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

• URL: http://arxiv.org/abs/2508.00628v1
• Date: Fri, 01 Aug 2025 13:40:10 GMT
• Title: Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs
• Authors: Xiong Xiong, Zhuo Zhang, Rongchun Hu, Chen Gao, Zichen Deng,
• Abstract summary: Separated-Variable Spectral Neural Networks (SV-SNN) is a novel framework that addresses the spectral bias problem in neural PDE solving.<n>We show that SV-SNN achieves 1-3 orders of magnitude of improvement in accuracy while reducing parameter count by over 90%.
• Score: 21.081644719506453
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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