Generalized Spherical Neural Operators: Green's Function Formulation

TL;DR

This paper introduces GSNO, a flexible spherical neural operator based on Green's functions, capable of modeling complex real-world systems while maintaining spectral efficiency and invariance properties.

Contribution

It presents a novel Green's function-based framework for spherical neural operators, enabling adaptable equivariance and invariance, and develops a hierarchical architecture SHNet for improved global feature learning.

Findings

GSNO outperforms existing methods on diffusion MRI, water dynamics, and weather forecasting.

The Green's function approach allows flexible balance of equivariance and invariance.

SHNet enhances global feature representation through multi-scale spectral modeling.

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.