U-FNO: U-Net Enhanced Fourier Neural Operator

Updated 16 December 2025

The paper presents a hybrid architecture combining spectral FNO and U-Net processing to capture both global low-frequency trends and local high-frequency details.
It achieves improved accuracy and data efficiency in modeling turbulent flows and multiphase dynamics compared to standalone FNO or CNN approaches.
The approach demonstrates significant computational speedup and enhanced stability in long-term surrogate simulations of complex PDE systems.

A U-Net Enhanced Fourier Neural Operator (U-FNO), also referred to in some works as a hybrid U-Net/FNO (HUFNO), integrates convolutional U-Net architectures with the Fourier Neural Operator (FNO) paradigm to construct data-driven surrogate models for complex partial differential equation (PDE) systems. This approach is motivated by the complementary inductive biases: FNO provides global, resolution-agnostic representations optimized for capturing low-frequency, long-range interactions via spectral convolution, while U-Nets recover local, high-frequency content through multi-scale, hierarchical, convolutional processing with skip connections. The U-FNO family—including several architectural variants for 2D and 3D flows, multiphase problems, and phase-field dynamics—systematically improves the fidelity and stability of machine learning–based solvers for turbulent, multiphase, and chemically reacting flows compared to either FNO or U-Net in isolation.

1. Network Architecture and Formulation

1.1 Hybrid Layout

U-FNO typically lifts input fields into a high-dimensional latent via a pointwise map or $1\times1\times1$ convolution. Within each U-Fourier layer, two branches operate in parallel:

Spectral branch (FNO):

$(\mathcal{K}v_l)(x) = \mathcal{F}^{-1}\bigl(R\,\widehat v_l\bigr)(x)$

where $R$ is a learnable, truncated Fourier kernel acting on the periodic dimensions, and $\widehat v_l$ is the discrete Fourier transform of $v_l$ .

U-Net branch: A multi-scale encoder–decoder convolutional path, with skip connections, either in all space (standard U-Net) or constrained to specific non-periodic directions (e.g., only the wall-normal direction in channel/hill flow).

The two outputs are combined (typically by addition) and passed forward with activation: $v^{\ell+1}(x) = \sigma\left(\mathcal{K}(v^{\ell})(x) + \mathcal{U}(v^{\ell})(x) + Wv^{\ell}(x)\right)$ where $W$ is a pointwise linear channel mixer, and $\sigma$ is a nonlinearity such as ReLU or GELU.

1.2 Domain-Specific Variants

HUFNO (Periodic Hill LES) (Wang et al., 17 Apr 2025): Applies FNO on only periodic dimensions (e.g., streamwise, spanwise), while a U-Net acts along nonperiodic (wall-normal) direction.
U-FNO (Multiphase, 2D-radial) (Wen et al., 2021, Abdellatif et al., 25 Nov 2025): Both branches run over the whole domain, with additional mechanisms—such as FiLM-modulation (Abdellatif et al., 25 Nov 2025)—for scalar input conditioning.
U-AFNO (Bonneville et al., 24 Jun 2024): Embeds an Adaptive Fourier Neural Operator (vision-transformer-style mixing in frequency space) at the bottleneck of a classical U-Net encoder–decoder for phase field models.
Implicit/U-FNO (IU-FNO, IUFNO) (Li et al., 2023, Wang et al., 5 Mar 2024, Zhang et al., 4 Nov 2024, Jiang et al., 22 Jan 2025): Repeatedly applies a single Fourier+U-Net block in a fixed-point/implicit iteration, reducing parameter count and enhancing stability.

v = P(A)  # Project to latent channels d_v=80
for ℓ in range(L):
    # FNO in x,z directions (for periodicity)
    Kxz = ifftx( R_x * ffts(v, dim='x') ) + ifftz( R_z * fftz(v, dim='z') )
    # Two-layer feed-forward
    h = relu( conv1(Kxz) )
    FF = relu( conv2(h) )
    # U-Net along non-periodic y
    Uout = U_Net_y(v - FF)
    # Residual update
    v = Uout + FF + v
Δv = Q(v)  # Recover velocity increment
v_next = v_last + Δv

2. Mathematical Formulation

U-FNO generalizes classical neural operators by learning mappings between function spaces of physical fields: $u_{t+\Delta t}(\mathbf{x}) = \mathcal{G}_\theta[u_t](\mathbf{x})$ A typical block update is: $v^{\ell+1}(\mathbf{x}) = \sigma\left( W\,v^{\ell}(\mathbf{x}) + \mathcal{F}^{-1}[R \cdot \widehat{v}^{\ell}](\mathbf{x}) + \mathcal{U}^*[s^{\ell}(\mathbf{x})] \right)$ where

$s^{\ell}(\mathbf{x}) = v^{\ell}(\mathbf{x}) - \mathcal{F}^{-1}[R \cdot \widehat{v}^{\ell}](\mathbf{x})$

In many variants, $s^{\ell}$ represents the small-scale (high-frequency) residual after spectral convolution, which the U-Net branch is specialized to recover.

In domain-decomposed settings (e.g., HUFNO), FNO convolutions are applied along periodic directions only: $(\mathcal{K}^f_{x,z}v_{\ell})(x,y,z) = \mathcal{F}_x^{-1}[R_x \cdot \mathcal{F}_x(v_\ell)] + \mathcal{F}_z^{-1}[R_z \cdot \mathcal{F}_z(v_\ell)]$ while a 1D U-Net addresses nonperiodic structure.

Loss functions are typically relative $\ell_2$ or composite, e.g.,

$\mathrm{Loss} = \frac{\|u^* - u\|_2}{\|u\|_2}$

with optional additional terms for gradient, front-tracking, or spatial weighting (Wang et al., 17 Apr 2025, Wen et al., 2021, Abdellatif et al., 25 Nov 2025).

3. Training Protocols and Data

Training utilizes large datasets of high-fidelity simulation, subject to downsampling or filtering to define the modeled (LES-scale) fields. Key points:

Input/Output: Stacks of recent field history, auxiliary masks or parameter maps, and the next-step field (or field increment).
Optimization: Adam optimizer, typical learning rate $10^{-3}$ , batch sizes 4–8 (limited by 3D memory), no explicit weight decay or dropout unless stated.
Epochs: Convergence usually in $50$–$140$ epochs, with early stopping on validation loss.
Loss shaping: Use of two-term relative losses for front-sharpness (Wen et al., 2021), gradient/Sobolev and stability regularization for turbulence (Gonzalez et al., 2023), and spatially weighted losses for application-specific error control (e.g., CO $_2$ plumes (Abdellatif et al., 25 Nov 2025)).

For some applications, scalars are injected as constant channels (traditional), or via FiLM modulation (channelwise affine transformations) to avoid spurious spectral content (Abdellatif et al., 25 Nov 2025).

4. Performance Benchmarks

U-FNO and its variants systematically outperform both standalone FNO and CNN/U-Net baselines in a variety of metrics:

Turbulent Flows (LES, HUFNO (Wang et al., 17 Apr 2025)):

$L_2$ $L_{2}$ relative velocity error after 400 steps:
- $Re=700:$ HUFNO 2.5%, U-Net 4.0%, FNO 5.2%
- $Re=1400:$ HUFNO 3.8%, U-Net 6.5%, FNO 8.0%
- $Re=5600:$ HUFNO 5.1% (FNO diverges, U-Net 12%)
Energy spectrum: HUFNO matches DNS up to $k\approx10$ ; classical SGS models over/under-predict the spectral content.
Computational speed: HUFNO on A100 is 30–60 $\times$ faster than Smagorinsky/WALE on 64 CPU cores for equivalent simulations.

Multiphase Porous Media (U-FNO (Wen et al., 2021)):

Model	MPE Gas Saturation (%)	$R^2_{\text{plume}}$
U-FNO	1.61 ± 1.05	0.981 ± 0.025
FNO	2.76 ± 1.60	0.961 ± 0.039
CNN	2.99 ± 1.75	0.955 ± 0.047

Data efficiency: U-FNO reaches CNN accuracy with $\sim$ 30% of training data.
Hybridization with FiLM (UFNO-FiLM (Abdellatif et al., 25 Nov 2025)): 21% further MAE reduction and better error localization via weighted loss.

Stiff Phase Field Problems (U-AFNO (Bonneville et al., 24 Jun 2024)):

U-AFNO achieves microstructure quantity-of-interest errors (mean curvature, perimeter, mass) matching HF solver discrepancies with $>10^4\times$ speedup per time interval.

Implicit Formulations (IUFNO/IU-FNO (Li et al., 2023, Wang et al., 5 Mar 2024, Zhang et al., 4 Nov 2024, Jiang et al., 22 Jan 2025)):

Implicit update confers long-term stability; IUFNO remains physically accurate over hundreds of large-eddy turnover times, with parameter count and memory usage reduced by up to $80\times$ compared to explicit deep stacks.

5. Advantages and Limitations

Key strengths:

Spectral–local fusion: U-FNO preserves global low-frequency structure (via FNO) and infuses high-frequency/local information (via U-Net), crucial for subgrid feature recovery in turbulence and sharp front propagation in multiphase flows.
Data efficiency: Architectural synergy enables high accuracy with fewer samples, reducing the need for expensive simulation data (Wen et al., 2021).
Transferability: Demonstrated capacity to generalize to unseen initial conditions and geometries (e.g., new hill shapes) (Wang et al., 17 Apr 2025).
Computational performance: Orders of magnitude speedup over physical solvers and classical SGS models, with negligible or modest overhead over FNO/CNN baselines.

Limitations:

U-Nets in the architecture hard-wire the model to fixed grid resolution, reducing flexibility vs. mesh-free FNO for variable meshes (Wen et al., 2021).
Handling of non-uniform or unstructured domains requires further extensions (e.g., geometry-adaptive neural operators) (Wang et al., 17 Apr 2025).
In many variants, no explicit enforcement of divergence-free, physical constraints, or boundary conditions—accuracy relies on training data and (where present) solver postprocessing.
Risk of overfitting/prediction drift in pure CNN U-Nets (without spectral coupling) or in unregularized models during long rollouts (Li et al., 2023, Gonzalez et al., 2023).

6. Extensions, Best Practices, and Emerging Directions

Best practices and future work identified in the literature include:

Physical periodicity: Partition FNO along periodic axes, restricting convolutions to nonperiodic directions for CNNs, as in HUFNO (Wang et al., 17 Apr 2025).
Residual updates, layer normalization: Structural choices such as residual updates and multi-layer local feed-forward paths improve training stability and predictive robustness.
Scalars and conditioning: Avoid duplicating scalar channels by FiLM modulation (Abdellatif et al., 25 Nov 2025).
Loss shaping: Use spatially weighted or composite gradient-front losses to target error in critical physical regions and improve sharpness (Wen et al., 2021, Abdellatif et al., 25 Nov 2025).
Implicit layers: Employ implicit/fixed-point update (IUFNO, IU-FNO) to reduce parameter count, enhance stability, and enable longer-term prediction with consistent statistics (Li et al., 2023, Wang et al., 5 Mar 2024, Zhang et al., 4 Nov 2024, Jiang et al., 22 Jan 2025).
Physics enforcement: Prospective advances include integrating physics-informed losses (divergence-free, boundary constraints), geometry-aware operators, and data assimilation/adversarial robustness (Wang et al., 17 Apr 2025).

7. Application Domains and Impact

U-FNO and its variants deliver state-of-the-art performance as surrogate models for:

Wall-bounded and separated turbulent flows in complex geometry (LES surrogate) (Wang et al., 17 Apr 2025, Wang et al., 5 Mar 2024).
Multiphase flow in porous media with sharp saturation/pressure fronts under heterogeneity and anisotropy (Wen et al., 2021, Abdellatif et al., 25 Nov 2025).
Chaotic phase-field evolution in solidification/corrosion, enabling large time-step acceleration (Bonneville et al., 24 Jun 2024).
Chemically reacting compressible turbulence, achieving accuracy and speed unattainable with classical LES (Zhang et al., 4 Nov 2024). Across these domains, U-FNO architectures outperform classical FNO/CNN surrogates in both one-step and long-term, auto-regressive prediction, and enable fast, high-fidelity emulation of multi-physics PDE systems for scientific and engineering applications.