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arXiv · 2508.00628

Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs

Xiong Xiong·Zhuo Zhang·Rongchun Hu·Chen Gao·Zichen Deng
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Abstract

Solving high-frequency oscillatory partial differential equations (PDEs) is a critical challenge in scientific computing, with applications in fluid mechanics, quantum mechanics, and electromagnetic wave propagation. Traditional physics-informed neural networks (PINNs) suffer from spectral bias, limiting their ability to capture high-frequency solution components. We introduce Separated-Variable Spectral Neural Networks (SV-SNN), a novel framework that addresses these limitations by integrating separation of variables with adaptive spectral methods. Our approach features three key innovations: (1) decomposition of multivariate functions into univariate function products, enabling independent spatial and temporal networks; (2) adaptive Fourier spectral features with learnable frequency parameters for high-frequency capture; and (3) theoretical framework based on singular value decomposition to quantify spectral bias. Comprehensive evaluation on benchmark problems including Heat equation, Helmholtz equation, Poisson equations and Navier-Stokes equations demonstrates that SV-SNN achieves 1-3 orders of magnitude improvement in accuracy while reducing parameter count by over 90\% and training time by 60\%. These results establish SV-SNN as an effective solution to the spectral bias problem in neural PDE solving. The implementation will be made publicly available upon acceptance at https://github.com/xgxgnpu/SV-SNN.

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Xiong Xiong, Zhuo Zhang, Rongchun Hu, Chen Gao, Zichen Deng. 2025-08-01. Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs. https://arxiv.org/abs/2508.00628

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