Spherical Fourier Neural Operators

• SFNOs are deep neural operator architectures that use spherical harmonics to represent functions on spherical domains and ensure rotation equivariance.
• They employ spectral convolution with learned degreewise weights, followed by lifting and nonlinearity to capture complex, nonlocal dynamics.
• Applications in weather forecasting and solar wind modeling demonstrate SFNO's capability to reduce computational cost while maintaining high predictive accuracy.

A Spherical Fourier Neural Operator (SFNO) is a deep neural operator architecture that models nonlocal mappings on spherical domains using an explicit spherical Fourier spectral representation. Designed to encode the intrinsic geometry and symmetries of $S^2$, SFNO leverages the spherical harmonic basis to learn global, rotation-equivariant convolutional operators for physical and geoscientific applications such as weather and solar wind modeling. Unlike planar neural operators based on the discrete Fourier transform (DFT), SFNO respects the compact, non-Euclidean topology of the sphere and ensures that learned mappings are equivariant with respect to the SO(3) group of rotations.

1. Mathematical Foundation: Spherical Harmonic Expansion and Spectral Filtering

The core mathematical underpinning of SFNO is the representation of scalar (or vector) fields on the sphere by an expansion in spherical harmonics $\{Y_{\ell}^m\}$, which are eigenfunctions of the spherical Laplacian: $$f(\theta, \phi) = \sum_{\ell=0}^{L_{\max}} \sum_{m=-\ell}^{\ell} a_{\ell m} Y_{\ell}^m(\theta, \phi)$$ with coefficients $a_{\ell m}$ computed as

$$a_{\ell m} = \int_{S^2} f(\theta, \phi) \overline{Y_{\ell}^m(\theta, \phi)} d\Omega$$

where $\ell$ is the angular degree, $m$ the order, and the integration is over the unit two-sphere. The spherical harmonic transform (SHT) and its inverse are central algorithmic building blocks.

SFNO leverages the diagonalization property of spherical convolution for zonal kernels: any isotropic integral operator $K$ defined by a kernel $k$ acts via

$$(K f)(\theta, \phi) = \sum_{\ell=0}^{L_{\max}} \sum_{m=-\ell}^{\ell} g_\ell a_{\ell m} Y_{\ell}^m(\theta, \phi)$$

where $g_\ell$ is a learned, degree-dependent spectral filter. This spectral parameterization ensures that SFNO implements operators that commute with SO(3) actions, making the model rotation-equivariant (Mansouri et al., 27 Nov 2025, Bonev et al., 2023, Mahesh et al., 6 Aug 2024, Mansouri et al., 25 Nov 2025).

2. Neural Operator Architecture and Forward Pass

An SFNO layer comprises three principal operations:

1. Lifting: A pointwise $1 \times 1$ linear map $W$ embeds the input field $f^{(0)}(\theta, \phi)$ to a higher-dimensional feature space, yielding $F^{(0)}(\theta, \phi)$.
2. Spectral Convolution: Forward SHT computes spectral coefficients, which are scaled by learned degreewise weights $g^{(k)}_\ell$ per layer, and inverse SHT reconstructs the spatial field.
3. Nonlinearity: A pointwise activation (e.g., GELU, ReLU) is applied.

Formally, for channel stacks $F^{(k)} \in \mathbb{R}^{C_k}$,

$$F^{(k+1)}(\theta, \phi) = \sigma \left[ W F^{(k)}(\theta, \phi) + K_{g^{(k)}} F^{(k)}(\theta, \phi) \right]$$

where $K_{g^{(k)}}$ denotes degree-diagonal spectral filtering. This update is iterated over $K$ layers before projecting back to physical space. Since filter weights depend solely on $\ell$, the symmetry properties of the sphere are preserved.

In practical implementations, the sphere is discretized using a Gauss–Legendre grid in latitude and uniform longitude sampling to maintain spectral orthogonality and accurate quadrature (Mansouri et al., 27 Nov 2025, Bonev et al., 2023).

3. Implementation Details and Scalability

SFNO networks are trained on gridded spherical datasets appropriate to the application (e.g., atmospheric state variables or solar wind velocities) (Mahesh et al., 6 Aug 2024, Mansouri et al., 27 Nov 2025, Mansouri et al., 25 Nov 2025). The forward and inverse SHTs are performed using FFTs in longitude and matrix multiplications in latitude, with aliasing control via a spectral cutoff $\ell_{\max}$.

Scalability is achieved through spatial model-parallel SHT, with latitude bands or tensor slices distributed across multiple GPUs, enabling efficient training at high resolutions. Parameter tensors for the spectral weights take the form $\{g_\ell\}_{\ell=0}^{L_{\max}}$, with possible extension to channel-mixing matrices per $\ell$. For large-scale applications, a “large” SFNO may contain $O(10^9)$ parameters and run efficiently on distributed GPU clusters with batched evaluation (Mahesh et al., 6 Aug 2024).

Training objectives are typically mean-squared error (MSE) losses, latitude-weighted for geophysical fields, with $L_2$ regularization and optimization via AdamW or similar adaptively scheduled optimizers (Mansouri et al., 27 Nov 2025, Mahesh et al., 6 Aug 2024).

4. Applications: Surrogate Modeling, Weather, and Solar Wind

SFNO has been successfully applied as a surrogate model for physical systems governed by PDEs on the sphere, including:

• Solar Wind Surrogates: SFNO outperforms or matches surrogate numerical schemes such as HUX on benchmarks of solar wind speed propagation, achieving superior RMSE (52 vs. 78 km/s), higher Pearson correlation ($\rho = 0.92$ vs. 0.85), and improved structural similarity (SSIM = 0.81) (Mansouri et al., 27 Nov 2025). Autoregressive SFNO propagates predictions from the solar photosphere to 1 AU in sequential shells, yielding significant speedups relative to MHD solvers and comparable forecast skill (Mansouri et al., 25 Nov 2025).
• Atmospheric and Climate Modeling: Large-scale SFNOs have been deployed for ensemble weather forecasting, enabling the simulation of “huge ensembles” (up to 7,424 members) at fractional computational cost compared to NWP models. SFNO-based ensembles approximate operational Integrated Forecasting System (IFS) metrics for mean, spectral, and extreme diagnostics, while maintaining physically plausible spectra throughout long-range autoregressive forecasts (Mahesh et al., 6 Aug 2024).
• Shallow Water and Stable Dynamics: SFNO demonstrates stable autoregressive rollouts for year-long simulations of shallow water equations and matched anomaly correlation skill against operational models, resolving classical pathologies of flat FNOs such as polar artifacts and spectral dissipation (Bonev et al., 2023).

The table summarizes performance metrics for SFNO and comparative methods:

Application	Metric	SFNO	Comparative	Reference
Solar wind (1 AU)	RMSE (km/s)	52	78 (HUX)	(Mansouri et al., 27 Nov 2025)
Solar wind (1 AU)	Pearson $\rho$	0.92	0.85 (HUX)	(Mansouri et al., 27 Nov 2025)
Weather forecasting (1d)	ACC (%)	89.9	91.5 (GSHNet)	(Tang et al., 11 Dec 2025)
Weather forecasting (3d)	ACC (%)	70.7	72.4 (GSHNet)	(Tang et al., 11 Dec 2025)
Shallow water (5h)	Mean Rel. Error	1.5e-1	1.3e-1 (GSHNet)	(Tang et al., 11 Dec 2025)

5. Equivariance, Limitations, and Extensions

SFNO’s spectral filtering ensures strict SO(3) equivariance: outputs co-rotate with inputs under arbitrary 3D rotations, and learned kernels are invariant to local coordinates. This is advantageous for planetary, atmospheric, and heliospheric systems lacking preferred orientation. However, this constraint can impair the modeling of boundary and anisotropic effects inherent in real-world Earth systems.

A generalization, the Green’s-function Spherical Neural Operator (GSNO), augments SFNO by introducing an absolute-position-dependent correction term in harmonic space. GSNO interpolates between strict equivariance (SFNO) and full invariance, capturing boundary-driven or heterogeneous effects, and demonstrates empirical superiority across PDE, weather, and biomedical imaging tasks (Tang et al., 11 Dec 2025). GSNO-based architectures such as GSHNet further integrate multi-scale spectral modeling and spherical up/downsampling for increased representation power.

SFNO’s main limitations include the resolution ceiling set by $\ell_{\max}$, inability to resolve fine-scale features beyond the chosen cutoff, and reliance on regular grids for efficient SHT computation. The overall computational complexity is $O(L^3)$ in the spectral cutoff. Addressing fine structure, adaptive resolution, and physics-informed regularization are identified research directions (Mansouri et al., 27 Nov 2025, Bonev et al., 2023, Tang et al., 11 Dec 2025).

6. Summary of Related Work and Future Outlook

Originated by Mansouri et al., Brandstetter et al., and collaborators, SFNO constitutes a rigorously constructed, geometry-aware neural operator for function-to-function learning on $S^2$ (Mansouri et al., 27 Nov 2025, Bonev et al., 2023, Mahesh et al., 6 Aug 2024, Mansouri et al., 25 Nov 2025). SFNO is referenced as a special, strictly equivariant instance in broader frameworks such as GSNO (Tang et al., 11 Dec 2025). SFNO’s integration into huge ensemble forecasting pipelines, solar wind surrogates, and autoregressive rollout models highlights its role as a scalable, interpretable, and physically consistent operator-learning mechanism for spherical systems.

A plausible implication is that future hybrid models could fuse SFNO’s spectral expressivity with localized, non-equivariant corrections and multi-resolution processing schemes to further enhance fidelity and real-world applicability, particularly in climate and space-weather forecasting.

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Key Citations:

• "Toward Data-Driven Surrogates of the Solar Wind with Spherical Fourier Neural Operator" (Mansouri et al., 27 Nov 2025)
• "Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere" (Bonev et al., 2023)
• "Huge Ensembles Part I: Design of Ensemble Weather Forecasts using Spherical Fourier Neural Operators" (Mahesh et al., 6 Aug 2024)
• "Autoregressive Surrogate Modeling of the Solar Wind with Spherical Fourier Neural Operator" (Mansouri et al., 25 Nov 2025)
• "Generalized Spherical Neural Operators: Green's Function Formulation" (Tang et al., 11 Dec 2025)