Enhancing Fourier Neural Operators with Local Spatial Features

Abstract. Partial differential equation (PDE) problems often exhibit strong local spatial structures, and accurately capturing these features is essential for high-quality solution approximation. The Fourier neural operator (FNO) has recently emerged as a powerful method for solving PDEs by leveraging frequency-domain parameterization to efficiently model global patterns. In addition to its high predictive accuracy, FNO is valued for its resolution-invariant property. However, this global emphasis comes at the expense of local spatial fidelity, as frequency-based convolutions inherently lack the ability to effectively encode fine-grained local dependencies. Although several studies have attempted to address this limitation without compromising the resolution invariance, their methods often fall short in capturing sufficient local spatial features (LSFs) and tend to significantly increase computational cost. To overcome this challenge, we propose a simple yet effective hybrid framework, termed Conv-FNO, which incorporates a convolutional neural network–based feature preextractor to capture LSFs directly from the input data. This architecture enhances the local representational capacity of FNO while maintaining its desirable global properties. To preserve resolution invariance, we introduce two resizing schemes for inputs with different resolutions. Theoretical analysis is provided to justify the effectiveness of our resizing schemes across varying resolutions. We conducted experiments on a wide range of representative PDEs to verify the effectiveness of Conv-FNO. Experimental results show that Conv-FNO not only significantly outperforms the standard FNO but also consistently surpasses other baseline methods. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/cyliu111/Conv-FNO and in the supplementary materials ( Conv-FNO-main.zip [342KB]). [Formula: see text]