mlsanityData to 14 Oct 2026

Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs

arXiv 2208.01565 ↗First posted 2 Aug 2022cs.LG cs.NA math.NAcited by 25

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Abstract (arXiv)

Neural operators are a type of deep architecture that learns to solve (i.e. learns the nonlinear solution operator of) partial differential equations (PDEs). The current state of the art for these models does not provide explicit uncertainty quantification. This is arguably even more of a problem for this kind of tasks than elsewhere in machine learning, because the dynamical systems typically described by PDEs often exhibit subtle, multiscale structure that makes errors hard to spot by humans. In this work, we first provide a mathematically detailed Bayesian formulation of the ''shallow'' (linear) version of neural operators in the formalism of Gaussian processes. We then extend this analytic treatment to general deep neural operators using approximate methods from Bayesian deep learning. We extend previous results on neural operators by providing them with uncertainty quantification. As a result, our approach is able to identify cases, and provide structured uncertainty estimates, where the neural operator fails to predict well.

Key facts. Cited by 25 papers in the map (data through 14 Oct 2026). Builds on 37 of its 73 references that are in the map. First posted on arXiv 2 Aug 2022.

Builds on

37 of 73 references.

References · 37

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,361
  2. 2015Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learningcited by 4,448
  3. 2016Simple and Scalable Predictive Uncertainty Estimation using Deep Ensemblescited by 3,926
  4. 2017Neural Message Passing for Quantum Chemistrycited by 3,487
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  12. 2019A Simple Baseline for Bayesian Uncertainty in Deep Learningcited by 560
25 more
  1. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  2. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  3. 2021Laplace Redux -- Effortless Bayesian Deep Learningcited by 365
  4. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 349
  5. 2018Why ReLU networks yield high-confidence predictions far away from the training data and how to mitigate the problemcited by 323
  6. 2015Scalable Bayesian Optimization Using Deep Neural Networkscited by 284
  7. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 262
  8. 2020Model Reduction and Neural Networks for Parametric PDEscited by 258
  9. 2021Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformerscited by 248
  10. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 234
  11. 2020Being Bayesian, Even Just a Bit, Fixes Overconfidence in ReLU Networkscited by 231
  12. 2022U-NO: U-shaped Neural Operatorscited by 230
  13. 2019Cyclical Stochastic Gradient MCMC for Bayesian Deep Learningcited by 203
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  21. 2022Generative Adversarial Neural Operatorscited by 52
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 5; 2025: 9; 2026: 11.

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  1. 2024Calibrated Uncertainty Quantification for Operator Learning via Conformal Predictioncited by 26
  2. 2025Operator Learning: A Statistical Perspectivecited by 17
  3. 2024Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEscited by 15
  4. 2024Linearization Turns Neural Operators into Function-Valued Gaussian Processescited by 12
  5. 2025Epistemic Artificial Intelligence is Essential for Machine Learning Models to Truly 'Know When They Do Not Know'cited by 8
  6. 2025Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operatorscited by 6
  7. 2024Learning Semilinear Neural Operators : A Unified Recursive Framework For Prediction And Data Assimilationcited by 4
  8. 2024Neural Operator induced Gaussian Process framework for probabilistic solution of parametric partial differential equationscited by 4
  9. 2025Locally Adaptive Conformal Inference for Operator Modelscited by 2
  10. 2026Structure-Aware Epistemic Uncertainty Quantification for Neural Operator PDE Surrogatescited by 2
  11. 2025ON-Traffic: An Operator Learning Framework for Online Traffic Flow Estimation and Uncertainty Quantification from Lagrangian Sensorscited by 1
  12. 2025Neural Measures for learning distributions of Random PDEscited by 1
13 more
  1. 2026Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospectscited by 1
  2. 2026Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regimecited by 1
  3. 2026Geometry-Aware Post-Hoc Uncertainty Quantification in Operator Learningcited by 1
  4. 2026Neural Operator Processes for Probabilistic Operator Learning under Partial Observationscited by 1
  5. 2026GeoQ: Geometry-Aware Conditional Quantile Error Estimation for Scientific Surrogate Modelscited by 1
  6. 2025Are Statistical Methods Obsolete in the Era of Deep Learning? A Study of ODE Inverse Problemscited by 0
  7. 2025From Local Interactions to Global Operators: Scalable Gaussian Process Operator for Physical Systemscited by 0
  8. 2025Infinite Neural Operators: Gaussian processes on functionscited by 0
  9. 2026Generative Neural Operators through Diffusion Last Layercited by 0
  10. 2026Posterior-First Neural PDE Simulation: Inferring Hidden Problem State from a Single Fieldcited by 0
  11. 2026Martingale Neural Operators: Learning Stochastic Marginals via Doob-Meyer Factorizationcited by 0
  12. 2026Conformal Uncertainty Quantification Guarantees for Neural Operatorscited by 0
  13. 2026Threshold-Aware Conformal Routingcited by 0