Title: Generalized Spherical Neural Operators: Green’s Function Formulation
Primary area: learning on graphs and other geometries & topologies
Keywords: Fourier neural operator, Green function, Spherical harmonic

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 introduce a rigorous formulation of spherical operators from spherical Green’s function and harmonic expansion, establishing a solid operator-theoretic foundation for spherical spectrum learning, and the proposed Green's Function Formulation imposing physical constraints. Based on this, we propose an absolute position-dependent constraint in the Green’s function that enables flexible balance of equivariance and nonlinear modeling. The resulting operator, Generalized Spherical Neural Operator with a novel spectral learning method (GSNO), can adapt to anisotropic, constraint-rich systems while retaining spectral efficiency. To exploit GSNO, we develop GSHNet, 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 GSHNet consistently outperform state-of-the-art methods. Our results position GSNO as a principled and general framework for spherical operator learning, bridging rigorous theory with real-world complexity.