Deep Parallel Spectral Neural Operators for Solving Partial Differential Equations with Enhanced Low-Frequency Learning Capability
This work aims to improve the accuracy of data-driven PDE solvers, specifically neural operators, for researchers and engineers working with complex physical simulations where low-frequency information is critical.
This paper addresses the challenge of enhancing low-frequency learning in neural operators for solving partial differential equations (PDEs). The proposed Deep Parallel Spectral Neural Operator (DPNO) uses parallel modules to improve low-frequency information learning and convolutional mappings to reduce high-frequency errors caused by truncation coefficients, demonstrating exceptional performance on several challenging PDE datasets.
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.