Generalized Spherical Neural Operators: Green's Function Formulation

AI summary

Overview

Research area: Machine learning for scientific computing, specifically neural operators for solving parametric partial differential equations (PDEs) on spherical domains. Applications span geophysics, weather prediction, and neuroimaging.

Technical level: Advanced. The paper derives operator solutions through Green's functions, spherical harmonic transforms (SHT), and the spherical convolution theorem, and assumes familiarity with spectral methods, equivariance, and PDE theory.

One-sentence scope: The paper builds a Green's-function-based theoretical framework for designing spherical neural operators, introduces the GSNO operator and SHNet architecture, and evaluates them on diffusion MRI, spherical shallow water equations, and global weather forecasting.

What This Paper Is About

Existing spherical neural operators, such as the Spherical Fourier Neural Operator (SFNO), are built by extending the Fourier Neural Operator (FNO) to the sphere using the Spherical Harmonic Transform and the spherical convolution theorem, rather than being derived from the integral solution of sphere-native PDEs. Because they enforce strict rotational equivariance for spectral efficiency, they struggle with real-world asymmetries such as boundary effects, position-dependent patterns, and heterogeneous media. This paper derives spherical operators from a designable Green's function, then designs an absolute and relative position-dependent Green's function to produce GSNO, an operator that flexibly balances equivariance and invariance while keeping spherical geometry and grid invariance, and wraps it in a multi-scale architecture called SHNet.

Key Contributions

1. A generalized theoretical framework for deriving system-based operator solutions from designable spherical Green's functions. The derivation extends the Green's function from the sphere into the spherical harmonic domain without disrupting spherical geometry (Sections 4.1, and Appendix B.1).

2. A new operator, GSNO, built from an absolute and relative position-dependent Green's function. It combines an original equivariant term with a learnable non-equivariant correction term, flexibly balancing equivariance and invariance while retaining grid invariance.

3. The SHNet architecture, a multi-scale spherical network based on GSNO that combines multi-scale spectral modeling with spherical up-down sampling. Scale transformation is achieved by modifying the number of sampling points along latitude and longitude and the degree l, avoiding distortions from traditional up- and down-sampling.

4. Evaluation on three spherical domains: diffusion MRI modeling of brain microstructure, spherical shallow water equations, and weather prediction, with reported improvements over other state-of-the-art models. Code is available at https://github.com/haot2025/GSNO.

Main Findings

• Spherical Shallow Water Equations (SSWE): SHNet achieved the best performance on all variables and time scales among the compared models, with a reported average increase of 7.5% at 5h and 9.7% at 10h. At 5h, SHNet reported MRE (×10⁻³) of 1.26 ± 0.07 (H), 134 ± 9.30 (V), 273 ± 5.91 (D), and average 136 ± 3.73, versus SFNONet's 1.39 ± 0.07, 145 ± 11.1, 295 ± 8.87, and 147 ± 4.89. At 10h, SHNet reported 1.49 ± 0.09, 201 ± 11.9, 326 ± 12.1, and average 176 ± 5.71, versus SFNONet's 1.68 ± 0.12, 229 ± 15.6, 353 ± 13.5, and 195 ± 6.86.

• WeatherBench forecasting accuracy: SHNet (with GSNO, 4.0 M parameters) achieved the best average ACC at every horizon: 91.5 ± 0.59 at 1 day, 72.4 ± 2.13 at 3 days, and 52.5 ± 3.99 at 5 days. For comparison, SFNONet (SFNO, 5.3 M parameters) reported 87.3 ± 0.83 at 1 day, 68.8 ± 2.40 at 3 days, and 49.2 ± 4.27 at 5 days.

• Operator-level comparison: GSNO with 64 channels and 4.00 M parameters reported 91.5%, 72.4%, and 52.5% ACC at 24 h, 72 h, and 120 h respectively, with 1024 s training time. This outperformed SFNO with 96 channels (8.30 M parameters, 1267 s, 89.6%/70.2%/50.7%) and a spatial position-embedding SNO (4.31 M, 1037 s, 90.1%/70.5%/50.3%).

• Diffusion MRI (HCP dataset): FOD-GSNO (1.21 M parameters) reported ACC of 0.9083 ± 0.0140 on white matter and 0.8517 ± 0.0139 on whole brain, exceeding FOD-SFNO (0.8995 ± 0.0151 and 0.8334 ± 0.0154), ESCNN (0.9006 ± 0.0142 and 0.8362 ± 0.0162), FOD-Net (0.8858 ± 0.0138 and 0.8250 ± 0.0159), and SSMT-CSD (0.7523 ± 0.0256 and 0.6640 ± 0.0145).

• Ablation results: Replacing the GSNO block with a classic SFNO block inside SHNet, or replacing the SFNO block with a GSNO block inside SFNONet, degraded performance across all variables and time horizons. The GSNO block variant (SFNONet with GSNO, 5.5 M parameters) reported average ACC of 88.7 ± 0.76 at 1 day, 70.5 ± 2.27 at 3 days, and 51.3 ± 4.11 at 5 days; SHNet with SFNO (3.7 M) reported 89.9 ± 0.68, 70.7 ± 2.24, and 50.9 ± 4.10.

• Long-range dynamics benefit: Performance gains from GSNO blocks were more pronounced in longer-term predictions, which the authors interpret as GSNO playing an increasingly important role in modelling long-range dynamics.

• Interpretability of the correction term: Even when the original term I(𝒯_orig) was frozen, using only the correction term I(𝒯_corr) outlined a temperature distribution framework linked to Earth's topography, distinguishing persistent low temperatures at the poles from high-temperature patterns such as the equator.

Methodology in Plain English

The authors start from the classical Green's function approach to solving PDEs, where a solution is written as a convolution integral with the Green's function as the kernel. They define a linear differential operator D on the sphere, define the associated Green's function G via a Dirac delta property, and write the solution g(u) as an integral of G against the input f.

They first show that if the Green's function is chosen to depend only on relative position, the whole construction reduces exactly to what existing spectral spherical operators (SFNOs) do. This confirms the framework is a generalization rather than a departure. They then re-derive the same result using a harmonic expansion of the Green's function, which shows the designable Green's function passes naturally into the spherical harmonic domain.

To go beyond strict equivariance, they design a Green's function that mixes a relative-position term with an absolute-position term. This splits the operator output into two parts: an original term that preserves rotational equivariance, and a correction term that is learnable, non-equivariant, and captures spatial constraints and heterogeneities. In the spectral domain the correction term reduces to a product involving the spherical integral of the input, a quantity they denote C_f. They simplify the formulation to avoid parameter redundancy, giving a compact expression where the input's spherical harmonic coefficients and the correction term are summed and then multiplied by an outer weight.

The GSNO block applies SHT, computes the correction term using the spherical integral, then applies inverse SHT to return to the sphere. Channel interaction uses a multi-layer perceptron with two 1×1 convolutional layers and GELU activation, with two additional light-weight convolutional layers for linear interaction, channel transformation, and skip connections. SHNet stacks these blocks in a U-Net-style structure with position embedding and geometric up- and down-sampling implemented by changing the number of sampling points along latitude and longitude and the degree l.

Why This Matters

Impact on research: The paper reframes spherical neural operator design as a question of choosing a Green's function, rather than extending an FNO to the sphere. This gives the learned integral kernels a physical meaning tied to a differential operator and provides a principled way to relax rotational equivariance rather than treating it as an architectural accident.

Real-world applications (as evaluated or discussed in the paper):

• Global weather forecasting, including prediction of variables such as 2T, 10U, 10V, U600, V600, and T600 on WeatherBench.
• Geophysical fluid dynamics, via the spherical shallow water equations modelling thin-layer fluid motion on a rotating sphere with geopotential height, vorticity, and divergence channels.
• Neuroimaging / brain microstructure, via diffusion MRI fiber orientation distribution angular super-resolution on the HCP dataset.
• Position-dependent geophysical modelling, where anisotropy, local heterogeneity, and non-periodic boundary conditions matter, such as weather patterns shaped by terrain.

Industry relevance: The results are reported at 16GB A5000 GPU scale with model sizes in the low single-digit millions of parameters, and GSNO is described as maintaining high computational efficiency and low parameter counts comparable to SFNO. This is relevant to operational forecasting and medical imaging pipelines where GPU memory and inference costs matter. The released code and the dMRI comparison (FOD-GSNO at 1.21 M parameters) target practitioners who need accuracy at modest parameter budgets.

Future Directions

• Designing other Green's functions: The framework is presented as general, so the paper's own stated suggestion is further designs for the development of spherical modeling beyond the specific absolute-plus-relative form used here.
• Extending the equivariance-invariance balance: Since GSNO relaxes strict SO(3) equivariance through one correction term, an open question is how this balance should be tuned or structured for other systems with different forms of asymmetry and boundary constraints.
• Scaling and resolution generalization: The paper reports additional SSWE experiments at 128 × 256 and weather experiments at 64 × 128 in the appendix, but the main tables use 256 × 512 and 5.625° (32 × 64). Broader resolution scaling is left as a reference rather than a full study.
• Interpretability of the learned terms: The authors show that the correction term alone outlines topography-linked temperature structure, but a systematic analysis of what the original and correction terms encode across tasks is only sketched through the appendix experiments (Figure 6 and Table 11).

Target Audience

Researchers and graduate students working on neural operators, PDE surrogates, geometric deep learning on manifolds, and spectral methods for spherical data. It will also interest applied machine learning practitioners in weather and climate modelling, geophysical fluid dynamics, and medical imaging, particularly those already familiar with FNO and SFNO baselines who need operators that can accommodate non-equivariant, position-dependent physical constraints.

Authors’ abstract

Neural operators offer powerful approaches for solving parametric partial differential equations, but extending them to spherical domains remains challenging due to the need to preserve intrinsic geometry while avoiding distortions that break rotational consistency. Existing spherical operators rely on rotational equivariance but often lack the flexibility for real-world complexity. We propose a generalized operator-design framework based on the designable spherical Green's function and its harmonic expansion, establishing a solid operator-theoretic foundation for spherical learning. Based on this, we propose an absolute and relative position-dependent Green's function that enables flexible balance of equivariance and invariance for real-world modeling. The resulting operator, Green's-function Spherical Neural Operator (GSNO) with a novel spectral learning method, can adapt to non-equivariant systems while retaining spectral efficiency and grid invariance. To exploit GSNO, we develop SHNet, a hierarchical architecture that combines multi-scale spectral modeling with spherical up-down sampling, enhancing global feature representation. Evaluations on diffusion MRI, shallow water dynamics, and global weather forecasting, GSNO and SHNet consistently outperform state-of-the-art methods. The theoretical and experimental results position GSNO as a principled and generalized framework for spherical operator design and learning, bridging rigorous theory with real-world complexity. The code is available at: https://github.com/haot2025/GSNO.