🤖 AI Summary
This study addresses the limitations of Fourier Neural Operators (FNOs) in learning high-frequency components due to frequency truncation, as well as the challenges in coordinate encoding design. To this end, we propose the CAFE+FNO framework, which introduces a novel CAFE+ mechanism that fuses Fourier-Chebyshev features via the Hadamard product to enhance frequency-domain interactions. By integrating implicit neural representations, parallel affine branches, and shared-kernel MLPs, the method generates Fourier kernels through explicit feature composition, enabling efficient PDE solving with model parameters independent of the number of modes. Experiments across five PDE benchmarks demonstrate that the proposed approach significantly outperforms existing FNO variants, fully validating its advantages in bandwidth learnability.
📝 Abstract
The Fourier Neural Operator (FNO) learns solution operators of partial differential equations (PDEs) through Fourier-space kernel parameterization, but frequency truncation can limit the learning of high-frequency variations. AM-FNO and SirenFNO generate kernels for all grid modes from spectral coordinates using shared networks, making coordinate encoding and generator design important. Recent work on implicit neural representations (INRs) has proposed constructing frequency interactions through explicit feature composition rather than relying on subsequent MLPs to form them implicitly. Building on this approach, we propose CAFE+FNO, which incorporates Content-Aware Frequency Encoding+ (CAFE+) into Fourier kernel generation. CAFE+ combines Fourier--Chebyshev features through parallel affine branches and a Hadamard product, forming interactions within and across the two feature families. A kernel MLP maps the resulting representation of each normalized spectral coordinate to a complex channel-mixing matrix. Each layer shares its generator across all stored modes, making the number of trainable parameters independent of the number of modes for a fixed architecture. We compare CAFE+FNO with existing FNO variants on five PDE benchmarks and conduct ablation studies on basis configuration, multiplicative composition, and bandwidth learnability. Code and experimental configurations are available at https://github.com/fabsk101/CAFEPlusFNO.git.