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
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 Function
Training Loss (MSE)
Test Loss (MSE)
tanh
1×10−4
2×10−4
Leaky ReLU (slope=0.05)
4×10−5
2×10−4
exp(−4x2)sin(3πx)
5×10−5
6×10−5
exp(−20x2)sin(10πx)
4×10−6
8×10−6
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
MODEL
Training Loss (MSE)
Test Loss (MSE)
FNO (Elementwise R )
4×10−4
4×10−4
FNO (Linear R )
6×10−5
7×10−5
WFP (Ours)
4×10−6
8×10−6
Appendix
Table 2: Comparison of training and test loss for different models on the 1D wave equation.
Neural surrogates provide computationally efficient auto-regressive modeling of wave propagation, enabling e.g. fast inverse solutions. Despite recent progress, the current surrogates are not yet sufficient for the needs of ultrasonic sensing, characterized by high frequency and heterogeneous propagation media. We show how methods building on Fourier Neural Operators (FNO) fail to capture essential signal features, such as reflections, whereas U-Net -based methods learn the wave fields well but lack the ability to transfer information across different spatial scales. We then propose a new surrogate that captures the best of the two worlds, by integrating FNOs and local-integral resampling into a U-Net architecture. We show it can be trained on limited data, and that the predictions are robust to input noise and work across spatial scales.
Burla Nur Korkmaz, Petro Moilanen, Ari Salmi +2
Department of Computer Science, University of Helsinki, Finland
Operator learning is an emerging field at the intersection of machine learning and scientific computing. By learning mappings between function spaces, neural operators provide data-driven surrogate models for families of partial differential equations (PDEs). Once trained, these models can evaluate solution operators efficiently, making them suitable for many-query applications such as real-time prediction and parameter sweeps. However, maintaining high approximation accuracy and stable long-term predictions remains challenging for complex forward and inverse problems. To address these challenges, we propose the Starter-Iterator Neural Operator (SINO), which incorporates the initialization and residual-correction structures of classical iterative solvers into neural operator learning. The frequency-domain Starter captures dominant global spectral features and provides an informed initial approximation, while the latent-space Iterator applies successive residual-based corrections to refine local and multiscale solution structures. Experiments on representative time-dependent PDEs, including the Navier-Stokes and acoustic wave equations, together with applications to image super-resolution and weather forecasting, show that SINO achieves competitive accuracy and stable performance across the benchmarks considered in this work.
Partial differential equations (PDEs) govern complex systems, yet neural operators often struggle to efficiently capture the long-range, nonlocal interactions inherent in their solution maps. We introduce Spectral Filtering Operator (SFO), a neural operator that parameterizes integral kernels using the Universal Spectral Basis (USB), a fixed, global orthonormal basis derived from the eigenmodes of the Hilbert matrix in spectral filtering theory. Motivated by our theoretical finding that the discrete Green's functions of shift-invariant PDE discretizations exhibit spatial Linear Dynamical System (LDS) structure, we prove that these kernels admit compact approximations in the USB. By learning only the spectral coefficients of rapidly decaying eigenvalues, SFO achieves a highly efficient representation. Across six benchmarks, including reaction-diffusion, fluid dynamics, and 3D electromagnetics, SFO achieves state-of-the-art accuracy, reducing error by up to 40% relative to strong baselines while using substantially fewer parameters.
Noam Koren, Rafael Moschopoulos, Kira Radinsky +1
Department of Computer Science Technion - Israel Institute of Technology · Department of Computer Science Princeton University