Spherical Fourier Networks Overview

• Spherical Fourier Networks are neural architectures that operate in the spectral domain using spherical harmonics to achieve exact SO(3) equivariance.
• They leverage spectral convolutions and Laplace–Beltrami eigen-decomposition to enhance data efficiency and rotational robustness for applications like climate modeling and 3D perception.
• Implementations such as SFNO and Clebsch–Gordan Net provide improved grid-invariance, stable dynamics, and superior performance in complex tasks across multiple domains.

Spherical Fourier Networks (SFNs) are neural architectures whose layers operate in the spectral domain defined by the harmonic analysis of the sphere, leveraging the spherical harmonics or derived bases to obtain exact or approximate equivariance to the rotation group SO(3). SFNs have emerged as a canonical solution for learning and processing data distributed on spherical geometries and more general homogeneous spaces, addressing longstanding challenges in equivariance, sample efficiency, and spectral expressivity for diverse domains such as climate modeling, 3D perception, and robotics.

1. Mathematical Foundations

SFNs exploit the eigenfunction decomposition of the Laplace–Beltrami operator on the 2-sphere $S^2$: any square-integrable scalar field $f \in L^2(S^2)$ can be expanded in the spherical harmonic basis $$\{Y_\ell^m(\theta, \varphi)\}_{\ell \ge 0, -\ell \le m \le \ell}$$. The spherical harmonics serve as the irreducible representations of SO(3) restricted to $S^2$. The forward spherical harmonic transform is

$$\hat f_{\ell,m} = \int_{S^2} f(\theta, \varphi)\, \overline{Y_\ell^m(\theta, \varphi)}\, \mathrm d\Omega \,,\quad \mathrm d\Omega = \sin\theta\,\mathrm d\theta\,\mathrm d\varphi\,,$$

with the inverse expansion (truncated at degree $L$)

$$f(\theta, \varphi) = \sum_{\ell=0}^L \sum_{m=-\ell}^{\ell} \hat f_{\ell,m}\; Y_\ell^m(\theta, \varphi)\,.$$

This foundation generalizes to spin-weighted spherical functions and tensor fields, as well as to volumetric domains using the spherical Fourier-Bessel basis, which incorporates both the angular (spherical harmonics) and radial (spherical Bessel functions) decomposition (Zhao et al., 26 Feb 2024).

Crucially, the spectral coefficients $(\ell, m)$ transform equivariantly under the action of SO(3) according to the Wigner D-matrix representations. Thus, SFNs, by parametrizing and manipulating features exclusively in this domain, guarantee rotation equivariance up to discretization and numerical error (Cohen et al., 2018, Bonev et al., 2023).

2. Spectral Convolution and Layer Design

The core operation of SFNs is a spectral or group convolution. On the sphere, zonal convolution with a filter $\kappa$ simplifies to a pointwise multiplication in the spectral domain (spherical convolution theorem):

$$\widehat{(\mathcal K_\vartheta f)}_{\ell, m} = \tilde\kappa_\vartheta(\ell)\,\hat f_{\ell, m},$$

where $\tilde\kappa_\vartheta(\ell)$ is a learned weight per band $\ell$ (Bonev et al., 2023). More general networks allow for anisotropic (i.e., non-zonal) filters, possibly dependent on $(\ell, m)$, and for vector/spin representations (Esteves et al., 2020).

Learning in the spectral domain enables efficient global convolution, parameter sharing, and enforcement of SO(3) equivariance. In architectures such as the Clebsch-Gordan Net (Kondor et al., 2018), layers consist of block-diagonal linear operations by degree $\ell$ and interleave the Clebsch-Gordan transform as the only nonlinearity, preserving equivariance without requiring transitions to the spatial domain. Other variants employ equivariant nonlinearities formulated via Mackey function lifting and projection (Xu et al., 2022).

Stacked spectral layers, possibly interleaved with spatial domain pointwise MLPs, nonlinearities, and normalization, constitute the bulk of modern SFN or SFNO architectures (Bonev et al., 2023, Esteves et al., 2020).

3. Generalizations: Spin, Tensor, and Homogeneous Spaces

Standard SFNs can be extended to process spin-weighted fields or tensor-valued features by adopting spin-weighted spherical harmonics $_sY^\ell_m$ as the basis, enabling the modeling of vector fields, tensor fields, and their rotations (Esteves et al., 2020). Spin-weighted convolutions, which respect the transformation law

$_s f(x)\; \mapsto\; _s f(g^{-1}x)\quad \Longrightarrow\quad (_s \hat f)^\ell_n \; \mapsto\; \sum_{m=-\ell}^{\ell} \overline{D^\ell_{m,n}(g)}\, (_s \hat f)^\ell_m\,,$

enable intrinsic equivariant feature extraction for arbitrary spin.

The most general formalism, unifying SFNs with other equivariant methods, is the treatment of features as tensor fields on homogeneous spaces $M = G/H$, handled via Mackey function lifting, group convolution, and Fourier decomposition with respect to the group's irreducible representations (Xu et al., 2022). Nonlinearities can be constructed equivariantly via pointwise nonlinear functions on the regular representation followed by equivariant projection.

In volumetric domains, expansion onto a spherical Fourier-Bessel product basis enables simultaneous angular and radial decomposition, crucial for 3D affine group equivariant networks and for combating the expressivity bottleneck of only angular parameterization (Zhao et al., 26 Feb 2024).

4. Representative Architectures and Implementation

Several influential architectures instantiate SFN principles:

• SFNO (Spherical Fourier Neural Operator): Implements blocks consisting of spatial MLP, spherical harmonic transform, spectral convolution (learned per $\ell$), inverse transform, and another MLP, with skip connections and grid-invariant implementations. This yields stable, resolution-independent, physically plausible dynamical rollouts for atmospheric forecasting (Bonev et al., 2023).
• Clebsch–Gordan Net: All operations remain in the Fourier domain; nonlinearities are realized as Clebsch–Gordan tensor products and mixing is handled by (learned) block-diagonal matrices per degree. This yields a fully SO(3)-equivariant CNN effective for spherical MNIST, QM7 molecular regression, and 3D model classification (Kondor et al., 2018).
• Spin-Weighted Spherical CNN: Allows for anisotropic filters and vector/tensor field processing on $S^2$. The convolution is implemented as a blockwise ($\ell, m$) product, and ReLU-like nonlinearities act on the magnitudes, preserving equivariance (Esteves et al., 2020).
• Spherical Fourier-Bessel CNN: Kernels are learned and adaptively fused over augmented (Monte Carlo sampled) Fourier–Bessel modes for 3D affine group equivariant architectures, empirically improving segmentation accuracy and group equivariance error in medical imaging (Zhao et al., 26 Feb 2024).

Training, inference, and grid sampling leverage fast algorithms for spherical and SO(3) Fourier transforms, including generalized fast Fourier transforms (GFFT) and efficient backpropagation in the spectral domain (Cohen et al., 2018, Bonev et al., 2023).

5. Applications and Empirical Performance

SFNs are currently applied in:

• Geophysical and Climate Modeling: SFNOs produce stable auto-regressive rollouts (over 1,460 steps—one simulated year) with physically plausible dynamics, matching or slightly trailing operational numerical models such as ECMWF IFS in anomaly correlation coefficients for 10–14 day weather forecasts, while producing a one-year forecast in 13 minutes on a single GPU (Bonev et al., 2023).
• 3D Shape Analysis and Classification: Spherical CNNs and Clebsch–Gordan networks achieve state-of-the-art mAP and NDCG on SHREC17 and competitive accuracy on ModelNet40, demonstrating exact SO(3) equivariance and strong generalization to rotated inputs (Kondor et al., 2018, Cohen et al., 2018, Esteves et al., 2020).
• Molecular Regression: Regression models using spherical spectral methods achieve RMSE 8.47 kcal/mol on QM7 for atomization energy, improving over kernel-based and sorted-Coulomb matrix baselines (Cohen et al., 2018, Kondor et al., 2018).
• Robotics and Policy Learning: Spherical Fourier Diffusion policies achieve robust generalization under SE(3)-transformed scenes, outperforming equivariant and non-equivariant policy baselines by 30–40% absolute in simulation and 61–71% absolute in physical multi-task robotics (Zhu et al., 2 Jul 2025).
• Medical Image Segmentation: Spherical Fourier–Bessel CNNs outperform standard as well as SE(3)-equivariant baselines, with statistically significant improvements in Dice coefficients and group equivariance errors (Zhao et al., 26 Feb 2024).
• Coordinate Encoding: SFN-style encodings, combining spherical harmonics with SIREN layers, yield superior accuracy and interpolation for global geospatial data, outperforming double-Fourier-sphere and MLP-based location encoders (Rußwurm et al., 2023).

6. Algorithmic and Computational Aspects

The key computational features of SFNs include:

• Spectral Efficiency: Fast spherical harmonic and Wigner D-matrix transforms, with the convolution theorem reducing global spherical convolutions to bandwise multiplications—$O(b^4)$ computational cost for typical bandwidth $b$—substantially faster than naive spatial convolutions (Cohen et al., 2018).
• Memory and Expressivity Tradeoffs: Bandlimit $L$ (maximum $\ell$) determines both expressivity and cost; $L\approx 10-20$ suffices for tasks with $256^2$ spherical samples (Kondor et al., 2018).
• Parameterization: Most architectures learn compact spectral weights (scalars, vectors, or block matrices per $\ell$) rather than full spatial kernels. For Fourier–Bessel layers, both angular and radial coefficients are learned (with affine augmentation) and adaptively fused per output channel (Zhao et al., 26 Feb 2024).
• Equivariance Guarantees: All layers preserve SO(3) (or SE(3)) equivariance up to numerical truncation and discretization. Empirical evaluation monitors group-equivariance error as a key metric.

7. Implications and Prospects

The SFN paradigm provides a theoretically principled and practical methodology for learning on spherical and 3D geometric domains where rotation or affine equivariance is essential. Its applications span from physical simulation at climate scales (Bonev et al., 2023), molecular modeling (Kondor et al., 2018, Xu et al., 2022), medical imaging (Zhao et al., 26 Feb 2024), geospatial representation (Rußwurm et al., 2023), and general policy learning in 3D spaces (Zhu et al., 2 Jul 2025).

Empirical evidence shows that spectral, equivariant architectures improve data efficiency, stability, and generalization on both synthetic and real-world rotated/augmented benchmarks, and that the spectral parameterization can yield robust, grid-invariant, and computationally efficient networks. Extensions to more general Lie groups, homogeneous spaces, and tensor field representations are ongoing, supported by the Fourier/Mackey framework for lifting, spectral filtering, and equivariant nonlinearity design (Xu et al., 2022).

Paper Title	arXiv ID	Notable Points
Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere	(Bonev et al., 2023)	SFNO architecture, stable climate rollouts, SO(3) equivariance
Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network	(Kondor et al., 2018)	Fully spectral, block-diagonal, Clebsch–Gordan nonlinearities, state-of-the-art on rotated tasks
Spherical CNNs	(Cohen et al., 2018)	Spherical group convolution, GFFTs, theoretical and implementation basics
Spin-Weighted Spherical CNNs	(Esteves et al., 2020)	Spin-weighted harmonics, anisotropic filters, vector field learning
Efficient 3D affinely equivariant CNNs with adaptive fusion of augmented spherical Fourier-Bessel bases	(Zhao et al., 26 Feb 2024)	Fourier–Bessel kernels, affine equivariance for volumetric data, medical imaging results
Geographic Location Encoding with Spherical Harmonics and Sinusoidal Representation Networks	(Rußwurm et al., 2023)	Spherical harmonics plus SIREN, global location encoding on S²
Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces	(Xu et al., 2022)	Unified kernel/nonlinearity construction for $G/H$, Mackey functions, fully Fourier-driven G-CNNs
SE(3)-Equivariant Diffusion Policy in Spherical Fourier Space	(Zhu et al., 2 Jul 2025)	SE(3)-equivariant diffusion policies, SFiLM layers, robotics applications with spherical Fourier basis