural Networkscited by 5
  10. 2024Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantificationcited by 4
  11. 2025Exploring the Noise Robustness of Online Conformal Predictioncited by 4
  12. 2025E-PINNs: Epistemic Physics-Informed Neural Networkscited by 4
16 more
  1. 2024Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanicscited by 3
  2. 2025Conformalized-KANs: Uncertainty Quantification with Coverage Guarantees for Kolmogorov-Arnold Networks (KANs) in Scientific Machine Learningcited by 3
  3. 2025iPINNER: An Iterative Physics-Informed Neural Network with Ensemble Kalman Filtercited by 3
  4. 2026Graph In-Context Operator Networks for Generalizable Spatiotemporal Predictioncited by 3
  5. 2024Large language models, physics-based modeling, experimental measurements: the trinity of data-scarce learning of polymer propertiescited by 2
  6. 2025Locally Adaptive Conformal Inference for Operator Modelscited by 2
  7. 2024Distribution free uncertainty quantification in neuroscience-inspired deep operatorscited by 1
  8. 2026Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regimecited by 1
  9. 2026GeoQ: Geometry-Aware Conditional Quantile Error Estimation for Scientific Surrogate Modelscited by 1
  10. 2025GeoFunFlow: Geometric Function Flow Matching for Inverse Operator Learning over Complex Geometriescited by 0
  11. 2026Softmax is not Enough (for Adaptive Conformal Classification)cited by 0
  12. 2026pADAM: A Plug-and-Play All-in-One Diffusion Architecture for Multi-Physics Learningcited by 0
  13. 2026Conformalized Quantum DeepONet Ensembles: Towards Scalable Operator Learning with Distribution-Free Guaranteescited by 0
  14. 2026Two-Step MV-DeepONet: Probabilistic Operator Learning for Uncertainty Propagation Driven by Random Input Fieldscited by 0
  15. 2026FAST-DeepONet: Factor-Augmented Branch Representations for High-Dimensional PDE Inputs in the Small-Sample Regimecited by 0
  16. 2026Conformal Uncertainty Quantification Guarantees for Neural Operatorscited by 0