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

arXiv: https://arxiv.org/abs/2205.06948 · HTML: https://mlsanity.com/paper/2205.06948/

## 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 (most cited first, 17 of 17)

- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Bayesian Neural Networks: An Introduction and Survey](https://mlsanity.com/paper/2006.12024/) (2020)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations](https://mlsanity.com/paper/1907.03507/) (2019)
- [Multi-fidelity Bayesian Neural Networks: Algorithms and Applications](https://mlsanity.com/paper/2012.13294/) (2020)
- [Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion](https://mlsanity.com/paper/2109.14860/) (2021)
- [Distributed physics informed neural network for data-efficient solution to partial differential equations](https://mlsanity.com/paper/1907.08967/) (2019)
- [PID-GAN: A GAN Framework based on a Physics-informed Discriminator for Uncertainty Quantification with Physics](https://mlsanity.com/paper/2106.02993/) (2021)
- [Bayesian deep learning with hierarchical prior: Predictions from limited and noisy data](https://mlsanity.com/paper/1907.04240/) (2019)
- [Temperature Field Inversion of Heat-Source Systems via Physics-Informed Neural Networks](https://mlsanity.com/paper/2201.06880/) (2022)

## Built on by (most cited first, 12 of 12)

- [Weak neural variational inference for solving Bayesian inverse problems without forward models: applications in elastography](https://mlsanity.com/paper/2407.20697/) (2024)
- [Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiology](https://mlsanity.com/paper/2408.07201/) (2024)
- [General Fourier Feature Physics-Informed Extreme Learning Machine (GFF-PIELM) for High-Frequency PDEs](https://mlsanity.com/paper/2510.12293/) (2025)
- [Neural Measures for learning distributions of Random PDEs](https://mlsanity.com/paper/2507.01687/) (2025)
- [Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEs](https://mlsanity.com/paper/2510.01039/) (2025)
- [Towards Fast Option Pricing PDE Solvers Powered by PIELM](https://mlsanity.com/paper/2510.04322/) (2025)
- [Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation](https://mlsanity.com/paper/2510.04490/) (2025)
- [A Critical Analysis of the Theoretical Framework of the Extreme Learning Machine](https://mlsanity.com/paper/2406.17427/) (2024)
- [Physics-Informed Extreme Learning Machine (PIELM) for Tunnelling-Induced Soil-Pile Interactions](https://mlsanity.com/paper/2510.00698/) (2025)
- [Variational Matrix-Learning Fourier Networks for Parametric Multiphysics Surrogates](https://mlsanity.com/paper/2605.02280/) (2026)
- [Physics-Informed Broad Learning System: An Efficient Backpropagation-Free Framework for Solving Partial Differential Equations](https://mlsanity.com/paper/2607.25608/) (2026)
- [Linearized PINN with pretrained nonlinear layers](https://mlsanity.com/paper/2609.14926/) (2026)

## 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.