Multi-scale low-frequency enhanced spectral neural operator for reducing low-frequency error in partial differential equations solving

Abstract

Designing universal artificial intelligence (AI) solver for partial differential equations (PDEs) is an open problem and a significant challenge in science and engineering. AI-inspired data-driven solvers, such as neural operators, have achieved great success in PDE solving. However, insufficient low-frequency learning ability and inability to utilize physical prior knowledge remain an obstacle for the PDEs solver which designed by neural operator. To tackle this challenge, we drew inspiration from the multigrid method and developed the multi-scale low-frequency enhanced spectral neural operator. Our approach broadens the neural operator’s learnable range in the frequency domain by folding the frequency spectrum, thereby reducing low-frequency error through a carefully designed residual structure. Additionally, to adapt to the varying spectral distributions of different PDEs, we propose a neural operator correction strategy based on the correspondence between the PDE spectrum distribution pattern and the neural operator learning pattern in the low-frequency region, which we summarized, to correct the results by utilizing the prior knowledge of the PDE. Extensive experiments on benchmark fluid datasets, including the Darcy equation, the Navier–Stokes equation, and their variants, demonstrate that our model achieves a 26.7% reduction in low-frequency error and a 25.6% improvement in accuracy, outperforming traditional neural operators and verifying the effectiveness of the proposed method.