Neural Operators for Biomedical Spherical Heterogeneity

TL;DR

This paper introduces a novel Green's function framework for spherical neural operators, enabling modeling of heterogeneity and anisotropy in biomedical applications while maintaining spectral efficiency.

Contribution

The paper proposes Green's-Function Spherical Neural Operator (GSNO), a new approach combining equivariant, invariant, and anisotropic solutions for better modeling of heterogeneous spherical systems.

Findings

GSNO outperforms existing methods on spherical MNIST and other benchmarks.

It effectively models heterogeneity and anisotropy in biomedical data.

Demonstrates superior accuracy in cortical parcellation and fiber prediction.

Abstract

Spherical deep learning has been widely applied to a broad range of real-world problems. Existing approaches often face challenges in balancing strong spherical geometric inductive biases with the need to model real-world heterogeneity. To solve this while retaining spherical geometry, we first introduce a designable Green's function framework (DGF) to provide new spherical operator solution strategy: Design systematic Green's functions under rotational group. Based on DGF, to model biomedical heterogeneity, we propose Green's-Function Spherical Neural Operator (GSNO) fusing 3 operator solutions: (1) Equivariant Solution derived from Equivariant Green's Function for symmetry-consistent modeling; (2) Invariant Solution derived from Invariant Green's Function to eliminate nuisance heterogeneity, e.g., consistent background field; (3) Anisotropic Solution derived from Anisotropic Green's Function to model anisotropic systems, especially fibers with preferred direction. Therefore, the resulting model, GSNO can adapt to real-world heterogeneous systems with nuisance variability and anisotropy while retaining spectral efficiency. Evaluations on spherical MNIST, Shallow Water Equation, diffusion MRI fiber prediction, cortical parcellation and molecule structure modeling demonstrate the superiority of GSNO.