mlsanityData to 7 Oct 2026

Bayesian deep learning framework for uncertainty quantification in high dimensions

arXiv 2210.11737 ↗First posted 21 Oct 2022stat.ML cs.AI cs.LGcited by 0

Not on the map. Why?

Abstract (arXiv)

We develop a novel deep learning method for uncertainty quantification in stochastic partial differential equations based on Bayesian neural network (BNN) and Hamiltonian Monte Carlo (HMC). A BNN efficiently learns the posterior distribution of the parameters in deep neural networks by performing Bayesian inference on the network parameters. The posterior distribution is efficiently sampled using HMC to quantify uncertainties in the system. Several numerical examples are shown for both forward and inverse problems in high dimension to demonstrate the effectiveness of the proposed method for uncertainty quantification. These also show promising results that the computational cost is almost independent of the dimension of the problem demonstrating the potential of the method for tackling the so-called curse of dimensionality.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 34 of its 61 references that are in the map. First posted on arXiv 21 Oct 2022.

Builds on

34 of 61 references.

References · 34

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  2. 2015Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learningcited by 4,416
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  4. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,597
  5. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,170
  6. 2015Variational Inference with Normalizing Flowscited by 2,161
  7. 2016Density estimation using Real NVPcited by 2,029
  8. 2018On the Spectral Bias of Neural Networkscited by 1,321
  9. 2019Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descentcited by 732
  10. 2017Masked Autoregressive Flow for Density Estimationcited by 680
  11. 2016Improving Variational Inference with Inverse Autoregressive Flowcited by 636
  12. 2019On Exact Computation with an Infinitely Wide Neural Netcited by 572
22 more
  1. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  2. 2016Stein Variational Gradient Descent: A General Purpose Bayesian Inference Algorithmcited by 498
  3. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  4. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
  5. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  6. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  7. 2017A Conceptual Introduction to Hamiltonian Monte Carlocited by 207
  8. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  9. 2019Towards Understanding the Spectral Bias of Deep Learningcited by 173
  10. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  11. 2019Variational Physics-Informed Neural Networks For Solving Partial Differential Equationscited by 144
  12. 2019The Convergence Rate of Neural Networks for Learned Functions of Different Frequenciescited by 134
  13. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 133
  14. 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
  15. 2018Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantificationcited by 81
  16. 2019Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networkscited by 63
  17. 2018Deep convolutional encoder-decoder networks for uncertainty quantification of dynamic multiphase flow in heterogeneous mediacited by 56
  18. 2021Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow modelscited by 34
  19. 2018Deep autoregressive neural networks for high-dimensional inverse problems in groundwater contaminant source identificationcited by 26
  20. 2018A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equationscited by 23
  21. 2018Dimensionally Tight Bounds for Second-Order Hamiltonian Monte Carlocited by 11
  22. 2020Deep neural network for solving differential equations motivated by Legendre-Galerkin approximationcited by 3

Built on by

Not cited yet.