Generalized Spherical Neural Operators: Green's Function Formulation
This work addresses the problem of solving parametric partial differential equations on spherical domains for applications such as medical imaging and climate modeling, offering a principled framework that balances theoretical rigor with practical flexibility.
The authors tackled the challenge of extending neural operators to spherical domains by proposing a generalized framework based on spherical Green's functions, resulting in GSNO and SHNet that outperform state-of-the-art methods in tasks like diffusion MRI and weather forecasting.
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.