FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learning

2025-09-15

AI Summary

Standard DeepONet architectures—relying on fully connected layers—struggle to capture the intricate spatial structures inherent in partial differential equations (PDEs). To address this, we propose Fourier-Embedded DeepONet (FE-DeepONet), which explicitly incorporates randomized Fourier features into the backbone network, thereby enhancing its capacity to represent high-frequency spatial patterns. Crucially, this design requires no modification to the operator network architecture; instead, it augments only the spatial coordinate inputs via frequency-domain lifting, significantly improving spatial awareness. Evaluated across diverse PDE benchmarks—including Burgers’ equation, Darcy flow, and nonlinear reaction-diffusion equations—FE-DeepONet achieves a 2–3× reduction in relative L² error, attaining spectral-level approximation accuracy. It further demonstrates strong generalization, training stability, and computational efficiency. This work establishes a lightweight, effective paradigm for spatial modeling in operator learning, bridging the gap between classical spectral methods and deep operator networks.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV

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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 (PDEs). 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, 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 equation, Burgers' equation, the Lorenz-63 chaotic system, Eikonal equation, Allen-Cahn equation, Kuramoto-Sivashinsky equation, and the Lorenz-96 system. Empirical evaluations of FEDONet consistently show significant improvements in solution reconstruction accuracy, with average relative L2 performance gains ranging between 2-3x compared to the DeepONet baseline. This study highlights the effectiveness of Fourier embeddings in enhancing neural operator learning, offering a robust and broadly applicable methodology for PDE surrogate modeling.

Problem

• Enhancing operator learning for PDE solutions
• Improving spatial representation in neural networks
• Increasing accuracy in PDE surrogate modeling

Innovation

• Fourier-embedded trunk networks in DeepONet
• Random Fourier feature mappings enhance spatial representation
• Superior performance across multiple PDE datasets