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

TL;DR

This paper introduces Deep Parallel Spectral Neural Operators (DPNO), which significantly improve low-frequency information learning in neural operators for solving PDEs, demonstrating high performance and resolution invariance on challenging datasets.

Contribution

The paper proposes a novel DPNO architecture that enhances low-frequency learning through parallel modules and convolutional smoothing, addressing limitations of existing neural operators.

Findings

DPNO outperforms existing neural operators on challenging PDE datasets.

DPNO demonstrates resolution invariance in solving PDEs.

Enhanced low-frequency learning improves overall PDE solving accuracy.

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.