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

arXiv 1907.03507 ↗First posted 8 Jul 2019cs.LG physics.comp-ph stat.MLcited by 66

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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.

Key facts. Cited by 66 papers in the map (data through 7 Oct 2026). First posted on arXiv 8 Jul 2019.

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Citing papers per year: 2021: 9; 2022: 9; 2023: 5; 2024: 11; 2025: 16; 2026: 16.

66 papers.

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  1. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  2. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  3. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 220
  4. 2024PirateNets: Physics-informed Deep Learning with Residual Adaptive Networkscited by 70
  5. 2025Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networkscited by 41
  6. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
  7. 2021Physics-Informed Neural Nets for Control of Dynamical Systemscited by 31
  8. 2022Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networkscited by 26
  9. 2021On Computing the Hyperparameter of Extreme Learning Machines: Algorithm and Application to Computational PDEs, and Comparison with Classical and High-Order Finite Elementscited by 25
  10. 2022Constructing coarse-scale bifurcation diagrams from spatio-temporal observations of microscopic simulations: A parsimonious machine learning approachcited by 25
  11. 2023An Extreme Learning Machine-Based Method for Computational PDEs in Higher Dimensionscited by 21
  12. 2022A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networkscited by 14
38 more
  1. 2021A Modified Batch Intrinsic Plasticity Method for Pre-training the Random Coefficients of Extreme Learning Machinescited by 13
  2. 2022Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Datacited by 12
  3. 2023Universal approximation property of Banach space-valued random feature models including random neural networkscited by 12
  4. 2022Enforcing continuous symmetries in physics-informed neural network for solving forward and inverse problems of partial differential equationscited by 11
  5. 2022Numerical Computation of Partial Differential Equations by Hidden-Layer Concatenated Extreme Learning Machinecited by 9
  6. 2022Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networkscited by 8
  7. 2025Curriculum Learning-Driven PIELMs for Fluid Flow Simulationscited by 8
  8. 2025Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradientscited by 7
  9. 2024Solving stochastic partial differential equations using neural networks in the Wiener chaos expansioncited by 6
  10. 2024Data-Guided Physics-Informed Neural Networks for Solving Inverse Problems in Partial Differential Equationscited by 5
  11. 2024Physics-Informed Neural Networks with Skip Connections for Modeling and Control of Gas-Lifted Oil Wellscited by 4
  12. 2024A Nonoverlapping Domain Decomposition Method for Extreme Learning Machines: Elliptic Problemscited by 4
  13. 2024Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiologycited by 3
  14. 2024RandNet-Parareal: a time-parallel PDE solver using Random Neural Networkscited by 3
  15. 2024Are Two Hidden Layers Still Enough for the Physics-Informed Neural Networks?cited by 3
  16. 2025ELM-DeepONets: Backpropagation-Free Training of Deep Operator Networks via Extreme Learning Machinescited by 3
  17. 2023PICS in Pics: Physics Informed Contour Selection for Rapid Image Segmentationcited by 2
  18. 2024Solving Partial Differential Equations with Equivariant Extreme Learning Machinescited by 2
  19. 2025General Fourier Feature Physics-Informed Extreme Learning Machine (GFF-PIELM) for High-Frequency PDEscited by 2
  20. 2025Iterative Training of Physics-Informed Neural Networks with Fourier-enhanced Featurescited by 2
  21. 2025Condition Numbers and Eigenvalue Spectra of Shallow Networks on Spherescited by 2
  22. 2026Do physics-informed neural networks (PINNs) need to be deep? Shallow PINNs using the Levenberg-Marquardt algorithmcited by 2
  23. 2021Cell-average based neural network method for hyperbolic and parabolic partial differential equationscited by 1
  24. 2021Numerical Solution of Stiff ODEs with Physics-Informed RPNNscited by 1
  25. 2022Transfer Learning as a Method to Reproduce High-Fidelity NLTE Opacities in Simulationscited by 1
  26. 2023Physics-informed Neural Network Combined with Characteristic-Based Split for Solving Navier-Stokes Equationscited by 1
  27. 2025MILP initialization for solving parabolic PDEs with PINNscited by 1
  28. 2025Fast, Convex and Conditioned Network for Multi-Fidelity Vectors and Stiff Univariate Differential Equationscited by 1
  29. 2025Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEscited by 1
  30. 2025Towards Fast Option Pricing PDE Solvers Powered by PIELMcited by 1
  31. 2025Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equationcited by 1
  32. 2025HEATNETs: Explainable Random Feature Neural Networks for High-Dimensional Parabolic PDEscited by 1
  33. 2021PCNN: A physics-constrained neural network for multiphase flowscited by 0
  34. 2023The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approachcited by 0
  35. 2024Extremization to Fine Tune Physics Informed Neural Networks for Solving Boundary Value Problemscited by 0
  36. 2024Parallel-in-Time Solutions with Random Projection Neural Networkscited by 0
  37. 2025A Neumann-Neumann Acceleration with Coarse Space for Domain Decomposition of Extreme Learning Machinescited by 0
  38. 2025Numerical simulation of transient heat conduction with moving heat source using Physics Informed Neural Networkscited by 0

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