FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learning

Abstract

Deep Operator Networks (DeepONets) have recently emerged as powerful data-driven frameworks for learning nonlinear operators, particularly suited for approximating solutions to partial differential equations. Despite their promising capabilities, the standard implementation of DeepONets, which typically employs fully connected linear layers in the trunk network, can encounter limitations in capturing complex spatial structures inherent to various PDEs. To address this limitation, we introduce Fourier-Embedded trunk networks within the DeepONet architecture, leveraging random fourier feature mappings to enrich spatial representation capabilities. Our proposed Fourier-Embedded DeepONet, FEDONet demonstrates superior performance compared to the traditional DeepONet across a comprehensive suite of PDE-driven datasets, including the two-dimensional Poisson, Burgers', Lorenz-63, Eikonal, Allen-Cahn, and the Kuramoto-Sivashinsky equation. FEDONet delivers consistently superior reconstruction accuracy across all benchmark PDEs, with particularly large relative $L^2$ error reductions observed in chaotic and stiff systems. This study highlights the effectiveness of Fourier embeddings in enhancing neural operator learning, offering a robust and broadly applicable methodology for PDE surrogate modeling.