用绿色函数设计球面神经算子，精准建模生物医学中的异质性与各向异性。
Neural Operators for Biomedical Spherical Heterogeneity
- 基于旋转群设计可定制绿函数，融合对称、不变与各向异性解法
- 在多个球面任务上优于现有方法，尤其在纤维方向预测中表现突出
- 适合需要保留球面几何且处理复杂生物结构的研究者
球面深度学习已广泛应用于各类实际问题。现有方法常难以平衡强球面几何先验与真实世界异质性的建模需求。为解决此问题并保持球面几何特性，我们提出可设计的绿函数框架（DGF），通过在旋转群下系统设计绿函数，提供新的球面算子求解策略。基于DGF，我们构建了绿函数球面神经算子（GSNO），融合三种解法：(1) 由等变绿函数导出的等变解，实现对称一致性建模；(2) 由不变绿函数导出的不变解，消除冗余异质性（如一致背景场）；(3) 由各向异性绿函数导出的各向异性解，建模具有主方向的系统（如纤维）。因此，GSNO可在保留谱效率的同时，适应包含冗余变异和各向异性的现实异质系统。在球面MNIST、浅水方程、扩散磁共振纤维预测、皮层分割及分子结构建模任务上的评估表明其显著优势。
原文摘要 · Abstract (English)
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.
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