mlsanityData to 7 Oct 2026

Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons

arXiv 2201.07766 ↗First posted 19 Jan 2022cs.LGcited by 169

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Abstract (arXiv)

Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with traditional methods. However, quantifying errors and uncertainties in NN-based inference is more complicated than in traditional methods. This is because in addition to aleatoric uncertainty associated with noisy data, there is also uncertainty due to limited data, but also due to NN hyperparameters, overparametrization, optimization and sampling errors as well as model misspecification. Although there are some recent works on uncertainty quantification (UQ) in NNs, there is no systematic investigation of suitable methods towards quantifying the total uncertainty effectively and efficiently even for function approximation, and there is even less work on solving partial differential equations and learning operator mappings between infinite-dimensional function spaces using NNs. In this work, we present a comprehensive framework that includes uncertainty modeling, new and existing solution methods, as well as evaluation metrics and post-hoc improvement approaches. To demonstrate the applicability and reliability of our framework, we present an extensive comparative study in which various methods are tested on prototype problems, including problems with mixed input-output data, and stochastic problems in high dimensions. In the Appendix, we include a comprehensive description of all the UQ methods employed, which we will make available as open-source library of all codes included in this framework.

Key facts. Cited by 169 papers in the map (data through 7 Oct 2026). Builds on 36 of its 63 references that are in the map. First posted on arXiv 19 Jan 2022.

Builds on

36 of 63 references.

References · 36

  1. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  2. 2016Variational Inference: A Review for Statisticianscited by 1,466
  3. 2019An Introduction to Variational Autoencoderscited by 997
  4. 2019Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methodscited by 903
  5. 2020A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challengescited by 813
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  8. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  9. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
  10. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  11. 2015Preconditioned Stochastic Gradient Langevin Dynamics for Deep Neural Networkscited by 121
  12. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 116
24 more
  1. 2021Priors in Bayesian Deep Learning: A Reviewcited by 101
  2. 2021A physics-informed variational DeepONet for predicting the crack path in brittle materialscited by 98
  3. 2018Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantificationcited by 81
  4. 2020Multi-fidelity Bayesian Neural Networks: Algorithms and Applicationscited by 59
  5. 2019Stochastic gradient Markov chain Monte Carlocited by 56
  6. 2021Meta-learning PINN loss functionscited by 51
  7. 2021Learning Functional Priors and Posteriors from Data and Physicscited by 37
  8. 2020Deeply Uncertain: Comparing Methods of Uncertainty Quantification in Deep Learning Algorithmscited by 34
  9. 2021Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow modelscited by 34
  10. 2013Langevin diffusions and the Metropolis-adjusted Langevin algorithmcited by 29
  11. 2021Valid prediction intervals for regression problemscited by 28
  12. 2020Multi-fidelity Generative Deep Learning Turbulent Flowscited by 21
  13. 2021PID-GAN: A GAN Framework based on a Physics-informed Discriminator for Uncertainty Quantification with Physicscited by 21
  14. 2019Conditional deep surrogate models for stochastic, high-dimensional, and multi-fidelity systemscited by 18
  15. 2020Efficient Uncertainty Quantification for Dynamic Subsurface Flow with Surrogate by Theory-guided Neural Networkcited by 17
  16. 2020Bayesian differential programming for robust systems identification under uncertaintycited by 16
  17. 2021Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 16
  18. 2020GINNs: Graph-Informed Neural Networks for Multiscale Physicscited by 12
  19. 2021Interval and fuzzy physics-informed neural networks for uncertain fieldscited by 9
  20. 2019A physics-aware, probabilistic machine learning framework for coarse-graining high-dimensional systems in the Small Data regimecited by 7
  21. 2019Inspecting adversarial examples using the Fisher informationcited by 7
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 16; 2023: 43; 2024: 40; 2025: 42; 2026: 28.

169 papers.

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  1. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  2. 2022Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures?cited by 124
  3. 2024A comprehensive and FAIR comparison between MLP and KAN representations for differential equations and operator networkscited by 103
  4. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
  5. 2023A Survey on Uncertainty Quantification Methods for Deep Learningcited by 78
  6. 2023Machine learning with data assimilation and uncertainty quantification for dynamical systems: a reviewcited by 74
  7. 2024From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learningcited by 73
  8. 2022Reliable extrapolation of deep neural operators informed by physics or sparse observationscited by 62
  9. 2022Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priorscited by 39
  10. 2023Explaining Predictive Uncertainty with Information Theoretic Shapley Valuescited by 36
  11. 2022NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operatorscited by 33
  12. 2023Correcting model misspecification in physics-informed neural networks (PINNs)cited by 33
38 more
  1. 2022Discovering and forecasting extreme events via active learning in neural operatorscited by 32
  2. 2023L-HYDRA: Multi-Head Physics-Informed Neural Networkscited by 29
  3. 2023A Review of Deep Learning for Video Captioningcited by 28
  4. 2023Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operatorscited by 28
  5. 2024Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networkscited by 28
  6. 2023Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorialcited by 27
  7. 2023Discovering a reaction-diffusion model for Alzheimer's disease by combining PINNs with symbolic regressioncited by 25
  8. 2024Active Learning for Neural PDE Solverscited by 22
  9. 2023IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learningcited by 18
  10. 2023D2NO: Efficient Handling of Heterogeneous Input Function Spaces with Distributed Deep Neural Operatorscited by 18
  11. 2024Physics-constrained polynomial chaos expansion for scientific machine learning and uncertainty quantificationcited by 17
  12. 2023Recent Advances and Applications of Machine Learning in Experimental Solid Mechanics: A Reviewcited by 15
  13. 2024Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEscited by 14
  14. 2025Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemblecited by 13
  15. 2022Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Datacited by 12
  16. 2024Explaining Bayesian Optimization by Shapley Values Facilitates Human-AI Collaborationcited by 12
  17. 2025Machine learning for modelling unstructured grid data in computational physics: a reviewcited by 12
  18. 2024Residual-based Attention Physics-informed Neural Networks for Spatio-Temporal Ageing Assessment of Transformers Operated in Renewable Power Plantscited by 11
  19. 2024Uncertainty Quantification of Surrogate Models using Conformal Predictioncited by 11
  20. 2023Deep Operator Learning-based Surrogate Models with Uncertainty Quantification for Optimizing Internal Cooling Channel Rib Profilescited by 10
  21. 2023Slow Invariant Manifolds of Singularly Perturbed Systems via Physics-Informed Machine Learningcited by 10
  22. 2024Composite Bayesian Optimization In Function Spaces Using NEON -- Neural Epistemic Operator Networkscited by 10
  23. 2022Quantification of Deep Neural Network Prediction Uncertainties for VVUQ of Machine Learning Modelscited by 9
  24. 2024Leveraging viscous Hamilton-Jacobi PDEs for uncertainty quantification in scientific machine learningcited by 9
  25. 2024Privacy Preserving Federated Learning in Medical Imaging with Uncertainty Estimationcited by 9
  26. 2023A Generative Modeling Framework for Inferring Families of Biomechanical Constitutive Laws in Data-Sparse Regimescited by 8
  27. 2023Uncertainty Quantification for Molecular Property Predictions with Graph Neural Architecture Searchcited by 8
  28. 2024Fast meta-solvers for 3D complex-shape scatterers using neural operators trained on a non-scattering problemcited by 8
  29. 2025Calibrated Physics-Informed Uncertainty Quantificationcited by 8
  30. 2025Position: Epistemic uncertainty estimation methods are fundamentally incompletecited by 8
  31. 2022Physics-informed Neural Networks with Unknown Measurement Noisecited by 7
  32. 2023Training, Architecture, and Prior for Deterministic Uncertainty Methodscited by 7
  33. 2024Randomized Physics-Informed Neural Networks for Bayesian Data Assimilationcited by 7
  34. 2023Efficient Bayesian Physics Informed Neural Networks for Inverse Problems via Ensemble Kalman Inversioncited by 6
  35. 2024NeuroSEM: A hybrid framework for simulating multiphysics problems by coupling PINNs and spectral elementscited by 6
  36. 2025Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operatorscited by 6
  37. 2022Structure and Distribution Metric for Quantifying the Quality of Uncertainty: Assessing Gaussian Processes, Deep Neural Nets, and Deep Neural Operators for Regressioncited by 5
  38. 2023Quantifying uncertainty for deep learning based forecasting and flow-reconstruction using neural architecture search ensemblescited by 5

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