cs.LGMar 15, 2026

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

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

Organizations: School of Mathematical Sciences, Fudan University, Shanghai, 200433, China · Shanghai Center for Mathematical Sciences, Fudan University, Shanghai, 200438, China · Center for Applied Mathematics, Fudan University, Shanghai, 200438, China

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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.

Figures & tables

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 [ 10 ] and emphasize relevant features.
Figure 5: Network Performance for 2D In Distribution Case - Limited Initial Frequencies
Figure 6: Network Performance for 2D In Distribution Case - Gaussian Beam
Figure 7: OOD Discontinuity Generalization: Performance on a velocity field with a sharp discontinuity. The model handles small jumps well but struggles as the discontinuity strength increases.
Figure 8: OOD Disturbance Generalization: Performance on velocity fields with perturbation strengths beyond the training distribution. The model shows graceful and predictable degradation as the input moves further from the training data.
Figure 9: L1 Error vs. Disturbance Strength: Error increases predictably as the perturbation strength exceeds the training distribution, demonstrating the model’s limited but reliable extrapolation capabilities.
Activation FunctionTraining Loss (MSE)Test Loss (MSE)
tanh⁡\tanh1×10−41\times 10^{-4}2×10−42\times 10^{-4}
Leaky ReLU (slope=0.05)4×10−54\times 10^{-5}2×10−42\times 10^{-4}
exp⁡(−4x2)sin⁡(3πx)\exp(-4x^{2})\sin(3\pi x)5×10−55\times 10^{-5}6×10−56\times 10^{-5}
exp⁡(−20x2)sin⁡(10πx)\exp(-20x^{2})\sin(10\pi x)4×10−6\mathbf{4\times 10^{-6}}8×10−6\mathbf{8\times 10^{-6}}
Table 1: Ablation Study: Impact of Activation Function on Training and Test Loss.
Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 10: Visualization of 1D Network Training Results
Figure 11: Effect of Different Activation Functions on Network Training Results
MODELTraining Loss (MSE)Test Loss (MSE)
FNO (Elementwise RR )4×10−44\times 10^{-4}4×10−44\times 10^{-4}
FNO (Linear RR )6×10−56\times 10^{-5}7×10−57\times 10^{-5}
WFP (Ours)4×10−6\mathbf{4\times 10^{-6}}8×10−6\mathbf{8\times 10^{-6}}
Appendix
Table 2: Comparison of training and test loss for different models on the 1D wave equation.

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