Axisymmetry-informed Neural Operators

Spherical neural operators have advanced operator learning on the sphere by exploiting group symmetries, with equivariance and invariance providing powerful inductive biases. However, such transformation laws are limited in characterizing the diverse physical responses. In this work, we explore a complementary principle: directly constructing neural operators from known physical response structures. We instantiate this principle through axisymmetry, a common physical structure in which system responses are organized around a distinguished axis. Starting from an axisymmetric Green's function, we derive its corresponding operator solution, which admits a structured spectral convolution characterized by a preferred axis and Legendre polynomials while retaining grid-invariance. This leads to the proposed Axisymmetry-informed Neural Operator (AsiNO). Across spherical physical dynamics, geophysical prediction and cortical modeling, AsiNO achieves superior operator learning, cross-scale generalization and spherical modeling with substantially improved parameter efficiency. These results demonstrate that physical response structure, beyond group symmetry, provides an effective principle for constructing neural operators on the sphere.