mlsanityData to 14 Oct 2026

NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators

arXiv 2208.11866 ↗First posted 25 Aug 2022cs.LG physics.comp-phcited by 33

Not on the map. Why?

Abstract (arXiv)

Uncertainty quantification (UQ) in machine learning is currently drawing increasing research interest, driven by the rapid deployment of deep neural networks across different fields, such as computer vision, natural language processing, and the need for reliable tools in risk-sensitive applications. Recently, various machine learning models have also been developed to tackle problems in the field of scientific computing with applications to computational science and engineering (CSE). Physics-informed neural networks and deep operator networks are two such models for solving partial differential equations and learning operator mappings, respectively. In this regard, a comprehensive study of UQ methods tailored specifically for scientific machine learning (SciML) models has been provided in [45]. Nevertheless, and despite their theoretical merit, implementations of these methods are not straightforward, especially in large-scale CSE applications, hindering their broad adoption in both research and industry settings. In this paper, we present an open-source Python library (https://github.com/Crunch-UQ4MI), termed NeuralUQ and accompanied by an educational tutorial, for employing UQ methods for SciML in a convenient and structured manner. The library, designed for both educational and research purposes, supports multiple modern UQ methods and SciML models. It is based on a succinct workflow and facilitates flexible employment and easy extensions by the users. We first present a tutorial of NeuralUQ and subsequently demonstrate its applicability and efficiency in four diverse examples, involving dynamical systems and high-dimensional parametric and time-dependent PDEs.

Key facts. Cited by 33 papers in the map (data through 14 Oct 2026). Builds on 41 of its 80 references that are in the map. First posted on arXiv 25 Aug 2022.

Builds on

41 of 80 references.

References · 41

  1. 2015Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learningcited by 4,448
  2. 2016Simple and Scalable Predictive Uncertainty Estimation using Deep Ensemblescited by 3,926
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  4. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  5. 2018GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Accelerationcited by 609
  6. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 608
  7. 2017Solving high-dimensional partial differential equations using deep learningcited by 556
  8. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 540
  9. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  10. 2018Accurate Uncertainties for Deep Learning Using Calibrated Regressioncited by 422
  11. 2020Universal Differential Equations for Scientific Machine Learningcited by 353
  12. 2017Snapshot Ensembles: Train 1, get M for freecited by 350
29 more
  1. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
  2. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 262
  3. 2017A Conceptual Introduction to Hamiltonian Monte Carlocited by 209
  4. 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 170
  5. 2020Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systemscited by 164
  6. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 147
  7. 2020Uncertainty Quantification and Deep Ensemblescited by 139
  8. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 135
  9. 2021A physics-informed variational DeepONet for predicting the crack path in brittle materialscited by 98
  10. 2019Evaluating and Calibrating Uncertainty Prediction in Regression Taskscited by 87
  11. 2018Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantificationcited by 81
  12. 2020Physics informed deep learning for computational elastodynamics without labeled datacited by 80
  13. 2020The Case for Bayesian Deep Learningcited by 69
  14. 2019Quality of Uncertainty Quantification for Bayesian Neural Network Inferencecited by 59
  15. 2020Multi-fidelity Bayesian Neural Networks: Algorithms and Applicationscited by 59
  16. 2020NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation frameworkcited by 58
  17. 2021Uncertainty Toolbox: an Open-Source Library for Assessing, Visualizing, and Improving Uncertainty Quantificationcited by 57
  18. 2022Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priorscited by 39
  19. 2019Deep learning observables in computational fluid dynamicscited by 38
  20. 2021Learning Functional Priors and Posteriors from Data and Physicscited by 37
  21. 2022Discovering and forecasting extreme events via active learning in neural operatorscited by 32
  22. 2013Langevin diffusions and the Metropolis-adjusted Langevin algorithmcited by 29
  23. 2016Edward: A library for probabilistic modeling, inference, and criticismcited by 28
  24. 2019Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architecturescited by 22
  25. 2021Accelerated replica exchange stochastic gradient Langevin diffusion enhanced Bayesian DeepONet for solving noisy parametric PDEscited by 22
  26. 2020Bayesian differential programming for robust systems identification under uncertaintycited by 16
  27. 2019PyDEns: a Python Framework for Solving Differential Equations with Neural Networkscited by 14
  28. 2020tfp.mcmc: Modern Markov Chain Monte Carlo Tools Built for Modern Hardwarecited by 10
  29. 2022Structure and Distribution Metric for Quantifying the Quality of Uncertainty: Assessing Gaussian Processes, Deep Neural Nets, and Deep Neural Operators for Regressioncited by 5

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 2; 2023: 12; 2024: 5; 2025: 8; 2026: 6.

33 papers.

Sort:

By area:

  1. 2023L-HYDRA: Multi-Head Physics-Informed Neural Networkscited by 29
  2. 2023Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operatorscited by 28
  3. 2023Discovering a reaction-diffusion model for Alzheimer's disease by combining PINNs with symbolic regressioncited by 25
  4. 2023Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoilscited by 19
  5. 2023IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learningcited by 18
  6. 2025Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemblecited by 13
  7. 2024Linearization Turns Neural Operators into Function-Valued Gaussian Processescited by 12
  8. 2024Leveraging viscous Hamilton-Jacobi PDEs for uncertainty quantification in scientific machine learningcited by 9
  9. 2023A Generative Modeling Framework for Inferring Families of Biomechanical Constitutive Laws in Data-Sparse Regimescited by 8
  10. 2024Fast meta-solvers for 3D complex-shape scatterers using neural operators trained on a non-scattering problemcited by 8
  11. 2025Calibrated Physics-Informed Uncertainty Quantificationcited by 8
  12. 2023Efficient Bayesian Physics Informed Neural Networks for Inverse Problems via Ensemble Kalman Inversioncited by 6
21 more
  1. 2023Leveraging Multi-time Hamilton-Jacobi PDEs for Certain Scientific Machine Learning Problemscited by 5
  2. 2022Error-Aware B-PINNs: Improving Uncertainty Quantification in Bayesian Physics-Informed Neural Networkscited by 4
  3. 2025E-PINNs: Epistemic Physics-Informed Neural Networkscited by 4
  4. 2025Torch-Uncertainty: A Deep Learning Framework for Uncertainty Quantificationcited by 4
  5. 2026Uncertainty Quantification in PINNs for Turbulent Flows: Bayesian Inference and Repulsive Ensemblescited by 4
  6. 2024Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiologycited by 3
  7. 2025Bilevel optimization for learning hyperparameters: Application to solving PDEs and inverse problems with Gaussian processescited by 3
  8. 2023Evaluation of machine learning architectures on the quantification of epistemic and aleatoric uncertainties in complex dynamical systemscited by 2
  9. 2024Towards certifiable AI in aviation: landscape, challenges, and opportunitiescited by 2
  10. 2025Physics-informed deep learning for infectious disease forecastingcited by 2
  11. 2025PINN - a Domain Decomposition Method for Bayesian Physics-Informed Neural Networkscited by 2
  12. 2022Physics-informed neural networks for operator equations with stochastic datacited by 1
  13. 2023PICProp: Physics-Informed Confidence Propagation for Uncertainty Quantificationcited by 1
  14. 2025JaxSGMC: Modular stochastic gradient MCMC in JAXcited by 1
  15. 2026Decision-Aware Evaluation of Physics-Informed Surrogatescited by 1
  16. 2023The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approachcited by 0
  17. 2023Evaluating Uncertainty Quantification approaches for Neural PDEs in scientific applicationscited by 0
  18. 2026Learning Surrogate LPV State-Space Models with Uncertainty Quantificationcited by 0
  19. 2026Neural Quantum Spectral Operator Learning for Solving Partial Differential Equationscited by 0
  20. 2026A Novel Tensor Product-Based Neural Network for Solving Partial Differential Equationscited by 0
  21. 2026Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solvercited by 0