用傅里叶嵌入提升DeepONet对偏微分方程的逼近精度。

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 use Fourier-Embedded trunk networks within the DeepONet architecture, leveraging random Fourier features to enrich spatial representation capabilities. The Fourier-Embedded DeepONet (FEDONet) demonstrates superior performance compared to the traditional DeepONet across a comprehensive suite of PDE-driven datasets, including the Burgers', 2D Poisson, Eikonal, Allen-Cahn, and the Kuramoto-Sivashinsky equation. To systematically evaluate the effectiveness of the architectures, we perform comparisons across multiple training dataset sizes and input noise levels. 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 work demonstrates the effectiveness of Fourier embeddings in enhancing neural operator learning, offering a robust and broadly applicable methodology for PDE surrogate modeling.