# Neural Operators for Biomedical Spherical Heterogeneity

arXiv: https://arxiv.org/abs/2601.03561 · HTML: https://mlsanity.com/paper/2601.03561/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 38 of its 80 references that are in the map.
- First posted on arXiv 7 Jan 2026.

## Builds on (most cited first, 30 of 38)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [WeatherBench: A benchmark dataset for data-driven weather forecasting](https://mlsanity.com/paper/2002.00469/) (2020)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [Spherical Transformer for LiDAR-based 3D Recognition](https://mlsanity.com/paper/2303.12766/) (2023)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Spherical CNNs on Unstructured Grids](https://mlsanity.com/paper/1901.02039/) (2019)
- [FAENet: Frame Averaging Equivariant GNN for Materials Modeling](https://mlsanity.com/paper/2305.05577/) (2023)
- [Equivariant Graph Mechanics Networks with Constraints](https://mlsanity.com/paper/2203.06442/) (2022)
- [DeepSphere: a graph-based spherical CNN](https://mlsanity.com/paper/2012.15000/) (2020)
- [DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications](https://mlsanity.com/paper/1810.12186/) (2018)
- [Spherical Kernel for Efficient Graph Convolution on 3D Point Clouds](https://mlsanity.com/paper/1909.09287/) (2019)
- [Kernel Transformer Networks for Compact Spherical Convolution](https://mlsanity.com/paper/1812.03115/) (2018)
- [Scale-Invariant Convolutional Neural Networks](https://mlsanity.com/paper/1411.6369/) (2014)
- [Spherical U-Net on Cortical Surfaces: Methods and Applications](https://mlsanity.com/paper/1904.00906/) (2019)
- [Efficient Generalized Spherical CNNs](https://mlsanity.com/paper/2010.11661/) (2020)
- [Improving Equivariant Model Training via Constraint Relaxation](https://mlsanity.com/paper/2408.13242/) (2024)
- [Scaling Spherical CNNs](https://mlsanity.com/paper/2306.05420/) (2023)
- [Machine learning in spectral domain](https://mlsanity.com/paper/2005.14436/) (2020)
- [HEAL-SWIN: A Vision Transformer On The Sphere](https://mlsanity.com/paper/2307.07313/) (2023)
- [Convolutional Networks for Spherical Signals](https://mlsanity.com/paper/1709.04893/) (2017)
- [Huge Ensembles Part I: Design of Ensemble Weather Forecasts using Spherical Fourier Neural Operators](https://mlsanity.com/paper/2408.03100/) (2024)
- [Sensitivity-Constrained Fourier Neural Operators for Forward and Inverse Problems in Parametric Differential Equations](https://mlsanity.com/paper/2505.08740/) (2025)
- [SphereUFormer: A U-Shaped Transformer for Spherical 360 Perception](https://mlsanity.com/paper/2412.06968/) (2024)

## Abstract (arXiv)

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.