Amaury Hayat - Academic webpage

Welcome on my academic webpage! Here you will find:

Some info and a CV

My current research subjects

My articles and preprints

My talks

Science outreach

Version française             

About

I am currently a Professor at Ecole des Ponts Paristech since April 2023 and a faculty member of the CERMICS (since 2019). I was a Research Associate at Rutgers University in 2019-2020 (on leave from Ecole des Ponts ParisTech), working with Benedetto Piccoli. Prior to this I was a Ph.D. candidate at Sorbonne Université (UPMC) in the Laboratory Jacques-Louis Lions, under the supervision of Jean-Michel Coron, and the joint supervision of Sébastien Boyaval (Ecole des Ponts Paristech, Laboratory of Hydraulic Saint-Venant).

I had the chance to have an invited stay at the Forschungsinstitut für Mathematik at ETH Zürich in 2017 (long visit), at the Department of Mathematics at Tongji University in 2018 (long visit), at the Department of Mathematics of EPFL in 2022 (short visit), at the Department of Mathematical Sciences of Rutgers University-Camden in 2022 (short visit), and at the Department of Mathematics of Peking University in 2024 (long visit) as laureate of the program Young Talent French-China. I was also an Oberwolfach Research Fellow / Research in Pairs in February 2022.

Here is a CV .  

You can contact me at: amaury.hayat "at" enpc.fr


Current research interests:

On the one hand, I study the stabilization of partial differential equations (PDE) and systems of PDEs. That is the ability to make a system stable when one has a means of control on it. I'm also studying the design and use of AI models to solve advanced mathematical problems, with the aim of helping mathematicians find solutions to open problems in mathematics.

The CERMICS is recruiting a Cauchy postdoctoral fellow ! More details on the opening here

Research subjects

The generic stabilization problem for evolution equations

Given a control system, is it possible to find an explicit feedback control that can stabilize it rapidly? This is a complicated question in general. We are looking for methods to solve this problem in a generic way. We use an approach called F-equivalence (or generalized backstepping). This consists in simultaneously constructing a feedback operator and an isomorphism that maps our system into an easier one that can be shown to be rapidly stable. An example on the heat equation. A general method for the skew-adjoint systems.

The stabilization of inhomogeneous systems.

Most physical systems are inhomogeneous, however this inhomogeneity leads to some difficulties when it couples equations with different propagation speeds. I studied this question with a focus on nonlinear hyperbolic systems with boundary controls. An example for the generic case in \(C^{1}\) norm. An exemple to guarantee in addition the Input-to-State Stability (ISS). An example for a global stabiliy in \(L^{2}\) norm for semilinear Lipschitz system.

Stability and stabilization of density velocity systems

These systems, consisting of a continuity equation and a dynamic equation, cover a large class of physical systems for Saint-Venant equations to osmosis phenomenon to isentropic Euler equations. We showed that these systems possess a kind of hidden energy that makes them easy to stabilize by the boundaries, even when part of the system is unknown. This opens the door to a wide range of applications. An example for the Saint-Venant equations, and for the general case.

Propotionnal integral (PI) controls for PDE.

PI controls are among the most used control tools in practice, but they are hard to handle mathematically when applied to nonlinear PDEs, and thus seldom studied. How to use them to their full potential? A method for a scalar equation. An example of system: Saint-Venant equations .

Smoothing traffic flows (in practice)

When congested, road traffic flow is unstable: stop-and-go waves form. Eliminating these stop-and-go waves would make the highways safer, and reduce traffic energy consumption and the associated CO2 emissions. To achieve this goal, we want to use a small proportion of autonomous vehicles (AV) to act as a means of control on the system. I am studying mathematical aspects of this question and the associated control algorithms within the project CIRCLES (2020-2022). An example on a ring-road. We conducted a large scale experiment in November 2022, here are some preliminary experimental results and, a summary of the approach of CIRCLES and a summary of the experiment.

Smoothing traffic flows (Hyperbolic PDE)

From a mathematical point of view, it is possible to view the traffic stabilization problem on several scales. On a macroscopic scale, the problem is modelled by hyperbolic PDEs. Surprisingly, entropic solutions, designed to be the physical solutions in fluid mechanics, are usually not the physical solutions of these systems. Non-classical shocks and fascinating new behaviors are emerging and force us to re-think many existing tools in analysis.

AI for advanced mathematical problems

Is it possible to teach advanced mathematical problems to an AI with no built-in math knowledge? It seems so. In the long run, we aim at using deep learning models to give insights on open math problems. An example where the model guesses the solution of a controllability problem. Another example relying on Reinforcement Learning to find an explicit control feedback law for a system modelling the population of mosquitoes. The difficulty of this system comes from the physical constraint on the measurements that prevents us from using the usual stabilization tools.

Automated theorem proving

Is it possible to train a neural network to prove theorems? To answer this question, we have worked with Meta AI's Evariste team to develop and train a neural network capable of proving small theorems (i.e. exercises) and providing the proof. Our approach is inspired by that of AlphaZero in chess. The article and the blog post. Some of the exercises proved by the neural network are taken from the International Mathematical Olympiads (IMO).

Students

Ph.D. students

  • Jean Cauvin-Vila (2020-2023, with Virginie Ehrlacher), Cross-diffusion systems in moving boundary domains
  • Nathan Lichtlé (2021-2024, with Alexandre Bayen), Reinforcement Learning to stabilize mathematical models
    • > Nathan was awarded the UC Berkeley Fellowship.
  • Epiphane Loko (since 2022, with Antoine Chaillet), Stability of infinite dimensional systems and disturbances
  • Yating Hu (since 2022, with Peipei Shang) Stabilization of networks of Saint-Venant equations
  • Fabian Glöckle (since 2023, with Timothy Gowers) Automated theorem proving using proof context
  • François Charton (since 2024) Language models for mathematics

Master students and research interns

  • Vincent Boulard (2024), master student internship.
  • Gwenaëlle Léon (2024), master student internship (with Fabian Glöckle).
  • Vincent Boulard (2023), undergraduate student (Ecole des Ponts Paristech).
    • > Vincent was awarded the Junior Fermat Prize 2023.
  • Fabian Glöckle (09/22-03/23), research associate (with Timothée Lacroix)
  • Tinhinane Mezair (2022), Master thesis (Algerian Ecole Nationale Polytechnique)
  • Epiphane Loko (2021), Master thesis (IMSP - Benin)
  • Nicolas Kardous (07/20-10/20), Master student intern (UC Berkeley, with Alexandre Bayen and Alexander Keimer)
  • Sydney Truong (11/19-06/20), undergraduate researcher (Rutgers University, with Benedetto Piccoli)
    • > Sydney was awarded the Dean's Undergraduate Research Prize and the Mathematical Sciences Scholarship Award for best undergraduate research 2020.

Talks

International conferences

Colloquium, invited or distinguished lectures, seminars and thematic days

Science outreach

You will find here the media where our work was mentionned.

In the media (in French)

In the media (in English)

Videos

And here are some popularization videos:

My portrait at CERMICS

TedX Ecole des Ponts - The next generation of mathematicians: in training ?

Video of Y. Kilcher on our article Learning advanced mathematical computations by examples


Articles and preprints

In mathematics:


Preprints

  1. Exponential stability of the viscous Saint-Venant equations using a quadratic Lyapunov function (with Nathan Lichtlé), 2024. Preprint.

  2. Fredholm backstepping for general linear systems (with Epiphane Loko), 2024. Preprint.

  3. The usefulness of viscosity for the robustness of boundary feedback control of an unstable fluid flow system (with Georges Bastin and Jean-Michel Coron), 2023. Preprint, abstract.

  4. Boundary Stabilization of Star-Shaped Saint-Venant Networks with Combined Subcritical and Supercritical Channels (with Yating Hu and Peipei Shang), 2023. Preprint, abstract.

  5. A multiscale second order model for the interaction between AV and traffic flows: analysis and existence of solutions (with Thibault Liard, Francesca Marcellini and Benedetto Piccoli), 2021. Preprint, abstract.

Published

  1. Boundary stabilization of one-dimensional cross-diffusion systems in a moving domain (with Jean Cauvin-Vila and Virginie Ehrlacher), Journal of Differential Equations 2023. Preprint, abstract.

  2. Fredholm backstepping for critical operators and application to rapid stabilization for the linearized water waves (with Ludovick Gagnon, Shengquan Xiang and Christophe Zhang), to appear in Annales de l'institut Fourier, 2023. Preprint, abstract.

  3. Diffusion and robustness of boundary feedback stabilization of hyperbolic systems (with Georges Bastin et Jean-Michel Coron), Mathematics of Control, Signals, and Systems, 2023. Preprint.

  4. PI control for the cascade channels modeled by general Saint-Venant equations (with Yating Hu et Peipei Shang), IEEE Transactions on Automatic Control, 2023, Preprint, article, abstract.

  5. Stability of multi-population traffic flows (with Benedetto Piccoli and Shengquan Xiang), Networks and Heterogeneous Media, 2023. Preprint, abstract.

  6. Fredholm transformation on Laplacian and rapid stabilization for the heat equation (with Ludovick Gagnon, Shengquan Xiang and Christophe Zhang), Journal of Functional Analysis, 2022. Preprint, abstract.

  7. Limitations and Improvements of the Intelligent Driver Model (IDM) (with Xiaoqian Gong, Alexander Keimer, Yilling You et al.), SIAM Journal on Applied Dynamical Systems, 2022. Article. Preprint.

  8. Boundary stabilization of 1D hyperbolic systems, Annual Reviews in Control, 2021. Article.

  9. Stabilization of the linearized water tank system (with Jean-Michel Coron, Shengquan Xiang and Christophe Zhang), Archive for Rational Mechanics and Analysis, 2022. Preprint, abstract.

  10. Dissipation of traffic jams using a single autonomous vehicle on a ring road (with Benedetto Piccoli and Sydney Truong), SIAM Journal on Applied Mathematics, 2023. Preprint, abstract.

  11. Global exponential stability and Input-to-State Stability of semilinear hyperbolic systems for the \(L^{2}\) norm, Systems and Control Letters, 2021. Article.

  12. Input-to-State Stability in sup norms for hyperbolic systems with boundary disturbances, (with Georges Bastin and Jean-Michel Coron), Nonlinear Analysis, 2021. Preprint, abstract.

  13. A Quantitative Study of Circadian Period and Phase of Entrainment (with Zheming An et al.), Bulletin of Mathematical Biology, 2020. Article.

  14. Feedforward boundary control of \(2 \times 2\) nonlinear hyperbolic systems with application to Saint-Venant equations, (with Georges Bastin and Jean-michel Coron), European Journal of Control, 2020. Preprint, abstract.

  15. Exponential stability of density-velocity systems with boundary conditions and source term for the \(H^{2}\) norm, (with Peipei Shang), Journal de Mathématiques Pures et Appliquées, 2021. Preprint, abstract.

  16. PI controller for the general Saint-Venant equations, Journal de l'École Polytechnique 2022. Preprint, abstract.

  17. Boundary feedback stabilization of hydraulic jumps, (with Georges Bastin, Jean-Michel Coron and Peipei Shang), IFAC Journal of systems and Control, 2018. Preprint.

  18. PI controllers for 1D nonlinear transport equation, (with Jean-Michel Coron), IEEE: Transactions on Automatic Control, 2018. Preprint, abstract.

  19. On boundary stability of inhomogeneous \(2 \times 2\) 1-D hyperbolic systems for the \(C^1\) norm, ESAIM: Control, Optimisation and Calculus of Variations, 2018. Preprint.

  20. Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation, (with Georges Bastin, Jean-Michel Coron and Peipei Shang), Mathematical Models and Methods in Applied Sciences, 2017. Preprint, abstract.

  21. Exponential stability of general 1-D quasilinear systems with source terms for the \(C^{1}\) norm under boundary conditions, 2019, SIAM J. Control. Optim. Preprint (2017), abstract.

  22. A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope (with Peipei Shang), Automatica, 2017. Preprint.

In machine learning

  1. Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers (with Alberto Alfarano and François Charton), accepted at NeurIPS, 2024. Preprint.

  2. ABEL: Sample Efficient Online Reinforcement Learning for Neural Theorem Proving (with Fabian Glöckle, Jannis Limperg and Gabriel Synnaeve), accepted at 4th Workshop on Math and AI at NeurIPS 2024. Preprint.

  3. A Novel Approach to Feedback Control with Deep Reinforcement Learning (with Kala Agbo Bidi, Jean-Michel Coron and Nathan Lichtlé), 2023. Preprint.

  4. Reinforcement Learning in Control Theory: A New Approach to Mathematical Problem Solving (with Kala Agbo Bidi, Jean-Michel Coron and Nathan Lichtlé), 3nd Workshop on Math and AI at NeurIPS, 2023. Preprint, abstract.

  5. Temperature-scaled large language models for Lean proofstep prediction (with Fabian Glöckle, Baptiste Roziere and Gabriel Synnaeve), 3nd Workshop on Math and AI at NeurIPS, 2023. Preprint, abstract.

  6. HyperTree Proof Search for Neural Theorem Proving (with Guillaume Lample, Marie-Anne Lachaux, Thibaut Lavril, Xavier Martinet, Gabriel Ebner, Aurélien Rodriguez, Timothée Lacroix), NeurIPS, 2022. Preprint, abstract.

  7. A deep language model to predict metabolic network equilibria (with François Charton, et al.), 2021. Preprint, abstract.

  8. Learning advanced computations from examples, (with François Charton and Guillaume Lample), ICLR, 2021. Preprint, abstract.

In applicative fields

  1. Traffic smoothing using explicit local controllers (with Arwa AlAnqary et al.), conditionnally accepted at IEEE Control and Systems Magazine, 2024. Preprint, abstract.

  2. Traffic Control via Connected and Automated Vehicles (with Jonathan Lee, Han Wang, Kathy Jang, Matthew Bunting et al.) accepted in IEEE Control and Systems Magazine, 2024. Preprint, abstract.

  3. Approaches for Synthesis and Deployment of Controller Models on Automated Vehicles for Car-following in Mixed Autonomy (with Rahul Bhadani et al.), Proceedings of CPS-IoT 2023. Paper, abstract.

  4. Integrated Framework of Vehicle Dynamics, Instabilities, Energy Models, and Sparse Flow Smoothing Controllers (with Jonathan Lee, George Gunter, Rabie Ramadan et al.), Proceedings of the ACM 1st Workshop on Data-Driven and Intelligent Cyber-Physical Systems, 2021. Preprint.

  5. A rigorous multi-population multi-lane hybrid traffic model and its mean-field limit for dissipation of waves via autonomous vehicles, (with Nicolas Kardous et al.), European Physical Journal - Special Topics, 2022. Preprint.

In plant science (at Cambridge University, DPMMS / Friend Group)

  1. Modeling Tree Growth Taking into Account Carbon Source and Sink Limitations, (with Andrew Friend, Andrew J. Hacket-Pain, Hans Pretzsch, Tim T. Rademacher) Front. Plant Sci. 2017, 8:182.doi: 10.3389/fpls.2017.00182. Article.

In physics (at Harvard University, Capasso Group)

  1. Lateral chirality-sorting optical forces, (with J.P. Balthasar Mueller and Federico Capasso) Proceedings of the National Academy of Sciences 2015, 112 (43) 13190-13194; DOI: 10.1073/pnas.1516704112. Article.

Book chapter

  1. A holistic approach to the energy-efficient smoothing of traffic via autonomous vehicles (with CIRCLES consortium), Intelligent Control and Smart Energy Management, Springer. Book chapter.

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