# DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications

arXiv: https://arxiv.org/abs/1810.12186 · HTML: https://mlsanity.com/paper/1810.12186/

## Key facts

- Cited by 62 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2018.

## Built on by (most cited first, 30 of 62)

- [Spherical Message Passing for 3D Graph Networks](https://mlsanity.com/paper/2102.05013/) (2021)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Surround-view Fisheye Camera Perception for Automated Driving: Overview, Survey and Challenges](https://mlsanity.com/paper/2205.13281/) (2022)
- [Applications and Techniques for Fast Machine Learning in Science](https://mlsanity.com/paper/2110.13041/) (2021)
- [GNN-Surrogate: A Hierarchical and Adaptive Graph Neural Network for Parameter Space Exploration of Unstructured-Mesh Ocean Simulations](https://mlsanity.com/paper/2202.08956/) (2022)
- [Guided Co-Modulated GAN for 360deg Field of View Extrapolation](https://mlsanity.com/paper/2204.07286/) (2022)
- [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022)
- [Conditional Local Convolution for Spatio-temporal Meteorological Forecasting](https://mlsanity.com/paper/2101.01000/) (2021)
- [Advancing Parsimonious Deep Learning Weather Prediction using the HEALPix Mesh](https://mlsanity.com/paper/2311.06253/) (2023)
- [Scalable and Equivariant Spherical CNNs by Discrete-Continuous (DISCO) Convolutions](https://mlsanity.com/paper/2209.13603/) (2022)
- [Scaling Spherical CNNs](https://mlsanity.com/paper/2306.05420/) (2023)
- [HEAL-SWIN: A Vision Transformer On The Sphere](https://mlsanity.com/paper/2307.07313/) (2023)
- [PDO-eS2CNNs: Partial Differential Operator Based Equivariant Spherical CNNs](https://mlsanity.com/paper/2104.03584/) (2021)
- [Scattering Networks on the Sphere for Scalable and Rotationally Equivariant Spherical CNNs](https://mlsanity.com/paper/2102.02828/) (2021)
- [Equivariant Networks for Pixelized Spheres](https://mlsanity.com/paper/2106.06662/) (2021)
- [A neural simulation-based inference approach for characterizing the Galactic Center γ-ray excess](https://mlsanity.com/paper/2110.06931/) (2021)
- [Diffusion MRI with Machine Learning](https://mlsanity.com/paper/2402.00019/) (2024)
- [Equivariant Spherical Deconvolution: Learning Sparse Orientation Distribution Functions from Spherical Data](https://mlsanity.com/paper/2102.09462/) (2021)
- [DeepAdversaries: Examining the Robustness of Deep Learning Models for Galaxy Morphology Classification](https://mlsanity.com/paper/2112.14299/) (2021)
- [Exploiting Code Symmetries for Learning Program Semantics](https://mlsanity.com/paper/2308.03312/) (2023)
- [HEAL-ViT: Vision Transformers on a spherical mesh for medium-range weather forecasting](https://mlsanity.com/paper/2403.17016/) (2024)
- [OSLO: On-the-Sphere Learning for Omnidirectional images and its application to 360-degree image compression](https://mlsanity.com/paper/2107.09179/) (2021)
- [Pseudocylindrical Convolutions for Learned Omnidirectional Image Compression](https://mlsanity.com/paper/2112.13227/) (2021)
- [Machine Learning and Cosmology](https://mlsanity.com/paper/2203.08056/) (2022)
- [Quantum algorithms for group convolution, cross-correlation, and equivariant transformations](https://mlsanity.com/paper/2109.11330/) (2021)
- [Estimating Depth of Monocular Panoramic Image with Teacher-Student Model Fusing Equirectangular and Spherical Representations](https://mlsanity.com/paper/2405.16858/) (2024)
- [PEAR: Equal Area Weather Forecasting on the Sphere](https://mlsanity.com/paper/2505.17720/) (2025)
- [Inferring dark matter substructure with astrometric lensing beyond the power spectrum](https://mlsanity.com/paper/2110.01620/) (2021)
- [Neural Networks Optimized by Genetic Algorithms in Cosmology](https://mlsanity.com/paper/2209.02685/) (2022)

## Abstract (arXiv)

Convolutional Neural Networks (CNNs) are a cornerstone of the Deep Learning toolbox and have led to many breakthroughs in Artificial Intelligence. These networks have mostly been developed for regular Euclidean domains such as those supporting images, audio, or video. Because of their success, CNN-based methods are becoming increasingly popular in Cosmology. Cosmological data often comes as spherical maps, which make the use of the traditional CNNs more complicated. The commonly used pixelization scheme for spherical maps is the Hierarchical Equal Area isoLatitude Pixelisation (HEALPix). We present a spherical CNN for analysis of full and partial HEALPix maps, which we call DeepSphere. The spherical CNN is constructed by representing the sphere as a graph. Graphs are versatile data structures that can act as a discrete representation of a continuous manifold. Using the graph-based representation, we define many of the standard CNN operations, such as convolution and pooling. With filters restricted to being radial, our convolutions are equivariant to rotation on the sphere, and DeepSphere can be made invariant or equivariant to rotation. This way, DeepSphere is a special case of a graph CNN, tailored to the HEALPix sampling of the sphere. This approach is computationally more efficient than using spherical harmonics to perform convolutions. We demonstrate the method on a classification problem of weak lensing mass maps from two cosmological models and compare the performance of the CNN with that of two baseline classifiers. The results show that the performance of DeepSphere is always superior or equal to both of these baselines. For high noise levels and for data covering only a smaller fraction of the sphere, DeepSphere achieves typically 10% better classification accuracy than those baselines. Finally, we show how learned filters can be visualized to introspect the neural network.