Multi-scale Spectral Mixture Neural Operator

Abstract

Seeking effective numerical approximations for partial differential equations (PDEs) is a major challenge in modern science and technology. Recently, AI-inspired data-driven solvers, such as neural operators, have achieved great success in quickly PDE solving. However, in the design of neural operators, the processing of frequency domain information is crucial, while the importance of low-frequency information is excessively overlooked. In order to reduce low-frequency error in PDE solving, we propose the multi scale spectral mixture neural operator (MSSMNO) architecture and design a residual learning structure that transfers residuals in multi-scale cycle by combining the frequency domain learning patterns of the FNO method. Meanwhile, we designed interpolation operator and restriction operator to effectively transmit and reconstruct high-frequency information in MSSMNO. Experimentally, MSSMNO achieves state-of-the-art and yields a relative error reduction of 21.9% averaged on four classical benchmarks.