mlsanityData to 14 Oct 2026

Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Data

arXiv 2205.06948 ↗First posted 14 May 2022cs.LGcited by 12

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Abstract (arXiv)

Physics-informed extreme learning machine (PIELM) has recently received significant attention as a rapid version of physics-informed neural network (PINN) for solving partial differential equations (PDEs). The key characteristic is to fix the input layer weights with random values and use Moore-Penrose generalized inverse for the output layer weights. The framework is effective, but it easily suffers from overfitting noisy data and lacks uncertainty quantification for the solution under noise scenarios.To this end, we develop the Bayesian physics-informed extreme learning machine (BPIELM) to solve both forward and inverse linear PDE problems with noisy data in a unified framework. In our framework, a prior probability distribution is introduced in the output layer for extreme learning machine with physic laws and the Bayesian method is used to estimate the posterior of parameters. Besides, for inverse PDE problems, problem parameters considered as new output layer weights are unified in a framework with forward PDE problems. Finally, we demonstrate BPIELM considering both forward problems, including Poisson, advection, and diffusion equations, as well as inverse problems, where unknown problem parameters are estimated. The results show that, compared with PIELM, BPIELM quantifies uncertainty arising from noisy data and provides more accurate predictions. In addition, BPIELM is considerably cheaper than PINN in terms of the computational cost.

Key facts. Cited by 12 papers in the map (data through 14 Oct 2026). Builds on 17 of its 40 references that are in the map. First posted on arXiv 14 May 2022.

Builds on

17 of 40 references.

References · 17

  1. 2020Bayesian Deep Learning and a Probabilistic Perspective of Generalizationcited by 480
  2. 2020Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Userscited by 338
  3. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
  4. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 262
  5. 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 170
  6. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 147
  7. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 135
  8. 2020Bayesian Neural Networks: An Introduction and Surveycited by 131
  9. 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
  10. 2020Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equationscited by 81
  11. 2019Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equationscited by 67
  12. 2020Multi-fidelity Bayesian Neural Networks: Algorithms and Applicationscited by 59
5 more
  1. 2021Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusioncited by 38
  2. 2019Distributed physics informed neural network for data-efficient solution to partial differential equationscited by 26
  3. 2021PID-GAN: A GAN Framework based on a Physics-informed Discriminator for Uncertainty Quantification with Physicscited by 21
  4. 2019Bayesian deep learning with hierarchical prior: Predictions from limited and noisy datacited by 8
  5. 2022Temperature Field Inversion of Heat-Source Systems via Physics-Informed Neural Networkscited by 8

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 3; 2025: 6; 2026: 3.

12 papers.

  1. 2024Weak neural variational inference for solving Bayesian inverse problems without forward models: applications in elastographycited by 5
  2. 2024Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiologycited by 3
  3. 2025General Fourier Feature Physics-Informed Extreme Learning Machine (GFF-PIELM) for High-Frequency PDEscited by 2
  4. 2025Neural Measures for learning distributions of Random PDEscited by 1
  5. 2025Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEscited by 1
  6. 2025Towards Fast Option Pricing PDE Solvers Powered by PIELMcited by 1
  7. 2025Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equationcited by 1
  8. 2024A Critical Analysis of the Theoretical Framework of the Extreme Learning Machinecited by 0
  9. 2025Physics-Informed Extreme Learning Machine (PIELM) for Tunnelling-Induced Soil-Pile Interactionscited by 0
  10. 2026Variational Matrix-Learning Fourier Networks for Parametric Multiphysics Surrogatescited by 0
  11. 2026Physics-Informed Broad Learning System: An Efficient Backpropagation-Free Framework for Solving Partial Differential Equationscited by 0
  12. 2026Linearized PINN with pretrained nonlinear layerscited by 0