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Green's-Function Spherical Neural Operators for Biological Heterogeneity

Hao Tang, Hao Chen, Hao Li, Chao Li

TL;DR

The paper tackles heterogeneity on spherical domains by marrying geometric inductive bias with real-world variability. It introduces the Designable Green's Function Framework (DGF) to construct programmable spherical Green's functions and builds Green's-Function Spherical Neural Operators (GSNO) by fusing Equivariant, Invariant, and Anisotropic components. These operators are derived from distinct Green's functions and enable efficient, spectrum-friendly modeling of heterogeneous biologically inspired systems. Extensive experiments across synthetic benchmarks, diffusion MRI, cortical parcellation, and molecular modeling show GSNO's superior generalization and robustness, validating the approach's practicality for real-world spherical data. The work provides a principled route to incorporate controlled symmetry breaking and directional bias while preserving sphere-native computation.

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 biological 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.

Green's-Function Spherical Neural Operators for Biological Heterogeneity

TL;DR

The paper tackles heterogeneity on spherical domains by marrying geometric inductive bias with real-world variability. It introduces the Designable Green's Function Framework (DGF) to construct programmable spherical Green's functions and builds Green's-Function Spherical Neural Operators (GSNO) by fusing Equivariant, Invariant, and Anisotropic components. These operators are derived from distinct Green's functions and enable efficient, spectrum-friendly modeling of heterogeneous biologically inspired systems. Extensive experiments across synthetic benchmarks, diffusion MRI, cortical parcellation, and molecular modeling show GSNO's superior generalization and robustness, validating the approach's practicality for real-world spherical data. The work provides a principled route to incorporate controlled symmetry breaking and directional bias while preserving sphere-native computation.

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 biological 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.
Paper Structure (18 sections, 19 equations, 3 figures, 6 tables)

This paper contains 18 sections, 19 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Results
  3. Study design
  4. Experimental design
  5. Synthetic Spherical Scenarios
  6. Spherical MNIST
  7. Shallow Water Equation
  8. Real-world Spherical Scenarios
  9. Diffusion MRI-based fiber estimation
  10. Cortical parcellation
  11. Molecule structure modeling
  12. Discussion
  13. Methods
  14. Designable Green's Function Framework
  15. Three Operators from three Green's Functions
  16. ...and 3 more sections

Figures (3)

  • Figure 1: The proposed DGF and GSNO. SHT and ISHT represent spherical harmonic transform and inverse transform. $w$ is the linear layer and $\sigma$ is the activation function.
  • Figure 2: FOD and Tractography on hcp dataset.
  • Figure 3: Cortical parcellation on Mindboggle-101 and NAMIC-39 datasets.