Frequency Shift Physics-Informed Extreme Learning Machine for Solving High-Frequency Partial Differential Equations

# Frequency Shift Physics-Informed Extreme Learning Machine for Solving High-Frequency Partial Differential Equations
Source: [https://arxiv.org/html/2607.01694](https://arxiv.org/html/2607.01694)
Xiong Xiong1,5, Ruonan Zhai2, Zheng Zeng3,5, Sheng Zhou3,5, Rongchun Hu3,5, Zichen Deng1,3,4,5 1\. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, 710072, China; 2\. College of Science, Shijiazhuang University, Shijiazhuang, 050035, China; 3\. Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an, 710072, China; 4\. Department of Aeronautical Engineering, Northwestern Polytechnical University, Xi’an, 710072, China; 5\. MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University, Xi’an, 710072, China

###### Abstract

Solving partial differential equations \(PDEs\) with high\-frequency solutions remains a central challenge in physics\-informed machine learning due to spectral bias—the tendency of neural networks to learn low\-frequency components preferentially\. This paper proposes a Frequency Shift Physics\-Informed Extreme Learning Machine \(FS\-PIELM\) framework that addresses this limitation through an additive mechanism for weight initialization\. Rather than multiplying random weights by a scaling factor, the method translates the mean of the Gaussian weight distribution while keeping the variance fixed at unity, thereby avoiding the variance amplification inherent in scaling\-based methods\. Two variants are developed: FS\-PIELM\-L assigns independent frequency magnitudes to individual neurons, while FS\-PIELM\-G groups neurons for improved robustness\. Theoretical analysis shows that the frequency variance under the proposed framework remains bounded and approaches unity regardless of target frequency, in contrast to the quadratic growth of conventional approaches\. The method preserves the computational efficiency of extreme learning machines, requiring only a single linear solve\. Experiments on seven benchmark problems spanning six equation types—Helmholtz, wave, Poisson, Klein\-Gordon, heat, and advection\-diffusion—on both regular and complex geometries show that the linear variant achieves the best accuracy in six of seven cases, with improvements of one to nearly five orders of magnitude over existing PIELM variants\. The code and data accompanying this manuscript will be made publicly available at[https://github\.com/xgxgnpu/Physics\-informed\-vibe\-coding/tree/main/FS\-PIELM](https://github.com/xgxgnpu/Physics-informed-vibe-coding/tree/main/FS-PIELM)\.

Keywords: Physics\-informed machine learning; Extreme learning machine; High\-frequency PDEs; Spectral bias; Helmholtz equation; Frequency shift

## 1Introduction

The numerical solution of partial differential equations \(PDEs\) underpins much of computational science and engineering\. Traditional mesh\-based methods—finite element, finite difference, and spectral methods\[[1](https://arxiv.org/html/2607.01694#bib.bib1),[2](https://arxiv.org/html/2607.01694#bib.bib2),[3](https://arxiv.org/html/2607.01694#bib.bib3)\]—have matured over decades, yet they can become impractical for complex geometries, high\-dimensional settings\[[4](https://arxiv.org/html/2607.01694#bib.bib4)\], or inverse problems where governing parameters must be inferred from data\[[5](https://arxiv.org/html/2607.01694#bib.bib5)\]\. Physics\-informed machine learning offers an alternative by embedding physical constraints directly into neural network training\[[6](https://arxiv.org/html/2607.01694#bib.bib6),[7](https://arxiv.org/html/2607.01694#bib.bib7),[8](https://arxiv.org/html/2607.01694#bib.bib8),[9](https://arxiv.org/html/2607.01694#bib.bib9),[10](https://arxiv.org/html/2607.01694#bib.bib10),[61](https://arxiv.org/html/2607.01694#bib.bib61)\]\.

Physics\-informed neural networks \(PINNs\)\[[7](https://arxiv.org/html/2607.01694#bib.bib7)\]approximate PDE solutions by training a neural network to satisfy the governing equations and boundary conditions at scattered collocation points\. This mesh\-free formulation handles irregular geometries naturally and can incorporate sparse measurement data\. PINNs have been applied successfully across fluid mechanics\[[11](https://arxiv.org/html/2607.01694#bib.bib11),[12](https://arxiv.org/html/2607.01694#bib.bib12)\], solid mechanics\[[13](https://arxiv.org/html/2607.01694#bib.bib13)\], and multiphysics problems\[[14](https://arxiv.org/html/2607.01694#bib.bib14)\]; see\[[10](https://arxiv.org/html/2607.01694#bib.bib10),[15](https://arxiv.org/html/2607.01694#bib.bib15),[59](https://arxiv.org/html/2607.01694#bib.bib59)\]for recent surveys\. Two practical limitations, however, restrict the applicability of standard PINNs: the gradient\-descent training can be expensive, often requiring hours even for moderately complex problems\[[16](https://arxiv.org/html/2607.01694#bib.bib16),[17](https://arxiv.org/html/2607.01694#bib.bib17)\], and neural networks exhibit*spectral bias*—a systematic tendency to learn low\-frequency components first while high\-frequency features converge much more slowly\[[18](https://arxiv.org/html/2607.01694#bib.bib18),[19](https://arxiv.org/html/2607.01694#bib.bib19),[20](https://arxiv.org/html/2607.01694#bib.bib20),[21](https://arxiv.org/html/2607.01694#bib.bib21),[65](https://arxiv.org/html/2607.01694#bib.bib65),[66](https://arxiv.org/html/2607.01694#bib.bib66)\]\.

The spectral bias phenomenon has been rigorously analyzed within the framework of neural tangent kernel \(NTK\) theory\[[22](https://arxiv.org/html/2607.01694#bib.bib22)\]by Wang et al\.\[[23](https://arxiv.org/html/2607.01694#bib.bib23)\], who demonstrated that the eigenvalues of the NTK associated with high\-frequency eigenfunctions are substantially smaller than those corresponding to low\-frequency modes\. This eigenvalue gap causes the network to converge exponentially slower for high\-frequency components, resulting in poor approximation accuracy for PDEs whose solutions exhibit rapid oscillations or multi\-scale behavior\[[24](https://arxiv.org/html/2607.01694#bib.bib24)\]\. Several strategies have been proposed to mitigate spectral bias in physics\-informed learning\[[25](https://arxiv.org/html/2607.01694#bib.bib25),[26](https://arxiv.org/html/2607.01694#bib.bib26),[62](https://arxiv.org/html/2607.01694#bib.bib62)\]\. Among the proposed remedies, Fourier feature mappings\[[27](https://arxiv.org/html/2607.01694#bib.bib27),[28](https://arxiv.org/html/2607.01694#bib.bib28)\]lift the input coordinates into a high\-dimensional sinusoidal space, enabling the network to represent rapid oscillations, while SIREN networks\[[29](https://arxiv.org/html/2607.01694#bib.bib29)\]achieve a similar effect through periodic activation functions\. Both families of methods, however, introduce frequency\-scale hyperparameters that require problem\-specific calibration and may fail when the solution spans a wide spectral range\. More recent remedies pursue adaptive spectral architectures, including separated\-variable spectral networks \(SV\-SNN\) with characteristic\-frequency multi\-level initialization\[[31](https://arxiv.org/html/2607.01694#bib.bib31)\], hierarchical adaptive Fourier feature encodings that allocate distinct frequency scales\[[30](https://arxiv.org/html/2607.01694#bib.bib30)\], and physics\-informed Kolmogorov–Arnold networks built on Jacobi orthogonal polynomial bases\[[32](https://arxiv.org/html/2607.01694#bib.bib32)\]\.

An alternative approach to improving the efficiency of physics\-informed learning is to replace deep neural networks with extreme learning machines \(ELMs\)\[[33](https://arxiv.org/html/2607.01694#bib.bib33),[34](https://arxiv.org/html/2607.01694#bib.bib34),[63](https://arxiv.org/html/2607.01694#bib.bib63),[64](https://arxiv.org/html/2607.01694#bib.bib64)\]\. The physics\-informed extreme learning machine \(PIELM\), developed by Dwivedi and Srinivasan\[[35](https://arxiv.org/html/2607.01694#bib.bib35)\], leverages the ELM architecture consisting of a single hidden layer with randomly initialized and fixed input weights\[[36](https://arxiv.org/html/2607.01694#bib.bib36),[37](https://arxiv.org/html/2607.01694#bib.bib37),[38](https://arxiv.org/html/2607.01694#bib.bib38)\], where only the output layer weights are determined analytically through least\-squares solution\. This formulation transforms the training process from iterative gradient descent to a single matrix inversion, reducing computational time by several orders of magnitude\[[35](https://arxiv.org/html/2607.01694#bib.bib35)\]while often achieving comparable or superior accuracy for smooth problems\. The PIELM framework has since been extended to sharp\-gradient elliptic problems\[[39](https://arxiv.org/html/2607.01694#bib.bib39)\], biharmonic equations\[[40](https://arxiv.org/html/2607.01694#bib.bib40)\], nonlinear PDEs\[[41](https://arxiv.org/html/2607.01694#bib.bib41)\], Stefan problems\[[42](https://arxiv.org/html/2607.01694#bib.bib42)\], fracture mechanics\[[43](https://arxiv.org/html/2607.01694#bib.bib43)\], domain decomposition\[[44](https://arxiv.org/html/2607.01694#bib.bib44),[57](https://arxiv.org/html/2607.01694#bib.bib57)\], hidden\-layer concatenation\[[45](https://arxiv.org/html/2607.01694#bib.bib45)\], and functional\-connection architectures\[[46](https://arxiv.org/html/2607.01694#bib.bib46),[47](https://arxiv.org/html/2607.01694#bib.bib47),[48](https://arxiv.org/html/2607.01694#bib.bib48)\]; see\[[49](https://arxiv.org/html/2607.01694#bib.bib49)\]for a comprehensive review\. A related line of inquiry concerns random feature methods for PDEs, where approximation\-theoretic foundations\[[55](https://arxiv.org/html/2607.01694#bib.bib55)\], high\-precision optimization\[[52](https://arxiv.org/html/2607.01694#bib.bib52)\], and extensions to time\-dependent\[[51](https://arxiv.org/html/2607.01694#bib.bib51)\], multiscale\[[56](https://arxiv.org/html/2607.01694#bib.bib56),[53](https://arxiv.org/html/2607.01694#bib.bib53),[58](https://arxiv.org/html/2607.01694#bib.bib58)\], and complex\-domain settings\[[54](https://arxiv.org/html/2607.01694#bib.bib54)\]have been developed\.

Despite these computational advantages, the conventional PIELM framework remains subject to spectral bias\. Because the input weights are drawn from a standard Gaussian distribution, the resulting basis functions concentrate their spectral energy near zero frequency, making it difficult to resolve highly oscillatory PDE solutions\. To address this limitation, Ren et al\.\[[50](https://arxiv.org/html/2607.01694#bib.bib50)\]recently proposed the general Fourier feature PIELM \(GFF\-PIELM\), which employs a cosine activation and multiplies the random pre\-activation by a linearly varying frequency scaling factor\. This multiplicative approach enables representation of higher frequencies while preserving ELM efficiency, but introduces an inherent coupling between the target frequency and the weight variance that limits precision at high wavenumbers, as we demonstrate formally in Section[3\.4](https://arxiv.org/html/2607.01694#S3.SS4)\.

We introduce a Frequency Shift Physics\-Informed Extreme Learning Machine \(FS\-PIELM\) that addresses spectral bias through a fundamentally different mechanism\. Instead of multiplying the network inputs by a scaling factor as in GFF\-PIELM, we shift the mean of the Gaussian distribution from which hidden layer weights are drawn\. Each neuron is assigned a random direction on the unit sphere and a scalar mean magnitude that varies linearly across the neuron ensemble, so that different neurons target different frequency bands\. Because only the mean is translated while the covariance matrix remains the identity, the variance of the effective frequency stays bounded regardless of the target frequency—in contrast to the quadratic variance growth inherent in scaling\-based methods\. Two variants are developed: FS\-PIELM\-L \(Linear\) assigns an independent mean magnitude to each neuron for fine frequency resolution, while FS\-PIELM\-G \(Grouped\) partitions neurons into groups sharing common mean magnitudes for improved stability\.

In summary, the contributions of this work are threefold\. First, we propose a frequency shift mechanism that controls the spectral properties of a single\-hidden\-layer ELM through the mean of the weight sampling distribution, rather than through input scaling, and we prove that the resulting frequency variance remains bounded regardless of target frequency\. Second, we develop two complementary variants—FS\-PIELM\-L for fine\-grained frequency resolution and FS\-PIELM\-G for improved robustness—and evaluate them on seven benchmark problems spanning elliptic, parabolic, and hyperbolic PDEs with both rectangular and irregular geometries\. Third, systematic parameter studies show that FS\-PIELM\-L achieves the best accuracy in six of seven cases, outperforming SIREN\-PIELM and GFF\-PIELM by one to nearly five orders of magnitude\.

The remainder of this paper is organized as follows\. Section[2](https://arxiv.org/html/2607.01694#S2)reviews PINNs, spectral bias, and the ELM framework including existing frequency\-enhanced PIELM variants\. Section[3](https://arxiv.org/html/2607.01694#S3)presents the FS\-PIELM framework: the core frequency shift mechanism, two architectural variants, and theoretical analysis\. Section[4](https://arxiv.org/html/2607.01694#S4)reports numerical experiments on seven benchmarks with detailed comparisons\. Section[5](https://arxiv.org/html/2607.01694#S5)summarizes findings and discusses limitations and future directions\.

## 2Background and Preliminaries

This section reviews the mathematical framework for physics\-informed learning and the existing approaches that motivate our work\. We first formulate the general problem and introduce the physics\-informed neural network paradigm along with its spectral bias limitation, and then present the extreme learning machine framework that forms the foundation of our approach\.

### 2\.1Physics\-Informed Learning and Spectral Bias

Consider a general partial differential equation \(PDE\) defined on a bounded domainΩ⊂ℝd\\Omega\\subset\\mathbb\{R\}^\{d\}with boundary∂Ω\\partial\\Omega:

𝒟​\[u​\(𝐱\)\]\\displaystyle\\mathcal\{D\}\[u\(\\mathbf\{x\}\)\]=f​\(𝐱\),𝐱∈Ω,\\displaystyle=f\(\\mathbf\{x\}\),\\quad\\mathbf\{x\}\\in\\Omega,\(2\.1\)ℬ​\[u​\(𝐱\)\]\\displaystyle\\mathcal\{B\}\[u\(\\mathbf\{x\}\)\]=g​\(𝐱\),𝐱∈∂Ω,\\displaystyle=g\(\\mathbf\{x\}\),\\quad\\mathbf\{x\}\\in\\partial\\Omega,\(2\.2\)whereu:Ω→ℝu:\\Omega\\to\\mathbb\{R\}denotes the unknown solution,𝒟\\mathcal\{D\}represents the differential operator characterizing the physical law,ℬ\\mathcal\{B\}denotes the boundary condition operator, andffandggare prescribed source and boundary data functions, respectively\. Throughout this paper, we adopt the following notation:𝐱=\(x1,…,xd\)T\\mathbf\{x\}=\(x\_\{1\},\\ldots,x\_\{d\}\)^\{\\mathrm\{T\}\}denotes the spatial coordinate vector,∥⋅∥\\\|\\cdot\\\|represents the Euclidean norm unless otherwise specified,𝒩​\(𝝁,𝚺\)\\mathcal\{N\}\(\\bm\{\\mu\},\\bm\{\\Sigma\}\)denotes the multivariate Gaussian distribution with mean𝝁\\bm\{\\mu\}and covariance𝚺\\bm\{\\Sigma\}, and𝒰​\(a,b\)\\mathcal\{U\}\(a,b\)denotes the uniform distribution on the interval\[a,b\]\[a,b\]\.

Physics\-informed neural networks \(PINNs\), introduced by Raissi et al\.\[[7](https://arxiv.org/html/2607.01694#bib.bib7)\], approximate the solutionu​\(𝐱\)u\(\\mathbf\{x\}\)using a deep neural networkuθ​\(𝐱\)u\_\{\\theta\}\(\\mathbf\{x\}\)parameterized by weights and biasesθ\\theta, optimizing a composite loss function that penalizes violations of both the governing equations and boundary conditions:

ℒ​\(θ\)=1NC​∑j=1NC\|𝒟​\[uθ​\(𝐱jC\)\]−f​\(𝐱jC\)\|2\+λNB​∑k=1NB\|ℬ​\[uθ​\(𝐱kB\)\]−g​\(𝐱kB\)\|2,\\mathcal\{L\}\(\\theta\)=\\frac\{1\}\{N\_\{C\}\}\\sum\_\{j=1\}^\{N\_\{C\}\}\\left\|\\mathcal\{D\}\[u\_\{\\theta\}\(\\mathbf\{x\}\_\{j\}^\{C\}\)\]\-f\(\\mathbf\{x\}\_\{j\}^\{C\}\)\\right\|^\{2\}\+\\frac\{\\lambda\}\{N\_\{B\}\}\\sum\_\{k=1\}^\{N\_\{B\}\}\\left\|\\mathcal\{B\}\[u\_\{\\theta\}\(\\mathbf\{x\}\_\{k\}^\{B\}\)\]\-g\(\\mathbf\{x\}\_\{k\}^\{B\}\)\\right\|^\{2\},\(2\.3\)where\{𝐱jC\}j=1NC\\\{\\mathbf\{x\}\_\{j\}^\{C\}\\\}\_\{j=1\}^\{N\_\{C\}\}are interior collocation points,\{𝐱kB\}k=1NB\\\{\\mathbf\{x\}\_\{k\}^\{B\}\\\}\_\{k=1\}^\{N\_\{B\}\}are boundary points, andλ\\lambdais a weighting parameter\. Derivatives are computed via automatic differentiation, and the mesh\-free nature provides advantages for complex geometries\[[60](https://arxiv.org/html/2607.01694#bib.bib60)\], high\-dimensional spaces, or inverse problems\[[10](https://arxiv.org/html/2607.01694#bib.bib10),[15](https://arxiv.org/html/2607.01694#bib.bib15)\]\.

However, PINNs suffer from*spectral bias*: neural networks preferentially learn low\-frequency components while high\-frequency features converge much more slowly\[[18](https://arxiv.org/html/2607.01694#bib.bib18),[19](https://arxiv.org/html/2607.01694#bib.bib19),[20](https://arxiv.org/html/2607.01694#bib.bib20),[21](https://arxiv.org/html/2607.01694#bib.bib21)\]\. This phenomenon has been rigorously analyzed through the neural tangent kernel \(NTK\) framework by Wang et al\.\[[23](https://arxiv.org/html/2607.01694#bib.bib23)\], who demonstrated that the NTK eigenvalue spectrum decays rapidly with increasing frequency, creating an exponentially growing gap in convergence rates between frequency modes\. Consequently, for PDEs with rapid oscillations or multi\-scale behavior, standard PINNs require prohibitively long training times and often fail to achieve acceptable accuracy\. Various remedies have been proposed, including Fourier feature mappings\[[27](https://arxiv.org/html/2607.01694#bib.bib27),[28](https://arxiv.org/html/2607.01694#bib.bib28)\]and SIREN networks with periodic activations\[[29](https://arxiv.org/html/2607.01694#bib.bib29)\], but these approaches introduce additional hyperparameters requiring careful tuning and may not generalize across problem classes\.

### 2\.2Extreme Learning Machine Framework

The physics\-informed extreme learning machine \(PIELM\), developed by Dwivedi and Srinivasan\[[35](https://arxiv.org/html/2607.01694#bib.bib35)\], addresses the computational efficiency limitations of PINNs while preserving their mesh\-free character\. The extreme learning machine \(ELM\) architecture consists of a single hidden layer feedforward network where the input weights𝐖=\[𝐰1,…,𝐰M\]∈ℝd×M\\mathbf\{W\}=\[\\mathbf\{w\}\_\{1\},\\ldots,\\mathbf\{w\}\_\{M\}\]\\in\\mathbb\{R\}^\{d\\times M\}and biases𝐛=\[b1,…,bM\]T\\mathbf\{b\}=\[b\_\{1\},\\ldots,b\_\{M\}\]^\{\\mathrm\{T\}\}are randomly initialized and remain fixed, while only the output weights𝜷=\[β1,…,βM\]T\\bm\{\\beta\}=\[\\beta\_\{1\},\\ldots,\\beta\_\{M\}\]^\{\\mathrm\{T\}\}are learned\[[33](https://arxiv.org/html/2607.01694#bib.bib33)\]\. The network output is:

u​\(𝐱;𝜷\)=∑m=1Mβm​ϕ​\(𝐰mT​𝐱\+bm\)=ϕ​\(𝐱\)T​𝜷,u\(\\mathbf\{x\};\\bm\{\\beta\}\)=\\sum\_\{m=1\}^\{M\}\\beta\_\{m\}\\phi\\left\(\\mathbf\{w\}\_\{m\}^\{\\mathrm\{T\}\}\\mathbf\{x\}\+b\_\{m\}\\right\)=\\bm\{\\phi\}\(\\mathbf\{x\}\)^\{\\mathrm\{T\}\}\\bm\{\\beta\},\(2\.4\)whereMMdenotes the number of hidden neurons andϕ:ℝ→ℝ\\phi:\\mathbb\{R\}\\to\\mathbb\{R\}is the activation function\. The ELM formulation transforms the nonlinear training problem into a linear least\-squares problem\. For linear PDEs, the differential operator can be applied analytically to Eq\. \([2\.4](https://arxiv.org/html/2607.01694#S2.E4)\), and evaluating the PDE residual and boundary conditions at collocation points produces the linear system𝐇​𝜷=𝐘\\mathbf\{H\}\\bm\{\\beta\}=\\mathbf\{Y\}, where𝐇∈ℝ\(NC\+NB\)×M\\mathbf\{H\}\\in\\mathbb\{R\}^\{\(N\_\{C\}\+N\_\{B\}\)\\times M\}is the collocation matrix\. The optimal output weights are obtained via the Moore\-Penrose pseudoinverse:

𝜷∗=\(𝐇T​𝐇\)−1​𝐇T​𝐘,\\bm\{\\beta\}^\{\*\}=\(\\mathbf\{H\}^\{\\mathrm\{T\}\}\\mathbf\{H\}\)^\{\-1\}\\mathbf\{H\}^\{\\mathrm\{T\}\}\\mathbf\{Y\},\(2\.5\)which can be computed efficiently using singular value decomposition \(SVD\)\. This closed\-form solution reduces computational time by several orders of magnitude compared to iterative PINN training\[[35](https://arxiv.org/html/2607.01694#bib.bib35)\]\.

Several frequency\-enhanced PIELM variants have been developed to address spectral bias within the ELM framework by introducing frequency\-controlling scaling factorsδm\\delta\_\{m\}\. The general form of existing scaling\-based methods is:

hm​\(𝐱\)=σ​\(δm​\(𝐰mT​𝐱\+bm\)\),𝐰m∼𝒩​\(𝟎,𝐈d\),h\_\{m\}\(\\mathbf\{x\}\)=\\sigma\\left\(\\delta\_\{m\}\(\\mathbf\{w\}\_\{m\}^\{\\mathrm\{T\}\}\\mathbf\{x\}\+b\_\{m\}\)\\right\),\\quad\\mathbf\{w\}\_\{m\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{d\}\),\(2\.6\)whereσ\\sigmadenotes the activation function andδm\\delta\_\{m\}varies linearly fromδmin\\delta\_\{\\min\}toδmax\\delta\_\{\\max\}across neurons\. Specific instantiations include Tanh\-PIELM with hyperbolic tangent activationhm​\(𝐱\)=tanh⁡\(δm​\(𝐰mT​𝐱\+bm\)\)h\_\{m\}\(\\mathbf\{x\}\)=\\tanh\(\\delta\_\{m\}\(\\mathbf\{w\}\_\{m\}^\{\\mathrm\{T\}\}\\mathbf\{x\}\+b\_\{m\}\)\), SIREN\-PIELM with sinusoidal activationhm​\(𝐱\)=sin⁡\(δm​\(𝐰mT​𝐱\+bm\)\)h\_\{m\}\(\\mathbf\{x\}\)=\\sin\(\\delta\_\{m\}\(\\mathbf\{w\}\_\{m\}^\{\\mathrm\{T\}\}\\mathbf\{x\}\+b\_\{m\}\)\)inspired by implicit neural representations\[[29](https://arxiv.org/html/2607.01694#bib.bib29)\], and GFF\-PIELM with cosine activationhm​\(𝐱\)=cos⁡\(δm​\(𝐰mT​𝐱\+bm\)\)h\_\{m\}\(\\mathbf\{x\}\)=\\cos\(\\delta\_\{m\}\(\\mathbf\{w\}\_\{m\}^\{\\mathrm\{T\}\}\\mathbf\{x\}\+b\_\{m\}\)\)integrating Fourier feature mappings\[[50](https://arxiv.org/html/2607.01694#bib.bib50)\]\. All these methods control the effective frequency through the productδm​‖ϵm‖\\delta\_\{m\}\\\|\\bm\{\\epsilon\}\_\{m\}\\\|, whereϵm\\bm\{\\epsilon\}\_\{m\}denotes the randomly drawn base weight vector\. While this scaling approach enables representation of higher frequencies, it couples frequency control with random weight magnitude, introducing a fundamental limitation:*frequency variance grows quadratically with the scaling factor*, as we shall demonstrate rigorously in Section[3](https://arxiv.org/html/2607.01694#S3)\.

## 3Frequency Shift Physics\-Informed ELM

This section develops the Frequency Shift PIELM \(FS\-PIELM\) framework, proceeding from the core sampling mechanism through the network architecture to a theoretical comparison with multiplicative scaling\.

### 3\.1Core Principle: Frequency Shift via Mean Translation

FS\-PIELM controls the effective frequency of basis functions through*additive mean shifting*rather than*multiplicative scaling*\. We propose the followingcore formulafor generating hidden layer weights:

𝐰m=𝝁m\+ϵm,ϵm∼𝒩​\(𝟎,𝐈d\)\\boxed\{\\mathbf\{w\}\_\{m\}=\\bm\{\\mu\}\_\{m\}\+\\bm\{\\epsilon\}\_\{m\},\\quad\\bm\{\\epsilon\}\_\{m\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{d\}\)\}\(3\.1\)where𝝁m∈ℝd\\bm\{\\mu\}\_\{m\}\\in\\mathbb\{R\}^\{d\}is the*mean shift vector*that determines the target frequency, andϵm\\bm\{\\epsilon\}\_\{m\}is a standard Gaussian random vector providing stochastic perturbation\. The key insight is that*only the mean is shifted while the variance remains constant at𝐈d\\mathbf\{I\}\_\{d\}*\.

This additive formulation stands in contrast to scaling\-based methods\. In the multiplicative scaling approach,𝐰m=δm⋅ϵm\\mathbf\{w\}\_\{m\}=\\delta\_\{m\}\\cdot\\bm\{\\epsilon\}\_\{m\}withϵm∼𝒩​\(𝟎,𝐈d\)\\bm\{\\epsilon\}\_\{m\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{d\}\), so that

𝐰m∼𝒩​\(𝟎,δm2​𝐈d\),variance scales as​δm2\.\\mathbf\{w\}\_\{m\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\delta\_\{m\}^\{2\}\\mathbf\{I\}\_\{d\}\),\\quad\\text\{variance scales as \}\\delta\_\{m\}^\{2\}\.\(3\.2\)In the proposed frequency shift approach,𝐰m=𝝁m\+ϵm\\mathbf\{w\}\_\{m\}=\\bm\{\\mu\}\_\{m\}\+\\bm\{\\epsilon\}\_\{m\}withϵm∼𝒩​\(𝟎,𝐈d\)\\bm\{\\epsilon\}\_\{m\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{d\}\), yielding

𝐰m∼𝒩​\(𝝁m,𝐈d\),variance remains constant at​𝐈d\.\\mathbf\{w\}\_\{m\}\\sim\\mathcal\{N\}\(\\bm\{\\mu\}\_\{m\},\\mathbf\{I\}\_\{d\}\),\\quad\\text\{variance remains constant at \}\\mathbf\{I\}\_\{d\}\.\(3\.3\)This distinction is central to the advantages demonstrated in Section[3\.4](https://arxiv.org/html/2607.01694#S3.SS4)\.

![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/mechanism.png)Figure 1:Comparison of frequency control mechanisms:\(Left\) Scaling\-based approach where variance grows quadratically with the scaling factor\. \(Right\) Frequency shift approach where variance remains constant regardless of target frequency\. The additive mean\-shifting preserves the Gaussian perturbation structure while enabling controlled frequency targeting\.###### Definition 3\.1\(Frequency Shift Sampling\)\.

The frequency shift mechanism generates hidden layer weights through the following three\-step procedure:

1. \(i\)Random direction sampling: Draw𝐝~m∼𝒩​\(𝟎,𝐈d\)\\tilde\{\\mathbf\{d\}\}\_\{m\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{d\}\)and normalize to obtain a uniformly distributed direction on the unit sphere: 𝐝m=𝐝~m‖𝐝~m‖∈𝕊d−1\.\\mathbf\{d\}\_\{m\}=\\frac\{\\tilde\{\\mathbf\{d\}\}\_\{m\}\}\{\\\|\\tilde\{\\mathbf\{d\}\}\_\{m\}\\\|\}\\in\\mathbb\{S\}^\{d\-1\}\.\(3\.4\)
2. \(ii\)Mean shift construction: Construct the mean shift vector by scaling the random direction with the frequency magnitude parameterμm≥0\\mu\_\{m\}\\geq 0: 𝝁m=μm​𝐝m\.\\bm\{\\mu\}\_\{m\}=\\mu\_\{m\}\\mathbf\{d\}\_\{m\}\.\(3\.5\)
3. \(iii\)Weight generation: Generate the final weight vector by adding the mean shift to a standard Gaussian sample: 𝐰m=𝝁m\+ϵm=μm​𝐝m\+ϵm,ϵm∼𝒩​\(𝟎,𝐈d\)\.\\mathbf\{w\}\_\{m\}=\\bm\{\\mu\}\_\{m\}\+\\bm\{\\epsilon\}\_\{m\}=\\mu\_\{m\}\\mathbf\{d\}\_\{m\}\+\\bm\{\\epsilon\}\_\{m\},\\quad\\bm\{\\epsilon\}\_\{m\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{d\}\)\.\(3\.6\)

The bias term is sampled uniformly asbm∼𝒰​\(0,2​π\)b\_\{m\}\\sim\\mathcal\{U\}\(0,2\\pi\), and the resulting basis function takes the formhm​\(𝐱\)=cos⁡\(𝐰mT​𝐱\+bm\)h\_\{m\}\(\\mathbf\{x\}\)=\\cos\(\\mathbf\{w\}\_\{m\}^\{\\mathrm\{T\}\}\\mathbf\{x\}\+b\_\{m\}\)\.

We highlight three properties of this construction\. First, the random direction𝐝m\\mathbf\{d\}\_\{m\}is uniformly distributed on𝕊d−1\\mathbb\{S\}^\{d\-1\}, ensuring isotropy so that no spatial direction is preferentially favored\. Second, the parameterμm\\mu\_\{m\}directly determines the expected effective frequency𝔼​\[‖𝐰m‖\]≈μm2\+d\\mathbb\{E\}\[\\\|\\mathbf\{w\}\_\{m\}\\\|\]\\approx\\sqrt\{\\mu\_\{m\}^\{2\}\+d\}, providing intuitive frequency targeting\. Third, the variance of‖𝐰m‖\\\|\\mathbf\{w\}\_\{m\}\\\|stays bounded asμm\\mu\_\{m\}grows \(Theorem[3\.5](https://arxiv.org/html/2607.01694#S3.Thmtheorem5)\)\.

### 3\.2Network Architecture

Building upon the core frequency shift principle, we develop two architectural variants that provide different trade\-offs between frequency resolution and robustness\.

![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/FS-PIELM-framework.png)Figure 2:FS\-PIELM framework:Schematic illustration of the Frequency Shift Physics\-Informed Extreme Learning Machine architecture\. The hidden layer weights are generated through the frequency shift mechanism with linearly distributed mean magnitudes, enabling adaptive spectral coverage for high\-frequency PDE solutions\.#### 3\.2\.1Linear Variant \(FS\-PIELM\-L\)

The linear variant assigns an independent frequency magnitude to each of theMMneurons, with magnitudes distributed linearly across the target frequency range:

μm=μmin\+m−1M−1​\(μmax−μmin\),m=1,2,…,M\.\\mu\_\{m\}=\\mu\_\{\\min\}\+\\frac\{m\-1\}\{M\-1\}\(\\mu\_\{\\max\}\-\\mu\_\{\\min\}\),\\quad m=1,2,\\ldots,M\.\(3\.7\)This provides maximum granularity in covering the frequency spectrum\[μmin,μmax\]\[\\mu\_\{\\min\},\\mu\_\{\\max\}\], making it well\-suited for problems requiring fine frequency resolution\.

#### 3\.2\.2Grouped Variant \(FS\-PIELM\-G\)

The grouped variant partitions theMMneurons intoKKgroups, where neurons within each group share a common frequency magnitude:

μk=μmin\+k−1K−1​\(μmax−μmin\),k=1,2,…,K\.\\mu\_\{k\}=\\mu\_\{\\min\}\+\\frac\{k\-1\}\{K\-1\}\(\\mu\_\{\\max\}\-\\mu\_\{\\min\}\),\\quad k=1,2,\\ldots,K\.\(3\.8\)Each neuron in groupkkusesμk\\mu\_\{k\}but receives an independent random direction𝐝m\\mathbf\{d\}\_\{m\}and Gaussian perturbationϵm\\bm\{\\epsilon\}\_\{m\}\. The redundancy within frequency bands improves robustness against unfavorable random realizations at the cost of coarser frequency resolution\.

#### 3\.2\.3Derivative Computation

For PDE applications, efficient computation of basis function derivatives is essential\. The frequency shift mechanism yields clean derivative expressions without multiplicative scaling factors\. For the cosine basishm​\(𝐱\)=cos⁡\(𝐰mT​𝐱\+bm\)h\_\{m\}\(\\mathbf\{x\}\)=\\cos\(\\mathbf\{w\}\_\{m\}^\{\\mathrm\{T\}\}\\mathbf\{x\}\+b\_\{m\}\):

∂hm∂xi\\displaystyle\\frac\{\\partial h\_\{m\}\}\{\\partial x\_\{i\}\}=−wm,i​sin⁡\(𝐰mT​𝐱\+bm\),\\displaystyle=\-w\_\{m,i\}\\sin\(\\mathbf\{w\}\_\{m\}^\{\\mathrm\{T\}\}\\mathbf\{x\}\+b\_\{m\}\),\(3\.9\)∂2hm∂xi2\\displaystyle\\frac\{\\partial^\{2\}h\_\{m\}\}\{\\partial x\_\{i\}^\{2\}\}=−wm,i2​cos⁡\(𝐰mT​𝐱\+bm\),\\displaystyle=\-w\_\{m,i\}^\{2\}\\cos\(\\mathbf\{w\}\_\{m\}^\{\\mathrm\{T\}\}\\mathbf\{x\}\+b\_\{m\}\),\(3\.10\)Δ​hm\\displaystyle\\Delta h\_\{m\}=−‖𝐰m‖2​cos⁡\(𝐰mT​𝐱\+bm\)\.\\displaystyle=\-\\\|\\mathbf\{w\}\_\{m\}\\\|^\{2\}\\cos\(\\mathbf\{w\}\_\{m\}^\{\\mathrm\{T\}\}\\mathbf\{x\}\+b\_\{m\}\)\.\(3\.11\)Since𝐰m=𝝁m\+ϵm\\mathbf\{w\}\_\{m\}=\\bm\{\\mu\}\_\{m\}\+\\bm\{\\epsilon\}\_\{m\}with unit\-variance perturbation, the weight magnitudes remain well\-controlled even for high target frequencies, avoiding the numerical amplification issues inherent in multiplicative scaling\.

### 3\.3Algorithm and Implementation

The complete FS\-PIELM training procedure is summarized in Algorithm[1](https://arxiv.org/html/2607.01694#alg1)\. The method requires only a single linear solve, maintaining the computational efficiency characteristic of the ELM framework\.

Algorithm 1Frequency Shift Physics\-Informed Extreme Learning Machine \(FS\-PIELM\)1:PDE operator

𝒟\\mathcal\{D\}; boundary operator

ℬ\\mathcal\{B\}; domain

Ω\\Omega; source

ff; boundary data

gg
2:Frequency bounds

μmin\\mu\_\{\\min\},

μmax\\mu\_\{\\max\}; neurons

MM; collocation points

NCN\_\{C\},

NBN\_\{B\}; variant \(Linear or Grouped with

KKgroups\)

3:Trained output weights

𝜷∗\\bm\{\\beta\}^\{\*\}
4:Step 1: Compute frequency magnitudes

5:ifLinear variantthen

6:

μm←μmin\+m−1M−1​\(μmax−μmin\)\\mu\_\{m\}\\leftarrow\\mu\_\{\\min\}\+\\frac\{m\-1\}\{M\-1\}\(\\mu\_\{\\max\}\-\\mu\_\{\\min\}\)for

m=1,…,Mm=1,\\ldots,M
7:else

8:

μk←μmin\+k−1K−1​\(μmax−μmin\)\\mu\_\{k\}\\leftarrow\\mu\_\{\\min\}\+\\frac\{k\-1\}\{K\-1\}\(\\mu\_\{\\max\}\-\\mu\_\{\\min\}\)for

k=1,…,Kk=1,\\ldots,K; assign

μm←μ⌈m​K/M⌉\\mu\_\{m\}\\leftarrow\\mu\_\{\\lceil mK/M\\rceil\}
9:endif

10:Step 2: Generate frequency\-shifted weights via core formula \([3\.1](https://arxiv.org/html/2607.01694#S3.E1)\)

11:for

m=1m=1to

MMdo

12:Sample direction:

𝐝~m∼𝒩​\(𝟎,𝐈d\)\\tilde\{\\mathbf\{d\}\}\_\{m\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{d\}\);

𝐝m←𝐝~m/‖𝐝~m‖\\mathbf\{d\}\_\{m\}\\leftarrow\\tilde\{\\mathbf\{d\}\}\_\{m\}/\\\|\\tilde\{\\mathbf\{d\}\}\_\{m\}\\\|
13:Construct mean shift:

𝝁m←μm​𝐝m\\bm\{\\mu\}\_\{m\}\\leftarrow\\mu\_\{m\}\\mathbf\{d\}\_\{m\}
14:Generate weight:

𝐰m←𝝁m\+ϵm\\mathbf\{w\}\_\{m\}\\leftarrow\\bm\{\\mu\}\_\{m\}\+\\bm\{\\epsilon\}\_\{m\}with

ϵm∼𝒩​\(𝟎,𝐈d\)\\bm\{\\epsilon\}\_\{m\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{d\}\);

bm∼𝒰​\(0,2​π\)b\_\{m\}\\sim\\mathcal\{U\}\(0,2\\pi\)
15:endfor

16:Step 3: Generate collocation points

17:Sample

\{𝐱jC\}j=1NC\\\{\\mathbf\{x\}\_\{j\}^\{C\}\\\}\_\{j=1\}^\{N\_\{C\}\}uniformly in

Ω\\Omega; sample

\{𝐱kB\}k=1NB\\\{\\mathbf\{x\}\_\{k\}^\{B\}\\\}\_\{k=1\}^\{N\_\{B\}\}on

∂Ω\\partial\\Omega
18:Step 4: Assemble collocation matrix

19:

HPDE​\[j,m\]←𝒟​\[cos⁡\(𝐰mT​𝐱jC\+bm\)\]H\_\{\\text\{PDE\}\}\[j,m\]\\leftarrow\\mathcal\{D\}\[\\cos\(\\mathbf\{w\}\_\{m\}^\{\\mathrm\{T\}\}\\mathbf\{x\}\_\{j\}^\{C\}\+b\_\{m\}\)\]for

j=1,…,NCj=1,\\ldots,N\_\{C\},

m=1,…,Mm=1,\\ldots,M
20:

HBC​\[k,m\]←cos⁡\(𝐰mT​𝐱kB\+bm\)H\_\{\\text\{BC\}\}\[k,m\]\\leftarrow\\cos\(\\mathbf\{w\}\_\{m\}^\{\\mathrm\{T\}\}\\mathbf\{x\}\_\{k\}^\{B\}\+b\_\{m\}\)for

k=1,…,NBk=1,\\ldots,N\_\{B\},

m=1,…,Mm=1,\\ldots,M
21:Form

𝐇←\[𝐇PDE;𝐇BC\]\\mathbf\{H\}\\leftarrow\[\\mathbf\{H\}\_\{\\text\{PDE\}\};\\mathbf\{H\}\_\{\\text\{BC\}\}\],

𝐘←\[f​\(𝐱jC\);g​\(𝐱kB\)\]T\\mathbf\{Y\}\\leftarrow\[f\(\\mathbf\{x\}\_\{j\}^\{C\}\);g\(\\mathbf\{x\}\_\{k\}^\{B\}\)\]^\{\\mathrm\{T\}\}
22:Step 5: Solve via truncated SVD

23:Compute SVD:

𝐇=𝐔​𝚺​𝐕T\\mathbf\{H\}=\\mathbf\{U\}\\bm\{\\Sigma\}\\mathbf\{V\}^\{\\mathrm\{T\}\}; set

τ←10−12⋅σmax\\tau\\leftarrow 10^\{\-12\}\\cdot\\sigma\_\{\\max\}
24:

𝜷∗←𝐕​𝚺\+​𝐔T​𝐘\\bm\{\\beta\}^\{\*\}\\leftarrow\\mathbf\{V\}\\bm\{\\Sigma\}^\{\+\}\\mathbf\{U\}^\{\\mathrm\{T\}\}\\mathbf\{Y\}with regularized pseudoinverse

25:return

𝜷∗\\bm\{\\beta\}^\{\*\}

###### Proposition 1\(Computational Complexity\)\.

The computational complexity of FS\-PIELM is: \(i\) matrix assembly:𝒪​\(\(NC\+NB\)⋅M⋅C𝒟\)\\mathcal\{O\}\(\(N\_\{C\}\+N\_\{B\}\)\\cdot M\\cdot C\_\{\\mathcal\{D\}\}\), whereC𝒟C\_\{\\mathcal\{D\}\}is the cost of evaluating the PDE operator; \(ii\) SVD decomposition:𝒪\(min\(N,M\)2⋅max\(N,M\)\)\\mathcal\{O\}\(\\min\(N,M\)^\{2\}\\cdot\\max\(N,M\)\)withN=NC\+NBN=N\_\{C\}\+N\_\{B\}\. The frequency shift mechanism introduces no additional asymptotic overhead compared to conventional scaling approaches\.

### 3\.4Theoretical Analysis

We now establish rigorous theoretical foundations that demonstrate the advantages of the frequency shift mechanism over scaling\-based approaches\. The analysis proceeds in three parts: we first formalize the concept of effective frequency, then analyze the statistical properties of both approaches, and finally provide a quantitative comparison\.

#### 3\.4\.1Effective Frequency

###### Definition 3\.2\(Effective Frequency\)\.

For a periodic basis functionh​\(𝐱\)=cos⁡\(𝐰T​𝐱\+b\)h\(\\mathbf\{x\}\)=\\cos\(\\mathbf\{w\}^\{\\mathrm\{T\}\}\\mathbf\{x\}\+b\)with weight vector𝐰∈ℝd\\mathbf\{w\}\\in\\mathbb\{R\}^\{d\}, the*effective frequency*is defined as:

ν=‖𝐰‖2\.\\nu=\\\|\\mathbf\{w\}\\\|\_\{2\}\.\(3\.12\)This quantity determines the spatial oscillation rate of the basis function along the direction𝐰/‖𝐰‖\\mathbf\{w\}/\\\|\\mathbf\{w\}\\\|\.

#### 3\.4\.2Analysis of Scaling\-Based Methods

For scaling\-based methods where𝐰m=δm​ϵm\\mathbf\{w\}\_\{m\}=\\delta\_\{m\}\\bm\{\\epsilon\}\_\{m\}withϵm∼𝒩​\(𝟎,𝐈d\)\\bm\{\\epsilon\}\_\{m\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{d\}\), the effective frequency isνm=δm​‖ϵm‖\\nu\_\{m\}=\\delta\_\{m\}\\\|\\bm\{\\epsilon\}\_\{m\}\\\|\.

###### Theorem 3\.3\(Frequency Statistics for Scaling\-Based Methods\)\.

Letϵ∼𝒩​\(𝟎,𝐈d\)\\bm\{\\epsilon\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{d\}\)and defineν=δ​‖ϵ‖\\nu=\\delta\\\|\\bm\{\\epsilon\}\\\|for scaling factorδ\>0\\delta\>0\. Then:

1. \(i\)𝔼​\[ν\]=δ⋅cd\\mathbb\{E\}\[\\nu\]=\\delta\\cdot c\_\{d\}, wherecd=2​Γ​\(d\+12\)/Γ​\(d2\)c\_\{d\}=\\sqrt\{2\}\\,\\Gamma\\left\(\\frac\{d\+1\}\{2\}\\right\)/\\Gamma\\left\(\\frac\{d\}\{2\}\\right\)\.
2. \(ii\)Var​\(ν\)=δ2⋅\(d−cd2\)\\mathrm\{Var\}\(\\nu\)=\\delta^\{2\}\\cdot\(d\-c\_\{d\}^\{2\}\)\.
3. \(iii\)Ford=2d=2:c2=π/2≈1\.253c\_\{2\}=\\sqrt\{\\pi/2\}\\approx 1\.253andVar​\(ν\)≈0\.429​δ2\\mathrm\{Var\}\(\\nu\)\\approx 0\.429\\delta^\{2\}\.

###### Proof\.

Sinceϵ∼𝒩​\(𝟎,𝐈d\)\\bm\{\\epsilon\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{d\}\), we have‖ϵ‖2∼χ2​\(d\)\\\|\\bm\{\\epsilon\}\\\|^\{2\}\\sim\\chi^\{2\}\(d\)\. The chi distribution has𝔼​\[‖ϵ‖\]=cd\\mathbb\{E\}\[\\\|\\bm\{\\epsilon\}\\\|\]=c\_\{d\}andVar​\(‖ϵ‖\)=d−cd2\\mathrm\{Var\}\(\\\|\\bm\{\\epsilon\}\\\|\)=d\-c\_\{d\}^\{2\}\. By linearity,𝔼​\[ν\]=δ⋅cd\\mathbb\{E\}\[\\nu\]=\\delta\\cdot c\_\{d\}andVar​\(ν\)=δ2​\(d−cd2\)\\mathrm\{Var\}\(\\nu\)=\\delta^\{2\}\(d\-c\_\{d\}^\{2\}\)\. ∎

#### 3\.4\.3Analysis of Frequency Shift Method

For the frequency shift approach where𝐰m=𝝁m\+ϵm\\mathbf\{w\}\_\{m\}=\\bm\{\\mu\}\_\{m\}\+\\bm\{\\epsilon\}\_\{m\}with‖𝝁m‖=μm\\\|\\bm\{\\mu\}\_\{m\}\\\|=\\mu\_\{m\}, the statistical properties differ qualitatively\.

###### Theorem 3\.4\(Frequency Control via Mean Shifting\)\.

Let𝐰=𝛍\+ϵ\\mathbf\{w\}=\\bm\{\\mu\}\+\\bm\{\\epsilon\}withϵ∼𝒩​\(𝟎,𝐈d\)\\bm\{\\epsilon\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{d\}\)and‖𝛍‖=μ\\\|\\bm\{\\mu\}\\\|=\\mu\. Then:

1. \(i\)𝔼​\[‖𝐰‖2\]=μ2\+d\\mathbb\{E\}\[\\\|\\mathbf\{w\}\\\|^\{2\}\]=\\mu^\{2\}\+d\.
2. \(ii\)𝔼​\[‖𝐰‖\]≈μ2\+d\\mathbb\{E\}\[\\\|\\mathbf\{w\}\\\|\]\\approx\\sqrt\{\\mu^\{2\}\+d\}forμ≫0\\mu\\gg 0, becoming exact asμ→∞\\mu\\to\\infty\.
3. \(iii\)For target effective frequencyν∗\\nu^\{\*\}, the required parameter isμ=\(ν∗\)2−d\\mu=\\sqrt\{\(\\nu^\{\*\}\)^\{2\}\-d\}\.

###### Proof\.

Expanding‖𝐰‖2=‖𝝁\+ϵ‖2=‖𝝁‖2\+2​𝝁T​ϵ\+‖ϵ‖2\\\|\\mathbf\{w\}\\\|^\{2\}=\\\|\\bm\{\\mu\}\+\\bm\{\\epsilon\}\\\|^\{2\}=\\\|\\bm\{\\mu\}\\\|^\{2\}\+2\\bm\{\\mu\}^\{\\mathrm\{T\}\}\\bm\{\\epsilon\}\+\\\|\\bm\{\\epsilon\}\\\|^\{2\}\. Taking expectations:𝔼​\[‖𝐰‖2\]=μ2\+0\+d=μ2\+d\\mathbb\{E\}\[\\\|\\mathbf\{w\}\\\|^\{2\}\]=\\mu^\{2\}\+0\+d=\\mu^\{2\}\+d\. For largeμ\\mu, concentration of measure implies𝔼​\[‖𝐰‖\]→μ2\+d\\mathbb\{E\}\[\\\|\\mathbf\{w\}\\\|\]\\to\\sqrt\{\\mu^\{2\}\+d\}\. ∎

###### Theorem 3\.5\(Bounded Frequency Variance for FS\-PIELM\)\.

Let𝐰=𝛍\+ϵ\\mathbf\{w\}=\\bm\{\\mu\}\+\\bm\{\\epsilon\}withϵ∼𝒩​\(𝟎,𝐈d\)\\bm\{\\epsilon\}\\sim\\mathcal\{N\}\(\\mathbf\{0\},\\mathbf\{I\}\_\{d\}\)and‖𝛍‖=μ\>0\\\|\\bm\{\\mu\}\\\|=\\mu\>0\. The variance of the effective frequency satisfies:

Var​\(‖𝐰‖\)≈2​μ2\+d2​\(μ2\+d\)→μ→∞1\.\\mathrm\{Var\}\(\\\|\\mathbf\{w\}\\\|\)\\approx\\frac\{2\\mu^\{2\}\+d\}\{2\(\\mu^\{2\}\+d\)\}\\xrightarrow\{\\mu\\to\\infty\}1\.\(3\.13\)

###### Proof\.

LetX=‖𝐰‖2=μ2\+2​𝝁T​ϵ\+‖ϵ‖2X=\\\|\\mathbf\{w\}\\\|^\{2\}=\\mu^\{2\}\+2\\bm\{\\mu\}^\{\\mathrm\{T\}\}\\bm\{\\epsilon\}\+\\\|\\bm\{\\epsilon\}\\\|^\{2\}\. ThenVar​\(X\)=4​μ2\+2​d\\mathrm\{Var\}\(X\)=4\\mu^\{2\}\+2d\. Applying the delta method withg​\(X\)=Xg\(X\)=\\sqrt\{X\}:Var​\(X\)≈\[g′​\(𝔼​\[X\]\)\]2​Var​\(X\)=4​μ2\+2​d4​\(μ2\+d\)=2​μ2\+d2​\(μ2\+d\)→1\\mathrm\{Var\}\(\\sqrt\{X\}\)\\approx\[g^\{\\prime\}\(\\mathbb\{E\}\[X\]\)\]^\{2\}\\mathrm\{Var\}\(X\)=\\frac\{4\\mu^\{2\}\+2d\}\{4\(\\mu^\{2\}\+d\)\}=\\frac\{2\\mu^\{2\}\+d\}\{2\(\\mu^\{2\}\+d\)\}\\to 1asμ→∞\\mu\\to\\infty\. ∎

#### 3\.4\.4Comparative Analysis

###### Corollary 3\.6\(Variance Ratio\)\.

For target effective frequencyν∗\\nu^\{\*\}, the ratio of frequency variances is:

Var​\(νscale\)Var​\(νshift\)≈\(ν∗\)2​\(d−cd2\)cd2\.\\frac\{\\mathrm\{Var\}\(\\nu^\{\\text\{scale\}\}\)\}\{\\mathrm\{Var\}\(\\nu^\{\\text\{shift\}\}\)\}\\approx\\frac\{\(\\nu^\{\*\}\)^\{2\}\(d\-c\_\{d\}^\{2\}\)\}\{c\_\{d\}^\{2\}\}\.\(3\.14\)Ford=2d=2andν∗=80\\nu^\{\*\}=80: this ratio equals approximately17461746, indicating that scaling\-based methods exhibit nearly three orders of magnitude higher frequency variance than FS\-PIELM\.

#### 3\.4\.5Summary of Theoretical Advantages

In summary, Theorems[3\.3](https://arxiv.org/html/2607.01694#S3.Thmtheorem3)–[3\.5](https://arxiv.org/html/2607.01694#S3.Thmtheorem5)establish that \(i\) the frequency variance under FS\-PIELM approaches unity asμ→∞\\mu\\to\\infty, unlike the𝒪​\(δ2\)\\mathcal\{O\}\(\\delta^\{2\}\)growth of scaling methods; \(ii\) the mappingμ↦μ2\+d\\mu\\mapsto\\sqrt\{\\mu^\{2\}\+d\}provides an intuitive frequency parameterization; and \(iii\) random directions𝐝m\\mathbf\{d\}\_\{m\}guarantee isotropic spatial coverage\. From a computational standpoint, the single\-layer closed\-form solution is fully preserved—the frequency shift sampling adds only𝒪​\(d\)\\mathcal\{O\}\(d\)operations per neuron\.

## 4Numerical Experiments

This section reports numerical experiments that evaluate FS\-PIELM against three baselines: Tanh\-PIELM, SIREN\-PIELM, and GFF\-PIELM\. Within each test case, all methods share identical network configurations to ensure fair comparison\. The code is implemented in Python with PyTorch using double precision arithmetic \(float64\)\. All computations are carried out on a standard CPU, and every PIELM method produces a solution in seconds owing to the closed\-form training procedure\.

The quality of approximate solutions is evaluated using the relativeL2L\_\{2\}error metric, defined as:

ϵL2=‖uexact−upred‖2‖uexact‖2=∑i=1Ntest\(uexact​\(𝐱i\)−upred​\(𝐱i\)\)2∑i=1Ntestuexact​\(𝐱i\)2,\\epsilon\_\{L\_\{2\}\}=\\frac\{\\\|u\_\{\\text\{exact\}\}\-u\_\{\\text\{pred\}\}\\\|\_\{2\}\}\{\\\|u\_\{\\text\{exact\}\}\\\|\_\{2\}\}=\\frac\{\\sqrt\{\\sum\_\{i=1\}^\{N\_\{\\text\{test\}\}\}\(u\_\{\\text\{exact\}\}\(\\mathbf\{x\}\_\{i\}\)\-u\_\{\\text\{pred\}\}\(\\mathbf\{x\}\_\{i\}\)\)^\{2\}\}\}\{\\sqrt\{\\sum\_\{i=1\}^\{N\_\{\\text\{test\}\}\}u\_\{\\text\{exact\}\}\(\\mathbf\{x\}\_\{i\}\)^\{2\}\}\},\(4\.1\)where the evaluation is performed on a uniform grid ofNtest=100×100=10000N\_\{\\text\{test\}\}=100\\times 100=10000test points covering the computational domain\. This metric provides a normalized measure of solution accuracy that facilitates comparison across different problem configurations\. To assess robustness with respect to random initialization, each configuration is tested with 5 independent random seeds, and both the best achieved error and the average error across seeds are reported\.

### 4\.1Two\-Dimensional Helmholtz Equation

The 2D Helmholtz equation is a canonical test for high\-frequency solvers: at large wavenumbers, inadequate spectral resolution produces phase errors analogous to the pollution effect in finite elements\. We consider the unit squareΩ=\[0,1\]2\\Omega=\[0,1\]^\{2\}with homogeneous Dirichlet conditions:

−Δ​u−κ2​u\\displaystyle\-\\Delta u\-\\kappa^\{2\}u=f​\(x,y\),\(x,y\)∈\[0,1\]2,\\displaystyle=f\(x,y\),\\quad\(x,y\)\\in\[0,1\]^\{2\},\(4\.2\)u​\(x,y\)\\displaystyle u\(x,y\)=0,\(x,y\)∈∂\[0,1\]2,\\displaystyle=0,\\quad\(x,y\)\\in\\partial\[0,1\]^\{2\},\(4\.3\)whereΔ=∂2/∂x2\+∂2/∂y2\\Delta=\\partial^\{2\}/\\partial x^\{2\}\+\\partial^\{2\}/\\partial y^\{2\}denotes the Laplacian operator and the wavenumber is set toκ=24​π≈75\.4\\kappa=24\\pi\\approx 75\.4, corresponding to a highly oscillatory solution with approximately 24 wavelengths along each spatial dimension\.

The exact solution and corresponding source term are constructed as:

uexact​\(x,y\)\\displaystyle u\_\{\\text\{exact\}\}\(x,y\)=sin⁡\(κ​x\)​sin⁡\(κ​y\),\\displaystyle=\\sin\(\\kappa x\)\\sin\(\\kappa y\),\(4\.4\)f​\(x,y\)\\displaystyle f\(x,y\)=κ2​sin⁡\(κ​x\)​sin⁡\(κ​y\)\.\\displaystyle=\\kappa^\{2\}\\sin\(\\kappa x\)\\sin\(\\kappa y\)\.\(4\.5\)The network configuration employsM=5000M=5000hidden neurons withNC=8000N\_\{C\}=8000interior collocation points sampled uniformly within the domain andNB=400N\_\{B\}=400boundary points per edge \(1600 total\)\. The frequency range parameters are systematically varied asμmin∈\{5,10,15,20,25,30\}\\mu\_\{\\min\}\\in\\\{5,10,15,20,25,30\\\}andμmax∈\{80,100,120,140,160,180\}\\mu\_\{\\max\}\\in\\\{80,100,120,140,160,180\\\}, yielding 36 configurations per method\. For FS\-PIELM\-G, the number of groups is set toK=10K=10\.

Table[1](https://arxiv.org/html/2607.01694#S4.T1)reports the results\. Tanh\-PIELM produces errors of𝒪​\(104\)\\mathcal\{O\}\(10^\{4\}\), indicating a complete inability to represent the target wavenumberκ=24​π\\kappa=24\\pi; the hyperbolic tangent activation acts as a low\-pass filter whose spectral energy decays exponentially beyond a cutoff\[[18](https://arxiv.org/html/2607.01694#bib.bib18)\]that lies far belowκ\\kappa, so the basis functions cannot resolve the oscillatory solution regardless of how the scaling factor is chosen\. SIREN\-PIELM and GFF\-PIELM, both employing periodic activations, reach𝒪​\(10−5\)\\mathcal\{O\}\(10^\{\-5\}\), while FS\-PIELM\-L attains1\.37×10−81\.37\\times 10^\{\-8\}—a6,366×6\{,\}366\\timesreduction relative to GFF\-PIELM\. FS\-PIELM\-L also shows the lowest seed\-to\-seed variability \(standard deviation1\.01×10−71\.01\\times 10^\{\-7\}\), compared with2\.19×10−42\.19\\times 10^\{\-4\}for SIREN\-PIELM and7\.07×10−47\.07\\times 10^\{\-4\}for GFF\-PIELM, reflecting the tighter concentration of effective frequencies aroundκ\\kappathat the mean\-shift mechanism provides\. FS\-PIELM\-G achieves an intermediate best error of1\.07×10−71\.07\\times 10^\{\-7\}, roughly one order of magnitude above FS\-PIELM\-L; the grouping reduces the number of distinct frequency magnitudes fromM=5000M=5000toK=10K=10, sacrificing fine spectral resolution for additional within\-group averaging\. All methods require comparable wall\-clock times \(14–17 s\), confirming that the frequency shift sampling introduces no measurable computational overhead\.

Table 1:Performance comparison on the 2D Helmholtz equation \(κ=24​π\\kappa=24\\pi\)\. Results show the optimal frequency parameters and corresponding relativeL2L\_\{2\}error statistics over 5 random seeds\.![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/case01_helmholtz2d_best_error.png)\(a\)Best relativeL2L\_\{2\}error \(color\-coded on logarithmic scale\) as a function of frequency parameters\(μmin,μmax\)\(\\mu\_\{\\min\},\\mu\_\{\\max\}\)for five PIELM methods\. Each cell represents the minimum error achieved over 5 independent random seeds\. Green colors indicate lower errors and better accuracy, while red colors indicate higher errors\.
![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/case01_helmholtz2d_lineplot_best.png)\(b\)Best relativeL2L\_\{2\}error versusμmax\\mu\_\{\\max\}for each method at its respective optimalμmin\\mu\_\{\\min\}\. Each point represents the minimum error achieved over 5 random seeds\.

Figure 3:2D Helmholtz equation:Comparison of five PIELM methods \(Tanh\-PIELM, SIREN\-PIELM, GFF\-PIELM, FS\-PIELM\-L, and FS\-PIELM\-G\) with wavenumberκ=24​π\\kappa=24\\pion the unit squareΩ=\[0,1\]2\\Omega=\[0,1\]^\{2\}\. Network configuration:M=5000M=5000neurons,NC=8000N\_\{C\}=8000collocation points,NB=400N\_\{B\}=400boundary points per edge\.The heatmaps in Fig\.[3\(a\)](https://arxiv.org/html/2607.01694#S4.F3.sf1)show that FS\-PIELM\-L maintains low errors \(green\) over a broad region of\(μmin,μmax\)\(\\mu\_\{\\min\},\\mu\_\{\\max\}\)values, whereas SIREN\-PIELM and GFF\-PIELM require narrow tuning near\[30,120\]\[30,120\]\. This parameter robustness is practically important: in real applications the dominant wavenumber may not be known precisely, and a method that tolerates a wider search range reduces the risk of suboptimal hyperparameter selection\. The lineplot in Fig\.[3\(b\)](https://arxiv.org/html/2607.01694#S4.F3.sf2)shows that FS\-PIELM\-L improves monotonically withμmax\\mu\_\{\\max\}and reaches𝒪​\(10−8\)\\mathcal\{O\}\(10^\{\-8\}\)forμmax≥140\\mu\_\{\\max\}\\geq 140, roughly10410^\{4\}times more accurate than both baselines\. The mean\-shifted weights concentrate near the target wavenumberκ=24​π\\kappa=24\\pi\(Theorem[3\.5](https://arxiv.org/html/2607.01694#S3.Thmtheorem5)\), while scaling\-based weights spread over a much wider frequency band\. Fig\.[4](https://arxiv.org/html/2607.01694#S4.F4)visualizes this concentration effect, where the effective frequency histogram of FS\-PIELM peaks sharply aroundκ\\kappawhile the scaling\-based histogram exhibits heavy tails that waste representational capacity on off\-target frequencies\.

![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/helmholtz2d_fig04_freq_analysis.png)Figure 4:2D Helmholtz equation \- Frequency analysis:Visualization of effective frequency distribution comparing scaling\-based methods and the proposed frequency shift mechanism\. The analysis demonstrates that FS\-PIELM maintains bounded frequency variance while scaling methods exhibit quadratic variance growth with increasing target frequency\.
### 4\.2One\-Dimensional Wave Equation with Time\-Varying Frequency

A more demanding scenario arises when the spectral content evolves in time\. We consider a wave equation whose spatial frequency increases linearly from2​π2\\pito16​π16\\piover the unit time interval, requiring the hidden layer to cover roughly an order\-of\-magnitude frequency sweep simultaneously\.

∂2u∂t2−∂2u∂x2\\displaystyle\\frac\{\\partial^\{2\}u\}\{\\partial t^\{2\}\}\-\\frac\{\\partial^\{2\}u\}\{\\partial x^\{2\}\}=f​\(x,t\),\(x,t\)∈\[0,1\]×\[0,1\],\\displaystyle=f\(x,t\),\\quad\(x,t\)\\in\[0,1\]\\times\[0,1\],\(4\.6\)u​\(0,t\)=u​\(1,t\)\\displaystyle u\(0,t\)=u\(1,t\)=g​\(t\),t∈\[0,1\],\\displaystyle=g\(t\),\\quad t\\in\[0,1\],\(4\.7\)u​\(x,0\)\\displaystyle u\(x,0\)=u0​\(x\),x∈\[0,1\],\\displaystyle=u\_\{0\}\(x\),\\quad x\\in\[0,1\],\(4\.8\)∂u∂t​\(x,0\)\\displaystyle\\frac\{\\partial u\}\{\\partial t\}\(x,0\)=v0​\(x\),x∈\[0,1\]\.\\displaystyle=v\_\{0\}\(x\),\\quad x\\in\[0,1\]\.\(4\.9\)
The exact solution exhibits linearly increasing spatial frequency:

uexact​\(x,t\)=sin⁡\[\(2​π\+14​π​t\)​x\]​cos⁡\(10​π​t\),u\_\{\\text\{exact\}\}\(x,t\)=\\sin\\left\[\(2\\pi\+14\\pi t\)x\\right\]\\cos\(10\\pi t\),\(4\.10\)where the time\-dependent spatial frequencyω​\(t\)=2​π\+14​π​t\\omega\(t\)=2\\pi\+14\\pi tvaries from2​π≈6\.282\\pi\\approx 6\.28att=0t=0to16​π≈50\.2716\\pi\\approx 50\.27att=1t=1\. The temporal oscillation frequency is10​π≈31\.4210\\pi\\approx 31\.42\. The initial conditions are:

u0​\(x\)\\displaystyle u\_\{0\}\(x\)=sin⁡\(2​π​x\),\\displaystyle=\\sin\(2\\pi x\),\(4\.11\)v0​\(x\)\\displaystyle v\_\{0\}\(x\)=14​π​x​cos⁡\(2​π​x\)\.\\displaystyle=14\\pi x\\cos\(2\\pi x\)\.\(4\.12\)
The source termf​\(x,t\)f\(x,t\)is derived analytically to ensure consistency with the exact solution\. The network employsM=5000M=5000neurons withNC=8000N\_\{C\}=8000interior collocation points,NB=400N\_\{B\}=400boundary points per temporal boundary, andNI=400N\_\{I\}=400initial condition points\. The frequency parameters areμmin∈\{1,4,7,10,13,16\}\\mu\_\{\\min\}\\in\\\{1,4,7,10,13,16\\\}andμmax∈\{60,70,80,90,100,110\}\\mu\_\{\\max\}\\in\\\{60,70,80,90,100,110\\\}\.

Table[2](https://arxiv.org/html/2607.01694#S4.T2)summarizes the results\. FS\-PIELM\-L reaches2\.00×10−82\.00\\times 10^\{\-8\}, a138×138\\timesreduction over GFF\-PIELM \(2\.76×10−62\.76\\times 10^\{\-6\}\) and62×62\\timesover SIREN\-PIELM \(1\.24×10−61\.24\\times 10^\{\-6\}\)\. Because the spatial frequency sweeps from2​π2\\pito16​π16\\piover the time interval, the network must represent a range of frequencies simultaneously; the linearly spaced mean magnitudes of FS\-PIELM\-L distribute neurons across the full\[μmin,μmax\]\[\\mu\_\{\\min\},\\mu\_\{\\max\}\]band, each with unit\-variance perturbation, providing dense and uniform spectral coverage\. FS\-PIELM\-G achieves6\.74×10−86\.74\\times 10^\{\-8\}, about3×3\\timesabove FS\-PIELM\-L; the grouped architecture again trades fine frequency resolution for within\-group averaging, which remains effective here because the frequency sweep is continuous rather than concentrated at discrete harmonics\. FS\-PIELM\-L also exhibits40×40\\timeslower seed\-to\-seed standard deviation than GFF\-PIELM, a direct consequence of the tighter frequency concentration afforded by the mean\-shift mechanism\. All methods require about 7 s of computation\.

Table 2:Performance comparison on the 1D wave equation with time\-varying frequency\. Results show optimal parameters and relativeL2L\_\{2\}error statistics over 5 random seeds\.![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/case02_wave1d_best_error.png)\(a\)Best relativeL2L\_\{2\}error as a function of\(μmin,μmax\)\(\\mu\_\{\\min\},\\mu\_\{\\max\}\); format as in Fig\.[3\(a\)](https://arxiv.org/html/2607.01694#S4.F3.sf1)\.
![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/case02_wave1d_lineplot_best.png)\(b\)Best relativeL2L\_\{2\}error versusμmax\\mu\_\{\\max\}at each method’s optimalμmin\\mu\_\{\\min\}; format as in Fig\.[3\(b\)](https://arxiv.org/html/2607.01694#S4.F3.sf2)\.

Figure 5:1D wave equation:Comparison of five PIELM methods with time\-varying spatial frequencyω​\(t\)=2​π\+14​π​t\\omega\(t\)=2\\pi\+14\\pi ton\[0,1\]×\[0,1\]\[0,1\]\\times\[0,1\]\. Network configuration:M=5000M=5000neurons,NC=8000N\_\{C\}=8000collocation points,NB=400N\_\{B\}=400boundary points,NI=400N\_\{I\}=400initial condition points\.The heatmaps in Fig\.[5\(a\)](https://arxiv.org/html/2607.01694#S4.F5.sf1)show that SIREN\-PIELM and GFF\-PIELM requireμmin≥13\\mu\_\{\\min\}\\geq 13andμmax≥90\\mu\_\{\\max\}\\geq 90to reach𝒪​\(10−6\)\\mathcal\{O\}\(10^\{\-6\}\), whereas FS\-PIELM\-L attains𝒪​\(10−8\)\\mathcal\{O\}\(10^\{\-8\}\)over a considerably wider parameter region\. The time\-varying frequency poses a particular challenge: fixed scaling factors must compromise between the low\-frequency early\-time regime \(ω≈6\.3\\omega\\approx 6\.3\) and the high\-frequency late\-time regime \(ω≈50\.3\\omega\\approx 50\.3\)\. In scaling\-based methods, matching the late\-time component inflates the weight variance for all neurons, degrading the accuracy of the early\-time low\-frequency component\. Mean\-shifted weights avoid this trade\-off because different neurons are assigned to different frequency bands across the full range\[μmin,μmax\]\[\\mu\_\{\\min\},\\mu\_\{\\max\}\], each retaining unit variance, providing simultaneous coverage of the entire evolving spectrum without mutual interference \(Fig\.[5\(b\)](https://arxiv.org/html/2607.01694#S4.F5.sf2)\)\. Compared with the Helmholtz equation, the improvement factor over GFF\-PIELM is smaller \(138×138\\timesversus6,366×6\{,\}366\\times\), which is expected because the maximum frequency here \(16​π≈5016\\pi\\approx 50\) is lower than the Helmholtz wavenumber \(24​π≈7524\\pi\\approx 75\), and the variance penalty of scaling grows quadratically with target frequency\.

### 4\.3One\-Dimensional Poisson Equation with Multi\-Scale Features

A stringent test of multi\-scale resolution is provided by a Poisson problem whose solution contains two widely separated frequency components \(ratio 15:1\)\. Resolving both scales simultaneously is notoriously difficult: basis functions tuned to one scale inevitably under\-resolve or over\-oscillate the other\. We consider:

d2​ud​x2\\displaystyle\\frac\{d^\{2\}u\}\{dx^\{2\}\}=f​\(x\),x∈\[0,1\],\\displaystyle=f\(x\),\\quad x\\in\[0,1\],\(4\.13\)u​\(0\)=u​\(1\)\\displaystyle u\(0\)=u\(1\)=0\.\\displaystyle=0\.\(4\.14\)
The exact solution contains two widely separated frequency scales:

uexact​\(x\)=sin⁡\(5​π​x\)\+0\.2​sin⁡\(75​π​x\),u\_\{\\text\{exact\}\}\(x\)=\\sin\(5\\pi x\)\+0\.2\\sin\(75\\pi x\),\(4\.15\)where the low\-frequency component has wavenumber5​π≈15\.715\\pi\\approx 15\.71and the high\-frequency component has wavenumber75​π≈235\.6275\\pi\\approx 235\.62, representing a frequency ratio of 15:1\. The corresponding source term is:

f​\(x\)=−\(5​π\)2​sin⁡\(5​π​x\)−0\.2​\(75​π\)2​sin⁡\(75​π​x\)\.f\(x\)=\-\(5\\pi\)^\{2\}\\sin\(5\\pi x\)\-0\.2\(75\\pi\)^\{2\}\\sin\(75\\pi x\)\.\(4\.16\)
Due to the one\-dimensional nature and relative simplicity of this problem, a smaller network withM=200M=200neurons suffices\. The training data consists ofNC=400N\_\{C\}=400interior collocation points andNB=2N\_\{B\}=2boundary points\. The frequency parameters span a wider range to accommodate the multi\-scale solution:μmin∈\{1,5,10,15,20,25\}\\mu\_\{\\min\}\\in\\\{1,5,10,15,20,25\\\}andμmax∈\{200,250,300,350,400,450\}\\mu\_\{\\max\}\\in\\\{200,250,300,350,400,450\\\}\.

This case produces the widest performance gap among all benchmarks \(Table[3](https://arxiv.org/html/2607.01694#S4.T3)\)\. SIREN\-PIELM and GFF\-PIELM achieve best\-case errors near10−810^\{\-8\}, but their average errors are𝒪​\(10−2\)\\mathcal\{O\}\(10^\{\-2\}\)with standard deviations exceeding10−210^\{\-2\}, revealing severe instability across random seeds\. The large discrepancy between best and average errors indicates a “lottery” effect: only particular random initializations happen to place enough weight vectors near both5​π5\\piand75​π75\\pi, while most draws leave one of the two frequency components poorly represented\. FS\-PIELM\-L, by contrast, reaches8\.22×10−128\.22\\times 10^\{\-12\}with a standard deviation of only1\.13×10−111\.13\\times 10^\{\-11\}—an8,977×8\{,\}977\\timesimprovement in best error and a10910^\{9\}\-fold improvement in reproducibility over GFF\-PIELM\. The 15:1 frequency ratio requires the network to resolve5​π5\\piand75​π75\\picomponents simultaneously; the additive mean shift distributes neurons across this range with linearly spaced mean magnitudes, guaranteeing that both scales are covered in every random draw\. FS\-PIELM\-G also shows strong performance \(5\.88×10−105\.88\\times 10^\{\-10\}\), though the coarser frequency discretization ofK=10K=10groups limits its resolution compared with FS\-PIELM\-L\.

Table 3:Performance comparison on the 1D Poisson equation with multi\-scale solution\. Results show optimal parameters and relativeL2L\_\{2\}error statistics over 5 random seeds\.![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/case03_poisson1d_best_error.png)\(a\)Best relativeL2L\_\{2\}error as a function of\(μmin,μmax\)\(\\mu\_\{\\min\},\\mu\_\{\\max\}\); format as in Fig\.[3\(a\)](https://arxiv.org/html/2607.01694#S4.F3.sf1)\.
![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/case03_poisson1d_lineplot_best.png)\(b\)Best relativeL2L\_\{2\}error versusμmax\\mu\_\{\\max\}at each method’s optimalμmin\\mu\_\{\\min\}; format as in Fig\.[3\(b\)](https://arxiv.org/html/2607.01694#S4.F3.sf2)\.

Figure 6:1D Poisson equation:Comparison of five PIELM methods with multi\-scale solution featuring frequency components at5​π5\\piand75​π75\\pi\(ratio 1:15\) on\[0,1\]\[0,1\]\. Network configuration:M=200M=200neurons,NC=400N\_\{C\}=400collocation points,NB=2N\_\{B\}=2boundary points\.The heatmaps in Fig\.[6\(a\)](https://arxiv.org/html/2607.01694#S4.F6.sf1)expose the instability of the multiplicative approaches on this problem: SIREN\-PIELM and GFF\-PIELM achieve low errors only in a narrow parameter band near\[25,350\]\[25,350\], degrading rapidly elsewhere\. Even within that band, the cell\-to\-cell variation is large, matching the high standard deviations reported in Table[3](https://arxiv.org/html/2607.01694#S4.T3)\. FS\-PIELM\-L, by contrast, maintains errors below10−1010^\{\-10\}over a broad region withμmin≤10\\mu\_\{\\min\}\\leq 10andμmax≥200\\mu\_\{\\max\}\\geq 200\(Fig\.[6\(b\)](https://arxiv.org/html/2607.01694#S4.F6.sf2)\)\. The near\-machine\-precision accuracy \(8\.22×10−128\.22\\times 10^\{\-12\}\) achieved at\[μmin,μmax\]=\[5,250\]\[\\mu\_\{\\min\},\\mu\_\{\\max\}\]=\[5,250\]indicates that the frequency shift mechanism distributes basis functions effectively across the 15:1 spectral gap without requiring an excessively largeμmax\\mu\_\{\\max\}\. The fact that the optimalμmax=250\\mu\_\{\\max\}=250is close to75​π≈23675\\pi\\approx 236—the wavenumber of the high\-frequency component—corroborating that𝔼​\[‖𝐰m‖\]≈μm2\+d\\mathbb\{E\}\[\\\|\\mathbf\{w\}\_\{m\}\\\|\]\\approx\\sqrt\{\\mu\_\{m\}^\{2\}\+d\}provides a practical guide for parameter selection\.

### 4\.4Klein\-Gordon Equation with Multiple Frequency Components

The Klein\-Gordon equation introduces a zeroth\-order reaction term that couples different frequency modes, amplifying approximation errors when the basis spectrum does not match the solution\. The exact solution here combines three qualitatively distinct components—a moderate\-frequency oscillation, a high\-frequency oscillation, and a linear baseline—testing the methods on mixed spectral content that includes a zero\-frequency contribution\. We consider:

∂2u∂t2−∂2u∂x2\+u\\displaystyle\\frac\{\\partial^\{2\}u\}\{\\partial t^\{2\}\}\-\\frac\{\\partial^\{2\}u\}\{\\partial x^\{2\}\}\+u=f​\(x,t\),\(x,t\)∈\[0,1\]2,\\displaystyle=f\(x,t\),\\quad\(x,t\)\\in\[0,1\]^\{2\},\(4\.17\)u​\(0,t\)=u​\(1,t\)\\displaystyle u\(0,t\)=u\(1,t\)=g​\(t\),t∈\[0,1\],\\displaystyle=g\(t\),\\quad t\\in\[0,1\],\(4\.18\)u​\(x,0\)\\displaystyle u\(x,0\)=u0​\(x\),x∈\[0,1\],\\displaystyle=u\_\{0\}\(x\),\\quad x\\in\[0,1\],\(4\.19\)∂u∂t​\(x,0\)\\displaystyle\\frac\{\\partial u\}\{\\partial t\}\(x,0\)=v0​\(x\),x∈\[0,1\]\.\\displaystyle=v\_\{0\}\(x\),\\quad x\\in\[0,1\]\.\(4\.20\)
The exact solution combines multiple frequency components spanning a broad spectrum:

uexact​\(x,t\)=x​sin⁡\(3​π​x\)​cos⁡\(7​π​t\)\+t​sin⁡\(19​π​x\)​cos⁡\(19​π​t\)\+x​t,u\_\{\\text\{exact\}\}\(x,t\)=x\\sin\(3\\pi x\)\\cos\(7\\pi t\)\+t\\sin\(19\\pi x\)\\cos\(19\\pi t\)\+xt,\(4\.21\)which contains three distinct frequency scales: a low\-mid frequency component with spatial frequency3​π≈9\.423\\pi\\approx 9\.42and temporal frequency7​π≈21\.997\\pi\\approx 21\.99, a high\-frequency component with both spatial and temporal frequencies at19​π≈59\.6919\\pi\\approx 59\.69, and a linear termx​txtrepresenting the zero\-frequency baseline\.

The initial conditions are:

u0​\(x\)\\displaystyle u\_\{0\}\(x\)=x​sin⁡\(3​π​x\),\\displaystyle=x\\sin\(3\\pi x\),\(4\.22\)v0​\(x\)\\displaystyle v\_\{0\}\(x\)=sin⁡\(19​π​x\)\+x\.\\displaystyle=\\sin\(19\\pi x\)\+x\.\(4\.23\)
The network configuration usesM=5000M=5000neurons withNC=8000N\_\{C\}=8000interior points,NB=400N\_\{B\}=400boundary points, andNI=400N\_\{I\}=400initial condition points\. The frequency parameters areμmin∈\{1,5,10,15,20,25\}\\mu\_\{\\min\}\\in\\\{1,5,10,15,20,25\\\}andμmax∈\{50,60,70,80,90,100\}\\mu\_\{\\max\}\\in\\\{50,60,70,80,90,100\\\}\.

Table[4](https://arxiv.org/html/2607.01694#S4.T4)lists the results\. FS\-PIELM\-L achieves4\.95×10−94\.95\\times 10^\{\-9\}, a19×19\\timesimprovement over GFF\-PIELM and36×36\\timesover SIREN\-PIELM\. The improvement is more modest than in the previous cases because the frequency ratio here is only about 6:1 \(19​π19\\piversus3​π3\\pi\), so the quadratic variance penalty of scaling remains moderate: to target the highest effective frequencyν∗=19​π​2≈84\\nu^\{\*\}=19\\pi\\sqrt\{2\}\\approx 84\(the 2D norm of the\(19​π,19​π\)\(19\\pi,19\\pi\)wavenumber vector\), the required scaling factor isδ=ν∗/c2≈67\\delta=\\nu^\{\*\}/c\_\{2\}\\approx 67, yieldingVar​\(νscale\)≈0\.429​δ2≈1930\\mathrm\{Var\}\(\\nu^\{\\text\{scale\}\}\)\\approx 0\.429\\delta^\{2\}\\approx 1930, which is large but not as extreme as in the Helmholtz \(≈1746\\approx 1746forκ=24​π\\kappa=24\\pi\) or Poisson equations\. The additional presence of the linear termx​txtdoes not appear to cause difficulty for any periodic\-activation method, which is expected because the ELM framework can approximate low\-order polynomials through superposition of cosine basis functions even when individual neurons target higher frequencies\. FS\-PIELM\-G achieves1\.01×10−71\.01\\times 10^\{\-7\}, comparable to SIREN\-PIELM and GFF\-PIELM; with onlyK=10K=10groups spanning the moderate frequency range\[10,100\]\[10,100\], the grouped variant offers limited additional resolution over the baselines for this problem\.

Table 4:Performance comparison on the Klein\-Gordon equation with multiple frequency components\. Results show optimal parameters and relativeL2L\_\{2\}error statistics over 5 random seeds\.![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/case04_kleingordon_best_error.png)\(a\)Best relativeL2L\_\{2\}error as a function of\(μmin,μmax\)\(\\mu\_\{\\min\},\\mu\_\{\\max\}\); format as in Fig\.[3\(a\)](https://arxiv.org/html/2607.01694#S4.F3.sf1)\.
![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/case04_kleingordon_lineplot_best.png)\(b\)Best relativeL2L\_\{2\}error versusμmax\\mu\_\{\\max\}at each method’s optimalμmin\\mu\_\{\\min\}; format as in Fig\.[3\(b\)](https://arxiv.org/html/2607.01694#S4.F3.sf2)\.

Figure 7:Klein\-Gordon equation:Comparison of five PIELM methods with multiple frequency components \(spatial:3​π3\\pi,19​π19\\pi; temporal:7​π7\\pi,19​π19\\pi\) on\[0,1\]2\[0,1\]^\{2\}\. Network configuration:M=5000M=5000neurons,NC=8000N\_\{C\}=8000collocation points,NB=400N\_\{B\}=400boundary points,NI=400N\_\{I\}=400initial condition points\.The heatmaps \(Fig\.[7\(a\)](https://arxiv.org/html/2607.01694#S4.F7.sf1)\) show that SIREN\-PIELM and GFF\-PIELM perform comparably in overlapping optimal regions nearμmax=100\\mu\_\{\\max\}=100, while FS\-PIELM\-L tolerates a broader range ofμmin\\mu\_\{\\min\}values \(down to 10\)\. Fig\.[7\(b\)](https://arxiv.org/html/2607.01694#S4.F7.sf2)shows a roughly one\-order\-of\-magnitude advantage for FS\-PIELM\-L over both baselines across mostμmax\\mu\_\{\\max\}values, commensurate with the moderate frequency ratio of this problem\. The observation that FS\-PIELM\-L achieves its optimum at\[μmin,μmax\]=\[10,100\]\[\\mu\_\{\\min\},\\mu\_\{\\max\}\]=\[10,100\]rather than \[25, 100\] suggests that including lower\-frequency neurons \(μm≈10\\mu\_\{m\}\\approx 10\) helps resolve the3​π3\\picomponent and the linear term, an option unavailable when the scaling factor simultaneously inflates variance for all neurons\.

### 4\.5Heat Equation with Multi\-Frequency Initial Condition

As a deliberate contrast to the preceding wave\-dominated cases, we consider a heat equation whose diffusive physics smooths high\-frequency content over time\. Because Fourier mode decay rates scale ask2k^\{2\}, the effective spectral complexity of the solution decreases with time, testing whether the variance\-reduction advantage of FS\-PIELM persists when the physics itself suppresses high frequencies\.

∂u∂t−α​∂2u∂x2\\displaystyle\\frac\{\\partial u\}\{\\partial t\}\-\\alpha\\frac\{\\partial^\{2\}u\}\{\\partial x^\{2\}\}=0,\(x,t\)∈\[−1,1\]×\[0,1\],\\displaystyle=0,\\quad\(x,t\)\\in\[\-1,1\]\\times\[0,1\],\(4\.24\)u​\(−1,t\)=u​\(1,t\)\\displaystyle u\(\-1,t\)=u\(1,t\)=0,t∈\[0,1\],\\displaystyle=0,\\quad t\\in\[0,1\],\(4\.25\)u​\(x,0\)\\displaystyle u\(x,0\)=u0​\(x\),x∈\[−1,1\],\\displaystyle=u\_\{0\}\(x\),\\quad x\\in\[\-1,1\],\(4\.26\)where the thermal diffusivity isα=1/\(20​π\)2\\alpha=1/\(20\\pi\)^\{2\}\.

The exact solution demonstrates the frequency\-dependent decay characteristic of parabolic equations:

uexact​\(x,t\)=e−λ1​t​sin⁡\(5​π​x\)\+0\.5​e−λ2​t​sin⁡\(10​π​x\)\+0\.2​e−λ3​t​sin⁡\(20​π​x\),u\_\{\\text\{exact\}\}\(x,t\)=e^\{\-\\lambda\_\{1\}t\}\\sin\(5\\pi x\)\+0\.5e^\{\-\\lambda\_\{2\}t\}\\sin\(10\\pi x\)\+0\.2e^\{\-\\lambda\_\{3\}t\}\\sin\(20\\pi x\),\(4\.27\)where the decay rates are determined byλk=α​\(k​π\)2\\lambda\_\{k\}=\\alpha\(k\\pi\)^\{2\}:

λ1\\displaystyle\\lambda\_\{1\}=α​\(5​π\)2=0\.0625,\\displaystyle=\\alpha\(5\\pi\)^\{2\}=0\.0625,\(4\.28\)λ2\\displaystyle\\lambda\_\{2\}=α​\(10​π\)2=0\.25,\\displaystyle=\\alpha\(10\\pi\)^\{2\}=0\.25,\(4\.29\)λ3\\displaystyle\\lambda\_\{3\}=α​\(20​π\)2=1\.0\.\\displaystyle=\\alpha\(20\\pi\)^\{2\}=1\.0\.\(4\.30\)The key spatial frequencies are5​π≈15\.715\\pi\\approx 15\.71,10​π≈31\.4210\\pi\\approx 31\.42, and20​π≈62\.8320\\pi\\approx 62\.83, with higher frequencies experiencing faster exponential decay\.

The initial condition is:

u0​\(x\)=sin⁡\(5​π​x\)\+0\.5​sin⁡\(10​π​x\)\+0\.2​sin⁡\(20​π​x\)\.u\_\{0\}\(x\)=\\sin\(5\\pi x\)\+0\.5\\sin\(10\\pi x\)\+0\.2\\sin\(20\\pi x\)\.\(4\.31\)
The network usesM=1200M=1200neurons withNC=8000N\_\{C\}=8000interior points,NB=400N\_\{B\}=400boundary points per boundary, andNI=1000N\_\{I\}=1000initial condition points to adequately resolve the high\-frequency initial data\. The frequency parameters areμmin∈\{5,10,15,20,25,30\}\\mu\_\{\\min\}\\in\\\{5,10,15,20,25,30\\\}andμmax∈\{50,60,70,80,90,100\}\\mu\_\{\\max\}\\in\\\{50,60,70,80,90,100\\\}\.

Table[5](https://arxiv.org/html/2607.01694#S4.T5)shows that all periodic\-activation methods reach the𝒪​\(10−9\)\\mathcal\{O\}\(10^\{\-9\}\)range, with GFF\-PIELM slightly leading at1\.41×10−91\.41\\times 10^\{\-9\}compared with5\.56×10−95\.56\\times 10^\{\-9\}for FS\-PIELM\-L\. This is the only case in which FS\-PIELM does not achieve the lowest error\. The result is consistent with the physics of the problem: the20​π20\\picomponent decays by a factor ofe−1≈0\.37e^\{\-1\}\\approx 0\.37over the time interval, and byt=1t=1the solution is dominated by the slowly decaying5​π5\\pimode \(e−0\.0625≈0\.94e^\{\-0\.0625\}\\approx 0\.94\)\. The effective frequency ratio at late times is therefore much smaller than the initial 4:1 \(20​π20\\pito5​π5\\pi\), placing all periodic\-activation methods within a comparable accuracy range\. The small4×4\\timesgap between GFF\-PIELM and FS\-PIELM\-L—both achieving𝒪​\(10−9\)\\mathcal\{O\}\(10^\{\-9\}\)—reflects the reduced spectral demands of this diffusion\-dominated problem rather than a deficiency in the frequency shift mechanism\. When the physics itself suppresses high\-frequency content, the variance reduction advantage becomes secondary, and all periodic\-activation methods perform comparably regardless of their frequency control strategy\. This observation delineates the regime where FS\-PIELM provides its greatest benefit: problems in which high\-frequency content persists throughout the domain or time interval, so that precise frequency targeting remains critical for solution accuracy\.

Table 5:Performance comparison on the heat equation with multi\-frequency initial condition\. Results show optimal parameters and relativeL2L\_\{2\}error statistics over 5 random seeds\.![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/case05_heatmultifreq_best_error.png)\(a\)Best relativeL2L\_\{2\}error as a function of\(μmin,μmax\)\(\\mu\_\{\\min\},\\mu\_\{\\max\}\); format as in Fig\.[3\(a\)](https://arxiv.org/html/2607.01694#S4.F3.sf1)\.
![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/case05_heatmultifreq_lineplot_best.png)\(b\)Best relativeL2L\_\{2\}error versusμmax\\mu\_\{\\max\}at each method’s optimalμmin\\mu\_\{\\min\}; format as in Fig\.[3\(b\)](https://arxiv.org/html/2607.01694#S4.F3.sf2)\.

Figure 8:Heat equation:Comparison of five PIELM methods with three\-frequency initial condition \(spatial frequencies:5​π5\\pi,10​π10\\pi,20​π20\\pi\) and diffusivityα=1/\(20​π\)2\\alpha=1/\(20\\pi\)^\{2\}on\[−1,1\]×\[0,1\]\[\-1,1\]\\times\[0,1\]\. Network configuration:M=1200M=1200neurons,NC=8000N\_\{C\}=8000collocation points,NB=400N\_\{B\}=400boundary points,NI=1000N\_\{I\}=1000initial condition points\.Fig\.[8\(a\)](https://arxiv.org/html/2607.01694#S4.F8.sf1)shows that all periodic\-activation methods yield similar error levels across the explored parameter space, with optimal regions concentrated nearμmin≈30\\mu\_\{\\min\}\\approx 30andμmax∈\[60,90\]\\mu\_\{\\max\}\\in\[60,90\]\. Tanh\-PIELM is the only method that achieves a finite, though large, error \(8\.29×10−28\.29\\times 10^\{\-2\}\), reflecting the diffusion\-induced smoothing that brings the solution closer to the tanh spectral window\. The lineplot \(Fig\.[8\(b\)](https://arxiv.org/html/2607.01694#S4.F8.sf2)\) shows that the roughly4×4\\timesgap between GFF\-PIELM and FS\-PIELM\-L is consistent acrossμmax\\mu\_\{\\max\}values—a minor difference compared with the orders\-of\-magnitude gaps observed in the earlier, more spectrally demanding cases\. Collectively, the heat equation results delineate the boundary of applicability for the frequency shift mechanism: when the physics itself suppresses high\-frequency content, the variance reduction advantage becomes secondary, and all periodic\-activation PIELM variants perform comparably\.

### 4\.6Two\-Dimensional Advection\-Diffusion on Pacman Domain

All preceding cases use rectangular domains\. To assess performance on irregular geometries—where rejection sampling replaces uniform grids and boundary representations are more involved—we solve an advection\-diffusion equation on a Pacman\-shaped domain\. The re\-entrant corners of the wedge opening stress global cosine basis functions, and the advection term breaks the spatial symmetry\.

∂u∂t\+4​∂u∂x\+4​∂u∂y−\(∂2u∂x2\+∂2u∂y2\)\\displaystyle\\frac\{\\partial u\}\{\\partial t\}\+4\\frac\{\\partial u\}\{\\partial x\}\+4\\frac\{\\partial u\}\{\\partial y\}\-\\left\(\\frac\{\\partial^\{2\}u\}\{\\partial x^\{2\}\}\+\\frac\{\\partial^\{2\}u\}\{\\partial y^\{2\}\}\\right\)=f​\(x,y,t\),\(𝐱,t\)∈ΩP×\[0,1\],\\displaystyle=f\(x,y,t\),\\quad\(\\mathbf\{x\},t\)\\in\\Omega\_\{P\}\\times\[0,1\],\(4\.32\)u​\(𝐱,t\)\\displaystyle u\(\\mathbf\{x\},t\)=g​\(𝐱,t\),𝐱∈∂ΩP,\\displaystyle=g\(\\mathbf\{x\},t\),\\quad\\mathbf\{x\}\\in\\partial\\Omega\_\{P\},\(4\.33\)u​\(𝐱,0\)\\displaystyle u\(\\mathbf\{x\},0\)=u0​\(𝐱\),𝐱∈ΩP,\\displaystyle=u\_\{0\}\(\\mathbf\{x\}\),\\quad\\mathbf\{x\}\\in\\Omega\_\{P\},\(4\.34\)whereΩP\\Omega\_\{P\}is a Pacman domain defined as a circle centered at\(0\.5,0\.5\)\(0\.5,0\.5\)with radius0\.40\.4, with a wedge of half\-angleπ/4\\pi/4removed \(the “mouth” opening to the right\)\.

The exact solution features oscillations in both spatial directions with temporal decay:

uexact​\(x,y,t\)=e−0\.4​t​sin⁡\(3​π​x\)​sin⁡\(10​π​y\),u\_\{\\text\{exact\}\}\(x,y,t\)=e^\{\-0\.4t\}\\sin\(3\\pi x\)\\sin\(10\\pi y\),\(4\.35\)with spatial frequencies3​π≈9\.423\\pi\\approx 9\.42in thexx\-direction and10​π≈31\.4210\\pi\\approx 31\.42in theyy\-direction\. The source term is derived analytically:

f​\(x,y,t\)=\(−0\.4\+109​π2\)​u\+12​π​e−0\.4​t​cos⁡\(3​π​x\)​sin⁡\(10​π​y\)\+40​π​e−0\.4​t​sin⁡\(3​π​x\)​cos⁡\(10​π​y\)\.f\(x,y,t\)=\\left\(\-0\.4\+109\\pi^\{2\}\\right\)u\+12\\pi e^\{\-0\.4t\}\\cos\(3\\pi x\)\\sin\(10\\pi y\)\+40\\pi e^\{\-0\.4t\}\\sin\(3\\pi x\)\\cos\(10\\pi y\)\.\(4\.36\)
The network employsM=5000M=5000neurons withNC=8000N\_\{C\}=8000interior points sampled via rejection sampling within the Pacman domain,NB=600N\_\{B\}=600boundary points distributed along the circular arc and straight edges, andNI=800N\_\{I\}=800initial condition points\. The frequency parameters areμmin∈\{1,4,7,10,13,16\}\\mu\_\{\\min\}\\in\\\{1,4,7,10,13,16\\\}andμmax∈\{35,45,55,65,75,85\}\\mu\_\{\\max\}\\in\\\{35,45,55,65,75,85\\\}\.

The irregular geometry raises the overall error floor for all methods \(Table[6](https://arxiv.org/html/2607.01694#S4.T6)\)\. The best errors here \(10−410^\{\-4\}–10−510^\{\-5\}\) are considerably above those achieved on rectangular domains for comparable frequency content, due to the difficulty of enforcing boundary conditions on the curved arc and straight edges of the Pacman shape\. FS\-PIELM\-L nonetheless reaches3\.87×10−53\.87\\times 10^\{\-5\}, a23×23\\timesreduction compared with GFF\-PIELM \(8\.96×10−48\.96\\times 10^\{\-4\}\)\. The standard deviation of FS\-PIELM\-L \(5\.99×10−55\.99\\times 10^\{\-5\}\) is55×55\\timessmaller than that of GFF\-PIELM \(3\.27×10−33\.27\\times 10^\{\-3\}\), indicating that mean\-shifted weights are less sensitive to the particular random realization—a useful property when the collocation point distribution is itself irregular\. FS\-PIELM\-G tracks FS\-PIELM\-L closely \(5\.42×10−55\.42\\times 10^\{\-5\}versus3\.87×10−53\.87\\times 10^\{\-5\}\), suggesting that the moderate frequency content of this problem does not require per\-neuron frequency resolution\. Computation times remain comparable across methods \(11–13 s\)\.

Table 6:Performance comparison on the 2D advection\-diffusion equation on Pacman domain\. Results show optimal parameters and relativeL2L\_\{2\}error statistics over 5 random seeds\.![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/case06_advdiff2d_best_error.png)\(a\)Best relativeL2L\_\{2\}error as a function of\(μmin,μmax\)\(\\mu\_\{\\min\},\\mu\_\{\\max\}\); format as in Fig\.[3\(a\)](https://arxiv.org/html/2607.01694#S4.F3.sf1)\.
![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/case06_advdiff2d_lineplot_best.png)\(b\)Best relativeL2L\_\{2\}error versusμmax\\mu\_\{\\max\}at each method’s optimalμmin\\mu\_\{\\min\}; format as in Fig\.[3\(b\)](https://arxiv.org/html/2607.01694#S4.F3.sf2)\.

Figure 9:2D advection\-diffusion equation \(Pacman domain\):Comparison of five PIELM methods on a Pacman\-shaped domain \(circle with wedge removed, center\(0\.5,0\.5\)\(0\.5,0\.5\), radius0\.40\.4\)\. Spatial frequencies:3​π3\\piinxxand10​π10\\piinyy, with advection velocity\(4,4\)\(4,4\)\. Network configuration:M=5000M=5000neurons,NC=8000N\_\{C\}=8000interior points,NB=600N\_\{B\}=600boundary points,NI=800N\_\{I\}=800initial condition points\.The heatmaps \(Fig\.[9\(a\)](https://arxiv.org/html/2607.01694#S4.F9.sf1)\) show that SIREN\-PIELM and GFF\-PIELM achieve acceptable accuracy only in a narrow region nearμmin=16\\mu\_\{\\min\}=16andμmax≤55\\mu\_\{\\max\}\\leq 55, while FS\-PIELM\-L extends its optimal region to higherμmax\\mu\_\{\\max\}values\. The higher error floor compared with rectangular\-domain cases reflects the added difficulty of boundary conformity on the Pacman geometry: the wedge opening creates re\-entrant corners where the solution gradient changes abruptly, and global cosine basis functions—which are smooth and periodic—must superpose a large number of terms to approximate such localized features\. The lineplot \(Fig\.[9\(b\)](https://arxiv.org/html/2607.01694#S4.F9.sf2)\) shows that FS\-PIELM\-L maintains a consistent one\-order\-of\-magnitude advantage over SIREN\-PIELM and GFF\-PIELM acrossμmax\\mu\_\{\\max\}values from 35 to 85, suggesting that the mean\-shift advantage persists even when geometric complexity, rather than spectral bias alone, dominates the error\.

### 4\.7Helmholtz Equation on Panda\-Shaped Domain

The final benchmark imposes both geometric complexity and broad spectral content simultaneously\. The Panda\-shaped domain is non\-convex with multiple curvature changes, and the solution contains four distinct frequency components spanning a 7\.5:1 ratio—the most demanding combination in this study\. We solve:

∂2u∂x2\+∂2u∂y2\+u\\displaystyle\\frac\{\\partial^\{2\}u\}\{\\partial x^\{2\}\}\+\\frac\{\\partial^\{2\}u\}\{\\partial y^\{2\}\}\+u=f​\(x,y\),\(x,y\)∈ΩPanda,\\displaystyle=f\(x,y\),\\quad\(x,y\)\\in\\Omega\_\{\\text\{Panda\}\},\(4\.37\)u​\(x,y\)\\displaystyle u\(x,y\)=g​\(x,y\),\(x,y\)∈∂ΩPanda,\\displaystyle=g\(x,y\),\\quad\(x,y\)\\in\\partial\\Omega\_\{\\text\{Panda\}\},\(4\.38\)whereΩPanda\\Omega\_\{\\text\{Panda\}\}is a panda\-head\-shaped domain constructed using a polar representation with modulated radius incorporating ear bumps and facial features\.

The exact solution exhibits multi\-scale oscillatory behavior in thexx\-direction:

uexact​\(x,y\)=sin⁡\(π​x\)​cos⁡\(5​π​x\)\+0\.5​sin⁡\(10​π​x\)​cos⁡\(20​π​x\)\.u\_\{\\text\{exact\}\}\(x,y\)=\\sin\(\\pi x\)\\cos\(5\\pi x\)\+0\.5\\sin\(10\\pi x\)\\cos\(20\\pi x\)\.\(4\.39\)Using the product\-to\-sum identitysin⁡\(a\)​cos⁡\(b\)=12​\[sin⁡\(a\+b\)\+sin⁡\(a−b\)\]\\sin\(a\)\\cos\(b\)=\\frac\{1\}\{2\}\[\\sin\(a\+b\)\+\\sin\(a\-b\)\], this solution can be decomposed into four frequency components:

uexact=12​\[sin⁡\(6​π​x\)−sin⁡\(4​π​x\)\]\+14​\[sin⁡\(30​π​x\)−sin⁡\(10​π​x\)\],u\_\{\\text\{exact\}\}=\\frac\{1\}\{2\}\\left\[\\sin\(6\\pi x\)\-\\sin\(4\\pi x\)\\right\]\+\\frac\{1\}\{4\}\\left\[\\sin\(30\\pi x\)\-\\sin\(10\\pi x\)\\right\],\(4\.40\)with key frequencies4​π≈12\.574\\pi\\approx 12\.57,6​π≈18\.856\\pi\\approx 18\.85,10​π≈31\.4210\\pi\\approx 31\.42, and30​π≈94\.2530\\pi\\approx 94\.25\.

The source term is computed analytically from the second derivatives ofuexactu\_\{\\text\{exact\}\}, noting thatuy​y=0u\_\{yy\}=0since the solution depends only onxx\.

The network usesM=5000M=5000neurons withNC=10000N\_\{C\}=10000interior points sampled within the Panda domain andNB=600N\_\{B\}=600boundary points\. The frequency parameters areμmin∈\{1,4,7,10,13,16\}\\mu\_\{\\min\}\\in\\\{1,4,7,10,13,16\\\}andμmax∈\{80,90,100,110,120,130\}\\mu\_\{\\max\}\\in\\\{80,90,100,110,120,130\\\}\.

The Helmholtz\-Panda problem produces the largest performance gap among all benchmarks \(Table[7](https://arxiv.org/html/2607.01694#S4.T7)\)\. SIREN\-PIELM and GFF\-PIELM reach only𝒪​\(10−3\)\\mathcal\{O\}\(10^\{\-3\}\), while FS\-PIELM\-L achieves6\.15×10−86\.15\\times 10^\{\-8\}—a roughly37,000×37\{,\}000\\timesreduction relative to GFF\-PIELM and nearly five orders of magnitude below both baselines\. The standard deviation of FS\-PIELM\-L \(1\.46×10−61\.46\\times 10^\{\-6\}\) is14,000×14\{,\}000\\timessmaller than that of GFF\-PIELM \(2\.07×10−22\.07\\times 10^\{\-2\}\), indicating that the scaling\-based methods produce highly variable errors across seeds, occasionally reaching𝒪​\(10−3\)\\mathcal\{O\}\(10^\{\-3\}\)but more often yielding𝒪​\(10−2\)\\mathcal\{O\}\(10^\{\-2\}\)\. The wide spectral gap \(4​π4\\pito30​π30\\pi, ratio≈7\.5:1\\approx 7\.5:1\) on an irregular domain amplifies the variance penalty of multiplicative scaling in two ways: the high target frequency30​π30\\piinflates the weight variance quadratically, and the non\-convex boundary requires the basis functions to maintain accurate phase relationships even near the ears and facial features of the Panda contour—a task that becomes harder when the effective frequencies are spread over a wide, uncontrolled band\. The mean\-shift mechanism keeps each neuron’s effective frequency concentrated near its assigned value \(Theorem[3\.5](https://arxiv.org/html/2607.01694#S3.Thmtheorem5)\), preserving both spectral accuracy and spatial coherence\. FS\-PIELM\-G reaches1\.59×10−61\.59\\times 10^\{\-6\}, roughly26×26\\timesabove FS\-PIELM\-L, consistent with the pattern that per\-neuron frequency assignment provides finer resolution for multi\-component solutions\. All methods require 8–10 s of computation\.

Table 7:Performance comparison on the Helmholtz equation on Panda\-shaped domain\. Results show optimal parameters and relativeL2L\_\{2\}error statistics over 5 random seeds\.![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/case07_helmholtzpanda_best_error.png)\(a\)Best relativeL2L\_\{2\}error as a function of\(μmin,μmax\)\(\\mu\_\{\\min\},\\mu\_\{\\max\}\); format as in Fig\.[3\(a\)](https://arxiv.org/html/2607.01694#S4.F3.sf1)\.
![Refer to caption](https://arxiv.org/html/2607.01694v1/myfig/case07_helmholtzpanda_lineplot_best.png)\(b\)Best relativeL2L\_\{2\}error versusμmax\\mu\_\{\\max\}at each method’s optimalμmin\\mu\_\{\\min\}; format as in Fig\.[3\(b\)](https://arxiv.org/html/2607.01694#S4.F3.sf2)\.

Figure 10:Helmholtz equation \(Panda domain\):Comparison of five PIELM methods with multi\-scale frequency content \(key frequencies:4​π4\\pi,6​π6\\pi,10​π10\\pi,30​π30\\pi\)\. The solution exhibits purexx\-dependence with complex oscillatory behavior\. Network configuration:M=5000M=5000neurons,NC=10000N\_\{C\}=10000interior points,NB=600N\_\{B\}=600boundary points\.The heatmaps \(Fig\.[10\(a\)](https://arxiv.org/html/2607.01694#S4.F10.sf1)\) illustrate the contrast: SIREN\-PIELM and GFF\-PIELM produce errors no better than10−310^\{\-3\}even at their optimal parameters, while FS\-PIELM\-L maintains errors below10−610^\{\-6\}across a wide region of\(μmin,μmax\)\(\\mu\_\{\\min\},\\mu\_\{\\max\}\)\. The lineplot \(Fig\.[10\(b\)](https://arxiv.org/html/2607.01694#S4.F10.sf2)\) shows that FS\-PIELM\-L improves steadily withμmax\\mu\_\{\\max\}and stabilizes near10−810^\{\-8\}forμmax≥90\\mu\_\{\\max\}\\geq 90, which corresponds roughly to30​π≈9430\\pi\\approx 94—the highest frequency component in the solution\. The combination of an irregular boundary with four distinct frequency components \(4​π4\\pithrough30​π30\\pi\) amplifies both the variance penalty and the directional sensitivity of scaling\-based weights, making this the scenario where the mean\-shift mechanism provides its greatest advantage\. The roughly37,000×37\{,\}000\\timesimprovement factor observed here is the largest across all seven benchmarks, underscoring that the mean\-shift mechanism yields its largest gains precisely when spectral demands are most stringent—whether through high absolute frequencies, wide frequency ratios, or geometric complexity that compounds the approximation challenge\.

### 4\.8Comprehensive Results and Analysis

Table[8](https://arxiv.org/html/2607.01694#S4.T8)summarizes the best achieved relativeL2L\_\{2\}errors for all methods across the seven test cases\. FS\-PIELM\-L achieves the lowest error in six out of seven cases, with improvement factors over GFF\-PIELM ranging from19×19\\times\(the Klein\-Gordon equation\) to roughly37,000×37\{,\}000\\times\(the Helmholtz\-Panda problem\)\. The sole exception is the heat equation, where GFF\-PIELM leads marginally\. Tanh\-PIELM fails across all cases with errors exceeding10−210^\{\-2\}, as expected for a non\-periodic activation applied to oscillatory solutions\. The largest gains arise in problems with wide frequency ratios \(the Poisson equation:8,977×8\{,\}977\\times\) or combined geometric\-spectral complexity \(the Helmholtz\-Panda problem\), where adaptive frequency coverage is most beneficial\.

Table 8:Best relativeL2L\_\{2\}errors for all methods across seven test cases\. Bold values indicate the best performance for each case\. Each result represents the minimum error over 36 frequency parameter combinations and 5 random seeds\.Table[9](https://arxiv.org/html/2607.01694#S4.T9)presents the optimal frequency range parameters\[μmin,μmax\]\[\\mu\_\{\\min\},\\mu\_\{\\max\}\]for each method\. FS\-PIELM\-L operates effectively with lowerμmin\\mu\_\{\\min\}values \(often 5–16\) than SIREN\-PIELM and GFF\-PIELM \(typically 16–30\), suggesting that the frequency shift mechanism compensates for suboptimal base frequency ranges\. The optimalμmax\\mu\_\{\\max\}scales approximately with the highest target frequency in each problem, ranging from 70–110 for moderate\-frequency cases to 250–350 for the multi\-scale Poisson equation\. Across methods, FS\-PIELM\-L attains its best results with diverse\[μmin,μmax\]\[\\mu\_\{\\min\},\\mu\_\{\\max\}\]combinations, whereas competing methods require more precise parameter tuning\.

Table 9:Optimal frequency range parameters\[μmin,μmax\]\[\\mu\_\{\\min\},\\mu\_\{\\max\}\]for each method achieving the best relativeL2L\_\{2\}error\.Table[10](https://arxiv.org/html/2607.01694#S4.T10)provides average error and standard deviation across random seeds for the FS\-PIELM variants at their optimal parameters\. Across all seven cases, FS\-PIELM\-L consistently achieves lower average errors and smaller standard deviations than FS\-PIELM\-G, with typical improvement factors of 2–10×10\\timesin both metrics\. The coefficient of variation \(standard deviation divided by mean\) remains below 1\.0 for FS\-PIELM\-L in most cases, indicating reliable results without excessive sensitivity to random initialization\. Both FS\-PIELM variants maintain computation times comparable to baseline methods, with no measurable overhead from the frequency shift sampling\.

Table 10:Detailed performance statistics for FS\-PIELM variants at optimal parameters\. Results show mean±\\pmstandard deviation over 5 random seeds\.Across the seven benchmarks, a consistent pattern emerges: the improvement of FS\-PIELM\-L over GFF\-PIELM correlates with the spectral complexity of the problem\. For moderate frequency ratios \(the Klein\-Gordon equation, ratio66:11\), the gain is modest \(19×19\\times\), and for the heat equation \(ratio44:11\) where diffusion suppresses high\-frequency content, no gain is observed\. For wide ratios or high absolute frequencies \(the Poisson equation, ratio1515:11; the Helmholtz\-Panda problem, frequencies up to30​π30\\pion an irregular domain\), the gain exceeds three to four orders of magnitude\. This trend is consistent with the theoretical prediction of Theorem[3\.5](https://arxiv.org/html/2607.01694#S3.Thmtheorem5): the variance penalty of scaling grows quadratically with the target frequency \(Theorem[3\.3](https://arxiv.org/html/2607.01694#S3.Thmtheorem3)\), so the advantage of mean\-shifted weights becomes increasingly pronounced as spectral demands increase\.

## 5Conclusions

We have presented FS\-PIELM, a framework that controls the spectral content of an extreme learning machine through additive mean translation of the weight sampling distribution\. Because only the distribution mean moves while the covariance remains fixed, each neuron’s effective frequency concentrates tightly around its assigned target—in contrast to the spreading effect of multiplicative scaling \(Corollary[3\.6](https://arxiv.org/html/2607.01694#S3.Thmtheorem6)\)\.

Two variants—FS\-PIELM\-L \(per\-neuron frequencies\) and FS\-PIELM\-G \(grouped frequencies\)—have been evaluated on seven benchmarks spanning elliptic, parabolic, and hyperbolic PDEs on both rectangular and irregular domains\. FS\-PIELM\-L achieves the lowest error in six of seven cases, with improvements over GFF\-PIELM ranging from roughly20×20\\timesto37,000×37\{,\}000\\times\. The gains are largest when the problem involves a wide frequency ratio or complex geometry, consistent with the theoretical variance analysis\. The sole exception is the heat equation, where diffusion smooths high\-frequency content and all periodic\-activation methods perform comparably\. Both variants retain the single\-linear\-solve efficiency of the ELM framework\.

Three open questions remain\. First, the frequency boundsμmin\\mu\_\{\\min\}andμmax\\mu\_\{\\max\}must currently be prescribed; an adaptive procedure that infers suitable bounds from the PDE structure would broaden practical applicability\. Second, extension to nonlinear PDEs—where the ELM framework requires iterative solvers—introduces coupling between the frequency parameterization and the nonlinear iteration that merits dedicated study\. Third, rigorous convergence rates and approximation\-theoretic error bounds for the frequency\-shifted architecture have yet to be established\.

## Acknowledgments

This work was supported by the MIIT Key Laboratory of Dynamics and Control of Complex Systems at Northwestern Polytechnical University\.

Declaration of Generative AI and AI\-Assisted Technologies in the Writing Process

During the preparation of this work the authors used Claude \(Anthropic\) in order to improve language and readability\. After using this tool, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication\.

## References

- \[1\]Thomas J\. R\. Hughes\.The Finite Element Method: Linear Static and Dynamic Finite Element Analysis\.Dover Publications, Mineola, NY, 2000\.
- \[2\]John C\. Strikwerda\.Finite Difference Schemes and Partial Differential Equations\.SIAM, Philadelphia, 2nd edition, 2004\.
- \[3\]Claudio Canuto, M\. Yousuff Hussaini, Alfio Quarteroni, and Thomas A\. Zang\.Spectral Methods: Fundamentals in Single Domains\.Scientific Computation\. Springer, Berlin, 2006\.
- \[4\]Jiequn Han, Arnulf Jentzen, and Weinan E\.Solving high\-dimensional partial differential equations using deep learning\.Proceedings of the National Academy of Sciences, 115\(34\):8505–8510, 2018\.
- \[5\]Maziar Raissi, Alireza Yazdani, and George Em Karniadakis\.Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations\.Science, 367\(6481\):1026–1030, 2020\.
- \[6\]Isaac E\. Lagaris, Aristidis Likas, and Dimitrios I\. Fotiadis\.Artificial neural networks for solving ordinary and partial differential equations\.IEEE Transactions on Neural Networks, 9\(5\):987–1000, 1998\.
- \[7\]M\. Raissi, P\. Perdikaris, and G\. E\. Karniadakis\.Physics\-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations\.Journal of Computational Physics, 378:686–707, 2019\.
- \[8\]Weinan E and Bing Yu\.The deep Ritz method: A deep learning\-based numerical algorithm for solving variational problems\.Communications in Mathematics and Statistics, 6\(1\):1–12, 2018\.
- \[9\]Justin Sirignano and Konstantinos Spiliopoulos\.DGM: A deep learning algorithm for solving partial differential equations\.Journal of Computational Physics, 375:1339–1364, 2018\.
- \[10\]George Em Karniadakis, Ioannis G\. Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang\.Physics\-informed machine learning\.Nature Reviews Physics, 3\(6\):422–440, 2021\.
- \[11\]Shengze Cai, Zhicheng Wang, Sifan Wang, Paris Perdikaris, and George Em Karniadakis\.Physics\-informed neural networks for heat transfer problems\.Journal of Heat Transfer, 143\(6\):060801, 2021\.
- \[12\]Shengze Cai, Zhiping Mao, Zhicheng Wang, Minglang Yin, and George Em Karniadakis\.Physics\-informed neural networks \(PINNs\) for fluid mechanics: A review\.Acta Mechanica Sinica, 37\(12\):1727–1738, 2021\.
- \[13\]Ehsan Haghighat, Maziar Raissi, Adrian Moure, Hector Gomez, and Ruben Juanes\.A physics\-informed deep learning framework for inversion and surrogate modeling in solid mechanics\.Computer Methods in Applied Mechanics and Engineering, 379:113741, 2021\.
- \[14\]Ameya D\. Jagtap and George Em Karniadakis\.Extended physics\-informed neural networks \(XPINNs\): A generalized space\-time domain decomposition based deep learning framework for nonlinear partial differential equations\.Communications in Computational Physics, 28\(5\):2002–2041, 2020\.
- \[15\]Salvatore Cuomo, Vincenzo Schiano Di Cola, Fabio Giampaolo, Gianluigi Rozza, Maziar Raissi, and Francesco Piccialli\.Scientific machine learning through physics\-informed neural networks: Where we are and what’s next\.Journal of Scientific Computing, 92\(3\):88, 2022\.
- \[16\]Sifan Wang, Yujun Teng, and Paris Perdikaris\.Understanding and mitigating gradient flow pathologies in physics\-informed neural networks\.SIAM Journal on Scientific Computing, 43\(5\):A3055–A3081, 2021\.
- \[17\]Aditi S\. Krishnapriyan, Amir Gholami, Shandian Zhe, Robert M\. Kirby, and Michael W\. Mahoney\.Characterizing possible failure modes in physics\-informed neural networks\.InAdvances in Neural Information Processing Systems, volume 34, 2021\.
- \[18\]Nasim Rahaman, Aristide Barber, Devansh Arpit, Felix Draxler, Min Lin, Fred Hamprecht, Yoshua Bengio, and Aaron Courville\.On the spectral bias of neural networks\.International Conference on Machine Learning, pages 5301–5310, 2019\.
- \[19\]Zhi\-Qin John Xu, Yaoyu Zhang, Tao Luo, Yanyang Xiao, and Zheng Ma\.Frequency principle: Fourier analysis sheds light on deep neural networks\.Communications in Computational Physics, 28\(5\):1746–1767, 2020\.
- \[20\]Zhi\-Qin John Xu, Yaoyu Zhang, and Tao Luo\.Overview frequency principle/spectral bias in deep learning\.Communications on Applied Mathematics and Computation, 7\(3\):827–864, 2025\.
- \[21\]Yuan Cao, Zhiying Fang, Yue Wu, Ding\-Xuan Zhou, and Quanquan Gu\.Towards understanding the spectral bias of deep learning\.arXiv preprint arXiv:1912\.01198, 2019\.
- \[22\]Arthur Jacot, Franck Gabriel, and Clément Hongler\.Neural tangent kernel: Convergence and generalization in neural networks\.InAdvances in Neural Information Processing Systems, volume 31, pages 8580–8589, 2018\.
- \[23\]Sifan Wang, Xinling Yu, and Paris Perdikaris\.When and why pinns fail to train: A neural tangent kernel perspective\.Journal of Computational Physics, 449:110768, 2022\.
- \[24\]Ziqi Liu, Wei Cai, and Zhi\-Qin John Xu\.Multi\-scale deep neural network \(MscaleDNN\) for solving Poisson\-Boltzmann equation in complex domains\.Communications in Computational Physics, 28\(5\):1970–2001, 2020\.
- \[25\]Ameya D\. Jagtap, Kenji Kawaguchi, and George Em Karniadakis\.Adaptive activation functions accelerate convergence in deep and physics\-informed neural networks\.Journal of Computational Physics, 404:109136, 2020\.
- \[26\]Levi D\. McClenny and Ulisses M\. Braga\-Neto\.Self\-adaptive physics\-informed neural networks\.Journal of Computational Physics, 474:111722, 2023\.
- \[27\]Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich\-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng\.Fourier features let networks learn high frequency functions in low dimensional domains\.Advances in Neural Information Processing Systems, 33:7537–7547, 2020\.
- \[28\]Sifan Wang, Hanwen Wang, and Paris Perdikaris\.On the eigenvector bias of fourier feature networks: From regression to solving multi\-scale pdes with physics\-informed neural networks\.Computer Methods in Applied Mechanics and Engineering, 384:113938, 2021\.
- \[29\]Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gordon Wetzstein\.Implicit neural representations with periodic activation functions\.Advances in Neural Information Processing Systems, 33:7462–7473, 2020\.
- \[30\]Xiong Xiong, Kang Lu, Zhuo Zhang, Zheng Zeng, Sheng Zhou, Rongchun Hu, and Zichen Deng\.High\-frequency flow field super\-resolution via physics\-informed hierarchical adaptive fourier feature networks\.Physics of Fluids, 37\(9\):097111, 2025\.
- \[31\]Xiong Xiong, Zhuo Zhang, Rongchun Hu, Chen Gao, and Zichen Deng\.Separated\-variable spectral neural networks: A physics\-informed learning approach for high\-frequency PDEs\.arXiv preprint arXiv:2508\.00628, 2025\.
- \[32\]Xiong Xiong, Kang Lu, Zhuo Zhang, Zheng Zeng, Sheng Zhou, Zichen Deng, and Rongchun Hu\.J\-PIKAN: A physics\-informed KAN network based on Jacobi orthogonal polynomials for solving fluid dynamics\.Communications in Nonlinear Science and Numerical Simulation, 152:109414, 2026\.
- \[33\]Guang\-Bin Huang, Qin\-Yu Zhu, and Chee\-Kheong Siew\.Extreme learning machine: Theory and applications\.Neurocomputing, 70\(1\-3\):489–501, 2006\.
- \[34\]Guang\-Bin Huang, Hongming Zhou, Xiaojian Ding, and Rui Zhang\.Extreme learning machine for regression and multiclass classification\.IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 42\(2\):513–529, 2012\.
- \[35\]Vikas Dwivedi and Balaji Srinivasan\.Physics informed extreme learning machine \(pielm\)–a rapid method for the numerical solution of partial differential equations\.Neurocomputing, 391:96–118, 2020\.
- \[36\]Ali Rahimi and Benjamin Recht\.Random features for large\-scale kernel machines\.InAdvances in Neural Information Processing Systems, volume 20, pages 1177–1184, 2007\.
- \[37\]Nicholas H\. Nelsen and Andrew M\. Stuart\.The random feature model for input\-output maps between Banach spaces\.SIAM Journal on Scientific Computing, 43\(5\):A3212–A3243, 2021\.
- \[38\]Jingrun Chen, Xurong Chi, Weinan E, and Zhouwang Yang\.Bridging traditional and machine learning\-based algorithms for solving PDEs: The random feature method\.Journal of Machine Learning, 1\(3\):268–298, 2022\.
- \[39\]Francesco Calabrò, Gianluca Fabiani, and Constantinos Siettos\.Extreme learning machine collocation for the numerical solution of elliptic pdes with sharp gradients\.Computer Methods in Applied Mechanics and Engineering, 387:114188, 2021\.
- \[40\]Vikas Dwivedi and Balaji Srinivasan\.Solution of biharmonic equation in complicated geometries with physics informed extreme learning machine\.Journal of Computing and Information Science in Engineering, 20\(6\):061004, 2020\.
- \[41\]Gianluca Fabiani, Francesco Calabrò, Lucia Russo, and Constantinos Siettos\.Numerical solution and bifurcation analysis of nonlinear partial differential equations with extreme learning machines\.Journal of Scientific Computing, 89:44, 2021\.
- \[42\]Fei Ren, Pei\-Zhi Zhuang, Xiaohui Chen, Hai\-Sui Yu, and He Yang\.Physics\-informed extreme learning machine \(PIELM\) for Stefan problems\.Computer Methods in Applied Mechanics and Engineering, 441:118015, 2025\.
- \[43\]Bokai Zhu, Hengguang Li, and Qinghui Zhang\.Extended physics\-informed extreme learning machine for linear elastic fracture mechanics\.Computer Methods in Applied Mechanics and Engineering, 435:117655, 2025\.
- \[44\]Suchuan Dong and Zongwei Li\.Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations\.Computer Methods in Applied Mechanics and Engineering, 387:114129, 2021\.
- \[45\]Naxian Ni and Suchuan Dong\.Numerical computation of partial differential equations by hidden\-layer concatenated extreme learning machine\.Journal of Scientific Computing, 95\(2\):35, 2023\.
- \[46\]Enrico Schiassi, Roberto Furfaro, Carl Leake, Mario De Florio, Hunter Johnston, and Daniele Mortari\.Extreme theory of functional connections: A fast physics\-informed neural network method for solving ordinary and partial differential equations\.Neurocomputing, 457:334–356, 2021\.
- \[47\]Ho Dac Quan and Hieu Trung Huynh\.Solving partial differential equation based on extreme learning machine\.Mathematics and Computers in Simulation, 205:697–708, 2023\.
- \[48\]Vikas Dwivedi and Balaji Srinivasan\.A normal equation\-based extreme learning machine for solving linear partial differential equations\.Journal of Computing and Information Science in Engineering, 22\(1\):014502, 2022\.
- \[49\]He Yang, Fei Ren, Francesco Calabrò, Hai\-Sui Yu, Xiaohui Chen, and Pei\-Zhi Zhuang\.Physics\-informed extreme learning machine \(PIELM\): Opportunities and challenges\.arXiv preprint arXiv:2510\.24577, 2025\.
- \[50\]Fei Ren, Sifan Wang, Pei\-Zhi Zhuang, Hai\-Sui Yu, and He Yang\.General fourier feature physics\-informed extreme learning machine \(GFF\-PIELM\) for high\-frequency PDEs\.arXiv preprint arXiv:2510\.12293, 2025\.
- \[51\]Jingrun Chen, Weinan E, and Yixin Luo\.The random feature method for time\-dependent problems\.East Asian Journal on Applied Mathematics, 13\(3\):435–463, 2023\.
- \[52\]Jingrun Chen, Weinan E, and Yifei Sun\.Optimization of random feature method in the high\-precision regime\.Communications on Applied Mathematics and Computation, 6:1490–1517, 2024\.
- \[53\]Chunyang Liao\.Solving partial differential equations with random feature models\.Communications in Nonlinear Science and Numerical Simulation, 152:109343, 2026\.
- \[54\]Yifei Sun and Jingrun Chen\.Two\-level random feature methods for elliptic partial differential equations over complex domains\.Computer Methods in Applied Mechanics and Engineering, 441:117961, 2025\.
- \[55\]Tim De Ryck, Siddhartha Mishra, Yunfei Shang, and Fei Wang\.Approximation theory and applications of randomized neural networks for solving high\-dimensional PDEs\.InInternational Conference on Learning Representations \(ICLR\), 2025\.
- \[56\]Yuhang Wu, Ziyuan Liu, Wenjun Sun, and Xu Qian\.Randomized radial basis function neural network for solving multiscale elliptic equations\.Machine Learning: Science and Technology, 6:015033, 2025\.
- \[57\]Vikas Dwivedi, Nishant Parashar, and Balaji Srinivasan\.Distributed learning machines for solving forward and inverse problems in partial differential equations\.Neurocomputing, 420:299–316, 2021\.
- \[58\]Hang Zhou, Zhuhong Wang, Geping Qi, and Yisheng Wang\.IPIRNNs: Integration\-based physics informed randomized neural networks for solving stiff ODEs\.Neurocomputing, 662:131911, 2026\.
- \[59\]Subarna Khanra, Vijay Kumar Kukreja, and Indu Bala\.Physics\-informed neural networks for differential equation solutions: A comprehensive review\.Neurocomputing, 680:133317, 2026\.
- \[60\]Jens Berg and Kaj Nyström\.A unified deep artificial neural network approach to partial differential equations in complex geometries\.Neurocomputing, 317:28–41, 2018\.
- \[61\]Guijun Ma, Zidong Wang, Weibo Liu, Zeyuan Yang, Desheng Huang, and Han Ding\.Closed\-loop parameter optimization for robotic machining using physics\-informed machine learning and multiobjective optimization\.IEEE Transactions on Automation Science and Engineering, 22:22410–22422, 2025\.
- \[62\]Min Zhao, Zidong Wang, Jin Wan, Guoping Lu, and Weibo Liu\.A novel neural network architecture utilizing parametric\-logarithmic\-modulus\-based activation function: Theory, algorithm, and applications\.Knowledge\-Based Systems, 303:112425, 2024\.
- \[63\]Monica Bianchini, Paolo Frasconi, and Marco Gori\.Learning without local minima in radial basis function networks\.IEEE Transactions on Neural Networks, 6\(3\):749–756, 1995\.
- \[64\]Monica Bianchini and Franco Scarselli\.On the complexity of neural network classifiers: A comparison between shallow and deep architectures\.IEEE Transactions on Neural Networks and Learning Systems, 25\(8\):1553–1565, 2014\.
- \[65\]Juan Pedro Dominguez\-Morales, Angel Jimenez\-Fernandez, Antonio Rios\-Navarro, Elena Cerezuela\-Escudero, Daniel Gutierrez\-Galan, Manuel J\. Dominguez\-Morales, and Gabriel Jimenez\-Moreno\.Multilayer spiking neural network for audio samples classification using SpiNNaker\.InInternational Conference on Artificial Neural Networks \(ICANN\), volume 9886 ofLecture Notes in Computer Science, pages 45–53\. Springer, 2016\.
- \[66\]Juan Pedro Dominguez\-Morales, Qian Liu, Robert James, Daniel Gutierrez\-Galan, Angel Jimenez\-Fernandez, Simon Davidson, and Steve B\. Furber\.Deep spiking neural network model for time\-variant signals classification: A real\-time speech recognition approach\.In2018 International Joint Conference on Neural Networks \(IJCNN\), pages 1–8\. IEEE, 2018\.