Generalized Spherical Neural Operators: Green's Function Formulation

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 designable 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 nonequivariant systems while retaining spherical geometry, 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 . However, FNOs rely on the standard Fourier transform and assume Euclidean geometry. On non-Euclidean manifolds such as the sphere (Bonev et al., 2023) , FFT-based representations introduce distortions: small polar displacements can map to large Cartesian displacements, breaking spatial coherence and degrading performance. To address this, Spherical Fourier Neural Operator (SFNO) is proposed (Bonev et al., 2023) , replacing the FFT with the Spherical Harmonic Transform (SHT). By projecting functions onto spherical harmonic bases, SFNO preserves rotational equivariance on the sphere, ensuring stability under arbitrary input rotations. SFNO-based methods have achieved strong performance on some spherical tasks, e.g., weather prediction (