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

arXiv: https://arxiv.org/abs/2508.00628 · HTML: https://mlsanity.com/paper/2508.00628/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 36 references that are in the map.
- First posted on arXiv 1 Aug 2025.

## Builds on (most cited first, 20 of 20)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [Spectral Bias in Practice: The Role of Function Frequency in Generalization](https://mlsanity.com/paper/2110.02424/) (2021)
- [Training-image based geostatistical inversion using a spatial generative adversarial neural network](https://mlsanity.com/paper/1708.04975/) (2017)
- [On the Activation Function Dependence of the Spectral Bias of Neural Networks](https://mlsanity.com/paper/2208.04924/) (2022)
- [Binary structured physics-informed neural networks for solving equations with rapidly changing solutions](https://mlsanity.com/paper/2401.12806/) (2024)
- [Physics-informed neural wavefields with Gabor basis functions](https://mlsanity.com/paper/2310.10602/) (2023)
- [Frequency-adaptive Multi-scale Deep Neural Networks](https://mlsanity.com/paper/2410.00053/) (2024)
- [Spectrum-Informed Multistage Neural Networks: Multiscale Function Approximators of Machine Precision](https://mlsanity.com/paper/2407.17213/) (2024)

## Built on by (most cited first, 3 of 3)

- [Gradient Scaling Effects in Adaptive Spectral PINNs for Stiff Nonlinear ODEs](https://mlsanity.com/paper/2605.04502/) (2026)
- [Multi-Scale Separable Fourier Neural Networks for Solving High-Frequency PDEs](https://mlsanity.com/paper/2605.31027/) (2026)
- [Frequency Shift Physics-Informed Extreme Learning Machine for Solving High-Frequency Partial Differential Equations](https://mlsanity.com/paper/2607.01694/) (2026)

## Abstract (arXiv)

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.