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FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learning

Arth Sojitra, Mrigank Dhingra, Omer San

TL;DR

The paper addresses neural operator learning for PDEs, where vanilla DeepONet struggles to represent high-frequency and multiscale features. It introduces FEDONet, which injects fixed random Fourier features into the trunk input, acting as a spectral preconditioner and effectively approximating shift-invariant kernels via Bochner’s theorem, thereby expanding the hypothesis class beyond the vanilla DeepONet. Empirical results across seven PDE families, including the 2D Poisson equation and the Kuramoto–Sivashinsky equation, show substantially improved reconstruction accuracy and spectral fidelity, with particularly large gains in chaotic and stiff regimes. FEDONet maintains the original architecture’s simplicity and incurs negligible runtime overhead, offering a robust and scalable approach for PDE surrogate modeling and scientific computing.

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.

FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learning

TL;DR

The paper addresses neural operator learning for PDEs, where vanilla DeepONet struggles to represent high-frequency and multiscale features. It introduces FEDONet, which injects fixed random Fourier features into the trunk input, acting as a spectral preconditioner and effectively approximating shift-invariant kernels via Bochner’s theorem, thereby expanding the hypothesis class beyond the vanilla DeepONet. Empirical results across seven PDE families, including the 2D Poisson equation and the Kuramoto–Sivashinsky equation, show substantially improved reconstruction accuracy and spectral fidelity, with particularly large gains in chaotic and stiff regimes. FEDONet maintains the original architecture’s simplicity and incurs negligible runtime overhead, offering a robust and scalable approach for PDE surrogate modeling and scientific computing.

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 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.

Paper Structure

This paper contains 17 sections, 49 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Methodology
  3. Fourier Embeddings as Spectral Preconditioners
  4. Training Objective
  5. Evaluation and Generalization
  6. Results
  7. 2D Poisson Equation
  8. Burgers' Equation
  9. Lorenz-63 System
  10. Eikonal Equation
  11. Allen–Cahn Equation
  12. Kuramoto--Sivashinsky Equation
  13. Summary
  14. Future Work
  15. Whitening Effect of Fourier Feature Embeddings
  16. ...and 2 more sections

Figures (19)

  • Figure 1: FEDONet: Fourier Embedded Deep operator Network
  • Figure 2: Solving a 2D Poisson equation: Comparison of predicted and reference solution fields of DeepONet and FEDONet
  • Figure 3: Energy spectrum comparison between the ground truth, vanilla and Fourier DeepOnet reconstructions for the 2D Poisson Dataset.
  • Figure 4: Solving a parametric Burgers' equation: Comparison of predicted spatio-temporal field for a random initial condition. Relative $L^2$ error: $4.09\%$ for DeepONet, $1.49\%$ for FEDONet.
  • Figure 5: Energy spectrum and pointwise absolute error across space and time for a representative sample for Burgers' Equation.
  • ...and 14 more figures