Fourier-Enhanced DeepONet

• Fourier-enhanced DeepONet is an operator learning framework that augments standard DeepONet with spectral components for improved performance in solving nonlinear PDEs.
• It employs Fourier feature embeddings and spectral convolution layers to accurately capture oscillatory and multiscale behaviors while reducing error.
• The architecture offers significant speedup, memory savings, and robust generalization across domains like geotechnics, seismic imaging, and nonlinear fiber optics.

Fourier-enhanced DeepONet refers to a class of operator learning architectures that augment the standard Deep Operator Network (DeepONet) framework with spectral (Fourier) components—either via spectral convolutional layers, Fourier feature embeddings, or hybrid modules with Fourier Neural Operators (FNO)—to improve accuracy, generalization, and computational efficiency across nonlinear PDE-driven domains. These models have demonstrated notable performance gains in highly oscillatory and multiscale regimes, as well as superior robustness and significant acceleration compared to conventional solvers.

1. Mathematical Foundation and General Formulation

The classical DeepONet approximates a nonlinear operator

$$\mathcal{G}: \mathcal{U} \to \mathcal{S}, \quad u \mapsto \mathcal{G}(u)$$

with two subnetworks: branch ($B_\theta(u)$) and trunk ($T_\theta(\zeta)$) such that the output at coordinate/query $\zeta$ is

$$\mathcal{G}_\theta(u)(\zeta) = \sum_{k=1}^p b_k(u)\, t_k(\zeta)$$

where $b_k(u)$ are branch outputs encoding the input function, and $t_k(\zeta)$ are trunk outputs that encode evaluation coordinates or physical parameters.

Fourier-enhanced DeepONet generalizes this by introducing spectral structure into either or both subnetworks:

• Fourier Feature Trunk: Coordinates or parameters are mapped through a high-dimensional Fourier embedding,

$$\phi(\zeta) = [\sin(2\pi B\zeta),\; \cos(2\pi B\zeta)]^\top$$

with frequency matrix $B$ (fixed or learnable), expanding the representational bandwidth for the trunk MLP (Sojitra et al., 15 Sep 2025, Choi et al., 14 Jul 2025).

• Spectral Convolutional Branch/Decoder: The branch net (or decoder head) implements Fourier spectral convolutions:

$\widehat{u}(k) = \sum_x u(x) e^{-2\pi i kx/N}$

Applied to a truncated mode set, filtered by learnable weights and inverted, yielding global coupling as in FNO (Zhu et al., 2023, Murugan et al., 17 Dec 2025).

• Hybrid/Multi-modal Designs: Separate networks for spatial (FNO) and temporal (MLP/KAN) learning, fused in a modular manner (Santos et al., 4 Nov 2025, Lee et al., 2024).

2. Spectral Augmentation Methods

Multiple architectural paradigms exist for incorporating Fourier enhancement:

• Fourier Feature Embedding (FFE) Trunk: Maps raw coordinates or physical parameters to a space of sinusoidal features. This preprocessing alleviates "spectral bias" in vanilla coordinate MLPs and enables the trunk net to approximate highly oscillatory basis functions (Sojitra et al., 15 Sep 2025, Choi et al., 14 Jul 2025). Typically, the embedding dimension is set equal to or greater than the trunk net output dimension, and frequencies are drawn from a Gaussian with scale $\sigma$.
• Fourier Spectral Convolution Layers: Deployed in the branch net, decoder, or both, these layers propagate information globally across the input grid by filtering learnable spectral modes and then locally merging spatial features with pointwise convolutions and nonlinearities. This approach provides efficient long-range coupling and multiscale representation (Zhu et al., 2023, Murugan et al., 17 Dec 2025).
• Hybrid FNO-DeepONet: Combines an FNO branch net for spatial dependence—leveraging stacked spectral convolutions—and a trunk net (MLP or KAN) for temporal or parametric input. The merged output is constructed by element-wise product (Hadamard fusion), enabling modular treatment of space and time and minimizing parameter count (Santos et al., 4 Nov 2025, Lee et al., 2024).

3. Representative Architectures and Implementation

Architectural instantiations span various domains:

• Bidirectional Fourier-enhanced DeepONet for multimode fiber propagation (MMF) (Murugan et al., 17 Dec 2025):
  • Dual-branch spectral stacks: 2D spatial and 1D temporal branches each use 2–4 spectral convolution layers.
  • Fourier-feature trunk with learnable frequency matrix; embedded parameters modulate the operator directionality.
  • Feature fusion and shared decoder with per-output heads.
  • Unified forward/inverse operator capability, loss defined on normalized outputs.
• Trunk-net Fourier-feature-enhanced DeepONet for 1D consolidation (Choi et al., 14 Jul 2025):
  • MLP branch net encodes input field (pore pressure).
  • Trunk net applies Fourier feature mapping to (depth, time, consolidation coefficient) before a standard MLP.
  • Inner-product merger yields solution at arbitrary query points.
  • Achieves real-time inference and order-of-magnitude speedup over classical solvers.
• FEDONet: Fourier-Embedded DeepONet (Sojitra et al., 15 Sep 2025):
  • Fixed random Fourier features for trunk net input.
  • Spectrally-augmented trunk network sharply reduces error for oscillatory/chaotic PDE problems.
• Fourier-DeepONet for full waveform inversion (FWI) (Zhu et al., 2023):
  • Branch net lifts seismic data; trunk net encodes source parameters.
  • Output passes through a stack of one Fourier convolutional layer plus multiple U-FNO blocks (Fourier-enhanced U-Net).
  • Demonstrated improved accuracy, generalization, and robustness under varied source parameters and noise regimes.
• Nested Fourier-DeepONet for 3D geologic carbon sequestration (Lee et al., 2024):
  • Multi-level nesting for refinement near wells; each level is an independent Fourier-DeepONet.
  • Trunk net treats time separately, reducing FFT dimensionality and GPU load.
  • Model achieves >80% reduction in memory and >2× speedup over nested FNO, maintaining comparable or improved accuracy especially for temporal extrapolation.

4. Training Methodologies and Performance Metrics

Training regimes follow supervised MSE minimization on input-output function pairs:

$$\mathcal{L}(\theta) = \frac{1}{N} \sum_{i=1}^N \frac{1}{Q} \sum_{j=1}^Q \| \mathcal{G}_\theta(u^{(i)})(\zeta_j^{(i)}) - s_j^{(i)} \|^2$$

Standard optimizers (Adam, AdamW) with learning-rate schedules are typical; hyperparameter selection (Fourier modes, frequency scaling, trunk/branch widths) is done via validation (Sojitra et al., 15 Sep 2025, Choi et al., 14 Jul 2025).

Empirical performance consistently reveals substantial improvement with Fourier enhancement:

Problem	DeepONet Rel L2 Error (%)	Fourier-Enhanced Rel L2 Error (%)
2D Poisson	5.80	2.32
Burgers (conv.-diff.)	4.01	2.63
Lorenz–63 (chaotic)	2.03	0.46
K-Sivashinsky (spatio-chaos)	75.11	20.62
1D consolidation (non-uniform)	9.38e-4 Pa²	3.69e-5 Pa²
FWI (FWI-L, FVB family)	0.5816	0.0700

Additional benchmarks repeatedly show order-of-magnitude reduction in relative error in highly oscillatory regimes, robustness to input noise and missing data, and strong extrapolative generalization (pressure and saturation errors <2% in extrapolated GCS regimes; FWI errors remain low for unseen source frequencies/locations) (Zhu et al., 2023, Lee et al., 2024).

5. Computational Efficiency, Scalability, and Generalization

Fourier-enhanced DeepONet frameworks attain significant computational advantages:

• Acceleration: Reported speedups of 1.5×–100× over classical solvers; e.g., single-fiber forward propagation reduced from 8.646 ms (SSFM) to 0.0965 ms (Murugan et al., 17 Dec 2025); consolidation surrogate is ∼100× faster than explicit solvers (Choi et al., 14 Jul 2025).
• Memory and Parameter Reduction: Nested Fourier-DeepONet for GCS requires ≈80% less GPU memory and ≈6× fewer parameters compared to nested FNO, enabling multi-level 3D surrogates on commodity hardware (Lee et al., 2024).
• Scalability Across Domains: Capable of handling spatially and temporally decoupled inputs, nonuniform output grids, and modular nesting. Generalization demonstrated for previously unseen problem parameters, including fiber powers/distances, consolidation coefficients, seismic source properties, reservoir geometry, and injection profiles.

6. Advantages, Limitations, and Domain-specific Observations

Core Advantages:

• Enhanced spectral capacity for trunk and/or branch nets yields improved representability of high-frequency, multiscale, and oscillatory features.
• Modular fusion with FNO or Fourier features circumvents spectral bias of shallow MLPs and enables global coupling.
• Unified forward/inverse operator learning and modular decoupling (space/time) minimize parameter overhead per module.
• Robustness to input perturbations, noise, and missing data exceeds that of pure MLP or convolutional surrogates.

Limitations and Open Challenges:

• Hyperparameter selection for frequency scaling and number of Fourier modes remains empirical; optimal choices depend on problem regularity (Sojitra et al., 15 Sep 2025, Choi et al., 14 Jul 2025).
• Simulation-trained surrogates may not capture experimental artifacts, noise, or real-world variability; phase information/preservation is typically not included (Murugan et al., 17 Dec 2025).
• Extension to arbitrary domains (e.g., 3D, extended temporal horizons, new physical parameters) often requires retraining, fine-tuning, or additional data augmentation (Lee et al., 2024, Santos et al., 4 Nov 2025).
• The theoretical limits of implicit operator learning for infinite-resolution support have not yet been fully characterized, especially in spatio-temporally continuous settings.

7. Domain Applications and Research Outlook

Fourier-enhanced DeepONet architectures have been successfully deployed for:

• Nonlinear fiber optics: Forward/inverse spatio-temporal field prediction in graded-index multimode fibers (Murugan et al., 17 Dec 2025).
• Geotechnics: Real-time surrogates for consolidation PDEs, applicable to soil mechanics (Choi et al., 14 Jul 2025).
• Seismic imaging (FWI): Full waveform inversion robust to variable source configurations, noise, and missing sensors (Zhu et al., 2023).
• Reservoir engineering: Modular surrogates for carbon sequestration and multiphase flow in porous media, scaling to 3D, multi-level domains with efficient memory (Lee et al., 2024, Santos et al., 4 Nov 2025).
• General nonlinear PDEs: FEDONet and related Fourier-feature DeepONets yield transferable spectral improvements for elliptic, parabolic, hyperbolic, and spatio-chaotic systems (Sojitra et al., 15 Sep 2025).

A plausible implication is that Fourier-enhanced DeepONets form a modular, spectrally capable operator-learning family suited for high-dimensional, multiscale, and parametric scientific machine learning tasks. Further advances are likely to focus on learnable or adaptive spectral embeddings, physics-informed losses, uncertainty quantification, and more rigorous theoretical understanding of their generalization on unseen domains.