Neural Operators for Biomedical Spherical Heterogeneity

2026-01-07

arXiv.org

AI Summary

This work addresses the challenge in spherical geometric modeling of simultaneously incorporating strong geometric inductive biases and capturing the heterogeneity inherent in real biological systems. To this end, the authors propose the Geometric Spherical Neural Operator (GSNO), which unifies equivariant, invariant, and anisotropic solution components within a novel Designable Green’s Function (DGF) framework. By integrating Green’s function design on rotation groups, spherical deep learning, equivariant/invariant neural operators, and spectral methods, GSNO effectively models complex systems exhibiting directional preferences and perturbative variability. The method demonstrates consistent and significant performance improvements over existing approaches across diverse tasks, including spherical MNIST classification, shallow water equation simulation, diffusion MRI fiber prediction, cortical parcellation, and molecular structure modeling.

Technology Category

Machine Learning: Learning with ManifoldsSearch and Optimization: Mixed Discrete/Continuous SearchKnowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsSystems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• spherical deep learning
• biological heterogeneity
• anisotropy
• nuisance variability
• geometric inductive bias

Innovation

• Green's function
• spherical neural operator
• biological heterogeneity
• equivariance
• anisotropy