Welcome to my webpage !
Here you will find all my research material

About me

I am Eloi Martinet, a postdoctoral researcher at the University of Würzburg and part of the team of Mathematics of Machine Learning. I am interested in using a variational approach to tackle machine learning problems, and using neural networks capabilities to solve shape optimization problems.

In October 2023, I defended my PhD Thesis about spectral shape optimization supervised by Dorin BUCUR and Edouard OUDET. You can download the manuscript here.

You can download my CV here.

Education

  • Postdoc in Variational Methods in Machine Learning.
    JMU (Würzburg, Germany)
  • PhD in Spectral Shape Optimization, 2019-2023
    LAMA (Chambéry), LJK (Grenoble)
  • Agrégation de Mathématiques, 2020-2021
    Institut Fourier (Grenoble)
  • Engineering School in Informatics and Applied Mathematics, 2016-2019
    ENSIMAG (Grenoble)
Picture of me

Research interests

  • Shape and topology optimization
  • Numerical methods for PDEs
  • Level set method
  • Neural Networks
  • Graph-based Machine Learning
  • Meshless methods

Teaching

Supervision

  • 2026 : Supervision of Jana Bardorz (student assistant) about Shape Optimization using Neural Networks
  • 2025 : Co-supervison of the MSc Internship of Nick Burk with Leon Bungert about Phase Field on Graphs (thesis, GitHub)
  • 2024 : Co-supervison of the MSc Internship of Nicolas Roblet with Romain Joly titled Spectrum of unbounded operators and applications to PDEs (thesis).

Publications

& preprints

Research Projects

(Non-)closedness of the set of ReLU networks

When is a limit of neural networks still a neural network of the same size? For one hidden layer, always in Sobolev spaces, which guarantees that Deep Ritz or Sobolev training have optimal networks. With two hidden layers, closedness breaks down, even for uniform convergence.

Piecewise affine surface of the limit function of the counterexample
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Numerical exploration of Blaschke-Santaló diagrams

Using a neural network parametrization of convex sets, we systematically explore the range of shape functionals.
Convex shape optimization using neural networks

How a new neural network architecture provides a constraint-free parametrization of convex sets. Try it yourself with example notebooks online!

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Neural-network-based level set method

Parametrizing a level set function with a neural network proved to be useful for overcoming certain burdens of mesh-based level set methods and computing easily common geometric quantities.
Solving PDEs with Graphs

Originally used in the context of unsupervised and semi-supervised learning, the graph laplacian can also be used as as a numerical PDE solver.

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Level set optimization of Neumann eigenvalues on the sphere

We perfom an analysis of the shape optimization problem of Neumann eigenvalues for domains on the sphere thanks to the level set method.
Density optimization of Neumann eigenvalues on the sphere

While the planar case has been investigated theoretically since the 50's, the optimization of Neumann eigenvalues on the sphere is still mysterious. We try to identify numerically some behaviour of the optimal densities.

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Optimization of Neumann eigenvalues in the plane

What is the shape of the drum that has the largest eigenfrencies ? We investigate this issue by implementing some density-based optimization procedure.

Teaching ressources

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Numerical Methods for PDEs
From the Finite Element Method to Neural Networks
Go to the course
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PDEs on Graphs
At the intersection of data science, PDEs and calculus of variations
Go to the course

Other Projects

QPong

A Pong game so small that the ball is alive and dead at the same time.

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Lenia: Artifical Life

Here we show how to implement a simple continuous cellular automaton that reproduces certain aspects of Conway's Game of Life.

Solving PDEs with neural networks

A folder of notebooks used in the context of my lecture on FEM and PINNs, showing different methods to solve PDEs with neural networks: PINNs, DRM, Natural Gradient Descent and Sampled Neural Networks.

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Image segmentation using Semi Supervised Learning

A notebook developed in the context of an outreach activity at the University of Würzburg. It shows how to perform semi-supervised segmentation using a graph approach.

A phase-field demonstration of the isoperimetric problem

The legendary isoperimetric problem is the most famous and ancient problem in shape optimization. It consists in finding the shape minimizing its perimeter while keeping its volume fixed. Here we simulate the solution to this problem using a relaxed formulation of the perimeter.

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Optimization of Neumann eigenvalues with FreeFem and Python

I provide the source code and some practical explanations on the density optimization of Neumann eigenvalues using python and FreeFem++.