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

2026-07-02

AI Summary

This work addresses the challenge in physics-informed machine learning where neural networks struggle to approximate high-frequency solutions of partial differential equations due to spectral bias. To overcome this limitation, the authors propose Frequency-Shifted Physics-Informed Extreme Learning Machines (FS-PIELM), which employ additive mean-shifting of Gaussian weights during initialization to embed a high-frequency prior while preserving constant variance—thereby avoiding the variance amplification inherent in conventional scaling approaches and retaining the computational efficiency of single-shot linear solves characteristic of extreme learning machines. Theoretical analysis establishes that the resulting frequency variance remains bounded and converges to one. Two variants are introduced: FS-PIELM-L, which allocates frequencies at the neuron level, and FS-PIELM-G, which enhances robustness through grouped frequency assignment. Evaluated across seven benchmark problems spanning six equation classes, the linear variant achieves state-of-the-art accuracy in six cases, outperforming existing PIELM methods by one to nearly five orders of magnitude.

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.

Problem

• high-frequency partial differential equations
• spectral bias
• physics-informed machine learning
• frequency shift
• extreme learning machine

Innovation

• Frequency Shift
• Physics-Informed Neural Networks
• Extreme Learning Machine
• Spectral Bias
• High-Frequency PDEs