Fourier Neural Operator Wave Forecast Model

• Fourier Neural Operator Wave Forecast Model is a surrogate machine learning framework that approximates wave equation solutions using spectral convolution.
• It leverages a sequence of Fourier layers and advanced deep learning architectures to provide rapid, mesh-independent predictions with uncertainty quantification.
• Demonstrated efficiency with up to 10⁴ speed-up over traditional PDE solvers, it supports real-time applications in seismology, ocean dynamics, and acoustics.

A Fourier Neural Operator (FNO) Wave Forecast Model refers to a Surrogate Machine Learning framework that utilizes the Fourier Neural Operator architecture to approximate the solution operator of wave equations—both linear and nonlinear—arising in geophysical, physical, and engineering contexts. FNOs parameterize infinite-dimensional mappings between input and output function-spaces, replacing conventional PDE solvers with mesh-independent, data-driven predictors leveraging spectral (Fourier-domain) convolution. These models enable rapid inference of full spatio-temporal wavefields, facilitate probabilistic and uncertainty quantification analyses, and support real-time or near-real-time applications in seismology, ocean dynamics, acoustics, and broader scientific machine learning.

1. Mathematical Formulation of Wave Equations and Solution Operators

FNO-based wave forecast models target time- and frequency-domain PDEs governing wave propagation. In seismic applications, the 3D elastic wave equation is posed in displacement form: $$\rho\,\partial^2_t u(x, t) = \nabla\cdot[\lambda(\nabla\cdot u) I] + \nabla\cdot[\mu(\nabla u + (\nabla u)^\top)] + (\lambda + 2\mu) \nabla(\nabla\cdot u) - \mu \nabla\times\nabla\times u,$$ where $u(x, t)\in\mathbb{R}^3$ is the displacement vector, $\rho$ density, $\lambda$, $\mu$ Lamé parameters, and the domain $D_\mathrm{a}\subset\mathbb{R}^3$ (Lehmann et al., 2023). Alternatively, frequency-domain formulations lead to the vector Helmholtz equation: $$\nabla\cdot\left[\lambda(\nabla\cdot U)I + 2\mu\varepsilon(U)\right] + \rho\omega^2 U = F,$$ where $U$ is the complex-valued wavefield at given $\omega$ (Kong et al., 3 Mar 2025, Li et al., 2022). In oceanic and photoacoustic settings, analogous acoustic or shallow-water wave equations appear (Jahanmard et al., 10 Oct 2025, Guan et al., 2021).

Given an input field (e.g., velocity model, source terms, physical parameters), the goal is to learn an operator

$\mathcal{G}: a(\cdot) \longmapsto u(\cdot),$

where $a$ encodes the coefficients or parameters of the PDE, and $u$ is the wavefield—either as a time series over receivers, a frequency-domain field, or the full spatio-temporal evolution.

2. Fourier Neural Operator Architecture and Variants

The canonical FNO architecture applies a sequence of Fourier integral layers interleaved with local linear transformations and nonlinearities:

• Lifting: Project input $a(x)$ to high-dimensional feature space via pointwise fully-connected (FC) layers (Lehmann et al., 2023, Guan et al., 2021).
• Fourier Layers: Each layer computes for feature $v^{(\ell)}(x)$:

$$v^{(\ell+1)}(x) = \sigma\left(W^{(\ell)}v^{(\ell)}(x) + \mathcal{F}^{-1}[R^{(\ell)}\circ \mathcal{F}(v^{(\ell)})](x)\right)$$

where $\mathcal{F}$/$\mathcal{F}^{-1}$ are (multi-dimensional) FFT/inverse FFT operators, $R^{(\ell)}$ are learnable complex-valued spectral weights (modes truncated for efficiency), $W^{(\ell)}$ local pointwise weights, and $\sigma$ a nonlinearity (commonly ReLU) (Lehmann et al., 2023, Kong et al., 3 Mar 2025).

• U-shaped Neural Operator (UNO): For 3D seismic applications, the UNO variant introduces encoder–decoder structure with down-/up-sampling, skip connections, and independent output projections for vector field components (Lehmann et al., 2023).
• Paralleled FNO (PFNO): Handles multi-source, multi-frequency problems by allocating separate sub-networks per frequency, concatenating the results (Li et al., 2022).
• Spatio-Temporal Kernel Variants (FNOtD): Incorporates temporal Fourier modes, internalizing wave dispersion relations for PDEs with rich frequency content, as demonstrated in ocean forecasting (Jahanmard et al., 10 Oct 2025).

3. Data Preparation, Training Protocols, and Loss Functions

Successful FNO model deployment depends on high-fidelity paired datasets of input coefficients (velocity models, sources, physical parameters) and corresponding wavefield solutions:

• Seismic 3D Ground Motions: Synthetic databases with up to 30,000 unique velocity structures, generated via spectral-element solvers (SEM3D), enable training on highly heterogeneous elastic media (Lehmann et al., 2023).
• Frequency-domain Helmholtz: Data comprising grids of complex wavefields at multiple discrete frequencies, with inputs encoding velocity fields, source locations, and physical coordinates (Kong et al., 3 Mar 2025, Li et al., 2022).
• Ocean and Acoustics: Reanalysis or pseudo-spectral simulations on spatial grids, often covering long time intervals and coarse-to-fine spatial resolutions (Jahanmard et al., 10 Oct 2025, Guan et al., 2021).

Training employs loss functions such as mean absolute error (MAE) summed across field components, sensors, and/or time: $$L(\theta) = \frac{1}{N}\sum_{i=1}^N \|G_\theta(a^{(i)}) - u^{(i)}\|_1$$ or relative $\ell^2$ loss when normalized to the magnitude of ground truth, with possible inclusion of spectral penalties or dispersion-relation regularizers for physically faithful output spectra (Lehmann et al., 2023, Jahanmard et al., 10 Oct 2025, Kong et al., 3 Mar 2025).

Multi-stage training protocols are used to improve high-frequency accuracy, successively training on residuals not captured by a baseline FNO and incorporating prior predictions as additional inputs (Kong et al., 3 Mar 2025).

4. Empirical Performance Evaluation and Bias Mitigation

Quantitative assessment of FNO wave forecast models proceeds along several axes:

• Accuracy Metrics:
  • Mean absolute error (MAE) and $\ell^2$ loss for time–space or frequency–space predictions (Lehmann et al., 2023, Kong et al., 3 Mar 2025).
  • Goodness-of-fit (GOF) scores evaluating envelope and phase coherence relative to ground truth, as in seismic validation (Lehmann et al., 2023).
  • Relative spectral errors comparing predicted and physical amplitude/frequency spectra, especially at high frequencies (Kong et al., 3 Mar 2025, Jahanmard et al., 10 Oct 2025).
• Generalization and Robustness:
  • Validation errors on out-of-distribution velocity models, unseen geologies, or experimentally perturbed data remain close to training errors for well-trained FNO/UNO/PFNO ensembles, even under label noise (Lehmann et al., 2023, Li et al., 2022).
  • Amplitude underestimation and frequency bias (tendency for FNO to fit low frequencies more accurately) are mitigated via dataset enlargement, multi-stage training, or by increasing retained Fourier modes (Lehmann et al., 2023, Kong et al., 3 Mar 2025).
• Speed and Computational Savings:
  • Individual FNO forward inference achieves $10^4\times$ speed-up over high-fidelity PDE solvers, making real-time applications feasible (e.g., $\mathcal{O}(0.1–1)$s per 3D seismic simulation on multi-GPU hardware) (Lehmann et al., 2023, Guan et al., 2021, Li et al., 2022).

5. Advanced Extensions: Spatio-Temporal Coupling and Physical Priors

Incorporating domain-specific physics into FNOs boosts stability and fidelity in long-term or multiscale wave propagation:

• Spatio-Temporal FNO kernels: FNOtD modifies the spectral kernel $\widehat{K}(k,\omega)$ to entangle spatial and temporal frequencies, capturing physical dispersion relations in wave PDEs such as shallow water equations:

$$\mathcal{K}[u](x, t) = \mathcal{F}_{k, \omega}^{-1}\left[\widehat{K}(k, \omega)\, \mathcal{F}_{x, t}[u](k, \omega)\right](x, t)$$

with learned or softly-constrained alignment to theoretical $\omega(k)$ as priors (Jahanmard et al., 10 Oct 2025).

• PINN–FNO Hybrids: Penalizing physical constraint violations (e.g., enforcing discrete PDE residuals) or introducing loss terms based on physical invariants remains an open extension for increasing robustness on out-of-distribution scenarios (Li et al., 2022).
• Activation and Depth Choice: Nonlinear activation functions (Swish, $x\tanh(x)$) and increased network depth demonstrably reduce worst-case and mean errors for highly nonlinear or dispersive wave systems (Zhong et al., 2022).

6. Applications and Practical Implications

FNO wave forecast models enable a paradigm shift in scientific and engineering workflows:

• Seismic Hazard and Site-Effect Assessment: FNO surrogates enable rapid, repeated assessment of ground motion scenarios over ensembles of geological models, supporting probabilistic seismic risk estimation, basin effects, and real-time earthquake response (Lehmann et al., 2023).
• High-Frequency Ocean and Weather Forecasting: Spatio-temporal kernel extensions improve the skill and dynamical consistency of long-range ocean state predictions versus state-of-the-art PDE solvers at dramatically reduced compute (Jahanmard et al., 10 Oct 2025).
• Acoustic Imaging and Inverse Problems: FNO-based forward models accelerate simulation-heavy modalities (e.g., photoacoustic imaging), maintain fidelity on natural and synthetic phantoms, and facilitate adjoint methods for gradient-based inversion (Guan et al., 2021, Li et al., 2022).

The generalization, mesh-independence, and speed of FNO-type models position them as foundations for probabilistic surrogate modeling, Bayesian inversion, and data assimilation in high-dimensional physical systems.

7. Limitations, Outlook, and Research Directions

Despite rapid progress, several challenges remain inherent to Fourier Neural Operator wave forecast models:

• Frequency Bias and Resolution Loss: Single-stage FNOs underfit high-frequency details; mitigations include multi-stage training, spectral loss augmentation, and increased mode retention (Kong et al., 3 Mar 2025, Lehmann et al., 2023).
• Dimensional Scaling: Full 3D and 4D (spatio-temporal) FNOs demand substantial memory and computational resources owing to the FFT size; hierarchically compressed or block-circulant FFTs are plausible remedies (Li et al., 2022).
• Lack of Explicit Physical Constraints: Purely data-driven FNOs may violate energy conservation or causality far from the training set; inclusion of physics-informed penalty terms or residual networks is an active research direction (Jahanmard et al., 10 Oct 2025, Li et al., 2022).
• Time- and Frequency-Domain Coupling: Standard FNOs generalize weakly beyond trained bands; further research is advancing joint time–frequency or hybrid operator architectures.

Overall, FNO wave forecast models represent a versatile and computationally efficient family of surrogate solvers for high-dimensional wave phenomena, with ongoing development focused on stability, spectral accuracy, interpretability, and direct coupling to data assimilation and inverse design methodologies (Lehmann et al., 2023, Jahanmard et al., 10 Oct 2025, Kong et al., 3 Mar 2025, Li et al., 2022, Zhong et al., 2022, Guan et al., 2021).