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Windowed Fourier Propagator: A Frequency-Local Neural Operator for Wave Equations in Inhomogeneous Media

Yiyang Cai, Zixuan Qiu, Yunlu Shu, Jiamao Wu, Yingzhou Li, Tianyu Wang, Xi Chen

Abstract

Wave equations are fundamental to describing a vast array of physical phenomena, yet their simulation in inhomogeneous media poses a computational challenge due to the highly oscillatory nature of the solutions. To overcome the high costs of traditional solvers, we propose the Windowed Fourier Propagator (WFP), a novel neural operator that efficiently learns the solution operator. The WFP's design is rooted in the physical principle of frequency locality, where wave energy scatters primarily to adjacent frequencies. By learning a set of compact, localized propagators, each mapping an input frequency to a small window of outputs, our method avoids the complexity of dense interaction models and achieves computational efficiency. Another key feature is the explicit preservation of superposition, which enables remarkable generalization from simple training data (e.g., plane waves) to arbitrary, complex wave states. We demonstrate that the WFP provides an explainable, efficient and accurate framework for data-driven wave modeling in complex media.

Windowed Fourier Propagator: A Frequency-Local Neural Operator for Wave Equations in Inhomogeneous Media

Abstract

Wave equations are fundamental to describing a vast array of physical phenomena, yet their simulation in inhomogeneous media poses a computational challenge due to the highly oscillatory nature of the solutions. To overcome the high costs of traditional solvers, we propose the Windowed Fourier Propagator (WFP), a novel neural operator that efficiently learns the solution operator. The WFP's design is rooted in the physical principle of frequency locality, where wave energy scatters primarily to adjacent frequencies. By learning a set of compact, localized propagators, each mapping an input frequency to a small window of outputs, our method avoids the complexity of dense interaction models and achieves computational efficiency. Another key feature is the explicit preservation of superposition, which enables remarkable generalization from simple training data (e.g., plane waves) to arbitrary, complex wave states. We demonstrate that the WFP provides an explainable, efficient and accurate framework for data-driven wave modeling in complex media.
Paper Structure (27 sections, 5 theorems, 37 equations, 11 figures, 2 tables)

This paper contains 27 sections, 5 theorems, 37 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Works
  3. The Method and Contributions
  4. The wave equation
  5. The principle of superposition
  6. Propagation of singularities
  7. Energy transfer among frequency components
  8. Problem Setup and Preliminaries
  9. Main Result
  10. Proof of Theorem \ref{['thm:main']}
  11. The windowed Fourier propagator
  12. The frequency window and extraction function
  13. Architecture of the WFP
  14. Neural Network Architecture for Learning the Propagator
  15. Numerical Experiments
  16. ...and 12 more sections

Key Result

Theorem 1

Consider the wave evolution over $t \in [0,T]$. For any initially unexcited frequency $\mathbf{n}_1 \neq \mathbf{k}_0$, we define the first-order approximate solution$\hat{u}_{\mathbf{n}_1}^{\text{approx}}(t)$ as the solution to the linearized inhomogeneous equation driven solely by the dominant mod with zero initial conditions. By Duhamel's principle, this is explicitly given by: If $\epsilon \l

Figures (11)

  • Figure 1: Comparison of Frequency Interaction Patterns: The left panel shows the fully connected network with full frequency coupling, the middle panel illustrates the standard FNO with independent and elementwise mode processing, the right panel demonstrates the WFP's localized windowed interactions.
  • Figure 2: Numerical evidence of frequency locality. The plot shows the Fourier spectrum of the wave solution at different time slices. Initially, only a single frequency is excited (back plane). As time progresses, energy spreads to adjacent frequencies, but remains highly localized in a growing area.
  • Figure 3: Overview of the WFP network architecture
  • Figure 4: The structure of the neural network. An input token is processed through two branches: a main processing path with a custom activation and a gate branch with a sigmoid. Their element-wise product yields the final output, allowing the gate to control information flow hochreiter1997long and emphasize relevant features.
  • Figure 5: Network Performance for 2D In Distribution Case - Limited Initial Frequencies
  • ...and 6 more figures

Theorems & Definitions (10)

  • Theorem 1: Frequency Locality
  • Lemma 2: Energy Conservation
  • Lemma 3: Uniform Bound on Spatial Gradient
  • proof
  • Lemma 4: Estimate of Dominant Frequency Perturbation
  • proof
  • Lemma 5: Energy Bound for Scattered Modes
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Definition 4.1