Fourier neural operator improves PDE solutions with feature composition
CAFE+FNO: Fourier Kernel Generation via Multiplicative Feature Composition
Machine Learning
Summary
Solving complicated physics problems often involves partial differential equations (PDEs). The Fourier Neural Operator helps by learning how to predict solutions quickly, but it can struggle to capture detailed small-scale patterns. The authors introduce a new method called CAFE+FNO that better combines different mathematical features to more accurately model these details. Their approach reduces model size while improving performance on several test problems.
What this means in practice
- •For computational engineers: Enhance accuracy and efficiency in numerical simulations of physical systems governed by PDEs using the improved kernel generation technique.
- •For software developers in simulation tools: Integrate the CAFE+FNO method into simulation software to reduce parameter counts and improve model scalability across different problem sizes.
Authors
Hyungjoon Juen, Minwoo Shin
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.