# General Fourier Feature Physics-Informed Extreme Learning Machine (GFF-PIELM) for High-Frequency PDEs

arXiv: https://arxiv.org/abs/2510.12293 · HTML: https://mlsanity.com/paper/2510.12293/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 40 references that are in the map.
- First posted on arXiv 14 Oct 2025.

## Builds on (most cited first, 18 of 18)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [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 networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Training behavior of deep neural network in frequency domain](https://mlsanity.com/paper/1807.01251/) (2018)
- [Towards Understanding the Spectral Bias of Deep Learning](https://mlsanity.com/paper/1912.01198/) (2019)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [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)
- [Learning in Sinusoidal Spaces with Physics-Informed Neural Networks](https://mlsanity.com/paper/2109.09338/) (2021)
- [On the Spectral Bias of Convolutional Neural Tangent and Gaussian Process Kernels](https://mlsanity.com/paper/2203.09255/) (2022)
- [Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2205.06948/) (2022)
- [Curriculum Learning-Driven PIELMs for Fluid Flow Simulations](https://mlsanity.com/paper/2503.06347/) (2025)
- [Physics-informed extreme learning machine for Terzaghi consolidation problems and interpretation of coefficient of consolidation based on CPTu data](https://mlsanity.com/paper/2506.08381/) (2025)
- [Fast, Convex and Conditioned Network for Multi-Fidelity Vectors and Stiff Univariate Differential Equations](https://mlsanity.com/paper/2508.05921/) (2025)

## Built on by (most cited first, 2 of 2)

- [Variational Matrix-Learning Fourier Networks for Parametric Multiphysics Surrogates](https://mlsanity.com/paper/2605.02280/) (2026)
- [Frequency Shift Physics-Informed Extreme Learning Machine for Solving High-Frequency Partial Differential Equations](https://mlsanity.com/paper/2607.01694/) (2026)

## Abstract (arXiv)

Conventional physics-informed extreme learning machine (PIELM) often faces challenges in solving partial differential equations (PDEs) involving high-frequency and variable-frequency behaviors. To address these challenges, we propose a general Fourier feature physics-informed extreme learning machine (GFF-PIELM). We demonstrate that directly concatenating multiple Fourier feature mappings (FFMs) and an extreme learning machine (ELM) network makes it difficult to determine frequency-related hyperparameters. Fortunately, we find an alternative to establish the GFF-PIELM in three main steps. First, we integrate a variation of FFM into ELM as the Fourier-based activation function, so there is still one hidden layer in the GFF-PIELM framework. Second, we assign a set of frequency coefficients to the hidden neurons, which enables ELM network to capture diverse frequency components of target solutions. Finally, we develop an innovative, straightforward initialization method for these hyperparameters by monitoring the distribution of ELM output weights. GFF-PIELM not only retains the high accuracy, efficiency, and simplicity of the PIELM framework but also inherits the ability of FFMs to effectively handle high-frequency problems. We carry out five case studies with a total of ten numerical examples to highlight the feasibility and validity of the proposed GFF-PIELM, involving high frequency, variable frequency, multi-scale behaviour, irregular boundary and inverse problems. Compared to conventional PIELM, the GFF-PIELM approach significantly improves predictive accuracy without additional cost in training time and architecture complexity. Our results confirm that that PIELM can be extended to solve high-frequency and variable-frequency PDEs with high accuracy, and our initialization strategy may further inspire advances in other physics-informed machine learning (PIML) frameworks.