# Physics Informed Extreme Learning Machine (PIELM) -- A rapid method for the numerical solution of partial differential equations

arXiv: https://arxiv.org/abs/1907.03507 · HTML: https://mlsanity.com/paper/1907.03507/

## Key facts

- Cited by 67 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jul 2019.

## Built on by (most cited first, 30 of 67)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [PirateNets: Physics-informed Deep Learning with Residual Adaptive Networks](https://mlsanity.com/paper/2402.00326/) (2024)
- [Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2501.16371/) (2025)
- [SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs](https://mlsanity.com/paper/2102.13037/) (2021)
- [Physics-Informed Neural Nets for Control of Dynamical Systems](https://mlsanity.com/paper/2104.02556/) (2021)
- [On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elements](https://mlsanity.com/paper/2110.14121/) (2021)
- [Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks](https://mlsanity.com/paper/2205.02902/) (2022)
- [Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approach](https://mlsanity.com/paper/2201.13323/) (2022)
- [An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensions](https://mlsanity.com/paper/2309.07049/) (2023)
- [A Modified Batch Intrinsic Plasticity Method for Pre-training the Random Coefficients of Extreme Learning Machines](https://mlsanity.com/paper/2103.08042/) (2021)
- [A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networks](https://mlsanity.com/paper/2210.04338/) (2022)
- [Universal approximation property of Banach space-valued random feature models including random neural networks](https://mlsanity.com/paper/2312.08410/) (2023)
- [Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2205.06948/) (2022)
- [Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equations](https://mlsanity.com/paper/2206.09299/) (2022)
- [Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networks](https://mlsanity.com/paper/2201.09989/) (2022)
- [Numerical Computation of Partial Differential Equations by Hidden-Layer Concatenated Extreme Learning Machine](https://mlsanity.com/paper/2204.11375/) (2022)
- [Curriculum Learning-Driven PIELMs for Fluid Flow Simulations](https://mlsanity.com/paper/2503.06347/) (2025)
- [Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradients](https://mlsanity.com/paper/2507.10241/) (2025)
- [Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion](https://mlsanity.com/paper/2411.03384/) (2024)
- [Data-Guided Physics-Informed Neural Networks for Solving Inverse Problems in Partial Differential Equations](https://mlsanity.com/paper/2407.10836/) (2024)
- [Physics-Informed Neural Networks with Skip Connections for Modeling and Control of Gas-Lifted Oil Wells](https://mlsanity.com/paper/2403.02289/) (2024)
- [A Nonoverlapping Domain Decomposition Method for Extreme Learning Machines: Elliptic Problems](https://mlsanity.com/paper/2406.15959/) (2024)
- [Iterative Training of Physics-Informed Neural Networks with Fourier-enhanced Features](https://mlsanity.com/paper/2510.19399/) (2025)
- [Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiology](https://mlsanity.com/paper/2408.07201/) (2024)
- [RandNet-Parareal: a time-parallel PDE solver using Random Neural Networks](https://mlsanity.com/paper/2411.06225/) (2024)
- [Are Two Hidden Layers Still Enough for the Physics-Informed Neural Networks?](https://mlsanity.com/paper/2412.19235/) (2024)
- [ELM-DeepONets: Backpropagation-Free Training of Deep Operator Networks via Extreme Learning Machines](https://mlsanity.com/paper/2501.09395/) (2025)
- [PICS in Pics: Physics Informed Contour Selection for Rapid Image Segmentation](https://mlsanity.com/paper/2311.07002/) (2023)

## Abstract (arXiv)

There has been rapid progress recently on the application of deep networks to the solution of partial differential equations, collectively labelled as Physics Informed Neural Networks (PINNs). In this paper, we develop Physics Informed Extreme Learning Machine (PIELM), a rapid version of PINNs which can be applied to stationary and time dependent linear partial differential equations. We demonstrate that PIELM matches or exceeds the accuracy of PINNs on a range of problems. We also discuss the limitations of neural network based approaches, including our PIELM, in the solution of PDEs on large domains and suggest an extension, a distributed version of our algorithm -{}- DPIELM. We show that DPIELM produces excellent results comparable to conventional numerical techniques in the solution of time-dependent problems. Collectively, this work contributes towards making the use of neural networks in the solution of partial differential equations in complex domains as a competitive alternative to conventional discretization techniques.