In many applications, data lie on non-Euclidean manifolds. Examples on the sphere include planetary data (such as temperature, wind, aerial imagery) and cosmological surveys (see DeepSphere [1, 2, 3] for details). While the physical processes that govern the data are often expensive to simulate or unknown, we can hope for a data-driven alternative when data is abundant. Designing Neural Networks that are efficient, while respecting the geometry and the physics of a problem, is a major challenge in Machine Learning.

Graph Neural Networks (GNNs) are a particularly interesting solution for this problem as they are a computationally efficient and flexible way to learn from structured data. In our setting, the samples are drawn from the manifold, and the structure is given by the manifold's geometry. As of today, the difficulty resides in building a graph such that the convolution operator, a function of the Laplacian operator, is equivariant to the symmetry group of the manifold. That is especially challenging when the sampling is not uniform.

**Project goal.**
Build a graph from a sampled manifold such that the graph convolution respects the symmetries of the manifold.
Test it in a graph neural network on multiple tasks involving data on non-Euclidean manifolds.
The student will interact on a weekly basis with the supervisors and receive ample directions and advices.
This project can lead to a publication.
The project can be steered towards more theory or practice, and the following tasks will be adapted upon the interests and strengths of the student.

- Study the literature in the relevant fields for a comparison: (i) graph signal processing, (ii) numerical mathematics (especially the finite element method and its variants), (iii) (discrete) differential geometry, (iv) (discrete) exterior calculus.
- Study the properties of the discrete Laplace-Beltrami operators developed in those fields. Relate them to get a global picture.
- Build a GNN and evaluate it on various tasks. Those can be chosen based on the interests of the student and may involve the collaboration with domain experts.

**Prerequisites.**
Highly motivated, ambitious, and independent student.
Experience with Python programming.
Experience in any of the mentioned field would be great.
Experience with (Deep) Machine Learning is desirable.

**What to expect.**
The student will interact on a weekly basis with the supervisors and receive ample directions and advices.
The student will produce midterm and final presentations.
The student will be evaluated on his work during the semester, a report, the final presentation, and the produced code.
This project can lead to a publication.

**Supervisors & Contact**

- Michaël Defferrard, michael.defferrard@epfl.ch, LTS2, EPFL
- Martino Milani, martino.milani@epfl.ch, EPFL
- Nathanaël Perraudin, nathanael.perraudin@sdsc.ethz.ch, Swiss Data Science Center

**References**

- Perraudin, N., Defferrard, M., Kacprzak, T., & Sgier, R. (2019). DeepSphere: Efficient spherical convolutional neural network with HEALPix sampling for cosmological applications. Astronomy and Computing, 27, 130-146.
- Defferrard, M., Perraudin, N., Kacprzak, T., & Sgier, R. (2019). DeepSphere: towards an equivariant graph-based spherical CNN.
- Slides presented at the AI methods in Cosmology workshop.