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Deep Parallel Spectral Neural Operators for Solving Partial Differential Equations with Enhanced Low-Frequency Learning Capability

Qinglong Ma, Peizhi Zhao, Sen Wang, Tao Song

TL;DR

The paper addresses the challenge of learning low-frequency components in neural operators for PDEs by introducing the Deep Parallel Spectral Neural Operator (DPNO). DPNO employs dual-branch parallel Fourier learning blocks and a projection network to smooth high-frequency content, achieving resolution-invariant performance across multiple PDE benchmarks. Empirical results on six PDE datasets show that DPNO improves over strong baselines (e.g., up to 36.3% relative reduction in MSE versus FNO) and demonstrates zero-shot super-resolution capabilities, validating its robust multi-scale frequency learning. The work highlights the practical potential of spectral, frequency-aware neural operators for efficient, mesh-invariant PDE solvers, with future directions toward physics-informed enhancements to further improve long-term accuracy and data efficiency.

Abstract

Designing universal artificial intelligence (AI) solver for partial differential equations (PDEs) is an open-ended problem and a significant challenge in science and engineering. Currently, data-driven solvers have achieved great success, such as neural operators. However, the ability of various neural operator solvers to learn low-frequency information still needs improvement. In this study, we propose a Deep Parallel Spectral Neural Operator (DPNO) to enhance the ability to learn low-frequency information. Our method enhances the neural operator's ability to learn low-frequency information through parallel modules. In addition, due to the presence of truncation coefficients, some high-frequency information is lost during the nonlinear learning process. We smooth this information through convolutional mappings, thereby reducing high-frequency errors. We selected several challenging partial differential equation datasets for experimentation, and DPNO performed exceptionally well. As a neural operator, DPNO also possesses the capability of resolution invariance.

Deep Parallel Spectral Neural Operators for Solving Partial Differential Equations with Enhanced Low-Frequency Learning Capability

TL;DR

The paper addresses the challenge of learning low-frequency components in neural operators for PDEs by introducing the Deep Parallel Spectral Neural Operator (DPNO). DPNO employs dual-branch parallel Fourier learning blocks and a projection network to smooth high-frequency content, achieving resolution-invariant performance across multiple PDE benchmarks. Empirical results on six PDE datasets show that DPNO improves over strong baselines (e.g., up to 36.3% relative reduction in MSE versus FNO) and demonstrates zero-shot super-resolution capabilities, validating its robust multi-scale frequency learning. The work highlights the practical potential of spectral, frequency-aware neural operators for efficient, mesh-invariant PDE solvers, with future directions toward physics-informed enhancements to further improve long-term accuracy and data efficiency.

Abstract

Designing universal artificial intelligence (AI) solver for partial differential equations (PDEs) is an open-ended problem and a significant challenge in science and engineering. Currently, data-driven solvers have achieved great success, such as neural operators. However, the ability of various neural operator solvers to learn low-frequency information still needs improvement. In this study, we propose a Deep Parallel Spectral Neural Operator (DPNO) to enhance the ability to learn low-frequency information. Our method enhances the neural operator's ability to learn low-frequency information through parallel modules. In addition, due to the presence of truncation coefficients, some high-frequency information is lost during the nonlinear learning process. We smooth this information through convolutional mappings, thereby reducing high-frequency errors. We selected several challenging partial differential equation datasets for experimentation, and DPNO performed exceptionally well. As a neural operator, DPNO also possesses the capability of resolution invariance.
Paper Structure (22 sections, 15 equations, 9 figures, 5 tables)

This paper contains 22 sections, 15 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Neural operators
  4. Spatial learning with attention mechanism
  5. Spectral learning via spectral analysis techniques
  6. Proposed method
  7. Problem setup
  8. Fourier neural operator
  9. DPNO architecture
  10. Experiments
  11. Experiment setup
  12. Benchmarks
  13. Baseline
  14. Dataset
  15. Results and discussion
  16. ...and 7 more sections

Figures (9)

  • Figure 1: DPNO overall architecture
  • Figure 2: The first row shows the true values (with color representing the magnitude of stress) and the predicted values, while the second row shows the model's absolute error.
  • Figure 3: Airfoil: The first row shows the true values (with color representing the magnitude of the velocity.) and the predicted values, while the second row shows the model's absolute error.
  • Figure 4: Pipe: The first row shows the true values (with color representing the magnitude of the velocity.) and the predicted values, while the second row shows the model's absolute error.
  • Figure 5: Some real values of the Darcy data and their frequency distributions, with the low frequencies being closer to the center point.
  • ...and 4 more figures