🤖 AI Summary
To address the weak low-frequency modeling capability and significant high-frequency truncation errors in data-driven PDE solvers (e.g., neural operators), this paper proposes the Deep Parallel Spectral Neural Operator (DPNO). Methodologically: (1) it introduces a novel parallel frequency-domain module that explicitly enhances multi-scale low-frequency feature learning; (2) it incorporates a lightweight convolutional mapping to smooth high-frequency reconstruction and mitigate spectral truncation artifacts while preserving resolution invariance. The contributions are threefold: DPNO achieves state-of-the-art performance across multiple PDE benchmarks—including Navier–Stokes and Darcy flow—reducing average low-frequency reconstruction error by 23.6% over existing neural operators; it demonstrates strong cross-resolution generalization; and it establishes a scalable architectural paradigm for universal AI-based PDE solvers.
📝 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.