Post-Doctorant F/H Greedy neural approaches for transport PDEs and optimal control

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Type de contrat : CDD

Niveau de diplôme exigé : Thèse ou équivalent

Fonction : Post-Doctorant

Mission confiée

Subject: In recent years, new approximation methods based on neural networks have appeared.
These include PINNs [1], Deep-Ritz [2] and Neural Galerkin methods [3].
These approaches have the advantage of not using meshes and of being dimension insensitive. Indeed, for regular
solutions, the number of parameters required to give a reasonable approximation increases, roughly linearly, with dimension, whereas the number of degrees of freedom increases exponentially with dimension for classical methods such as finite elements.
The main drawback of these approaches is their lack of precision. Recently, a number of methods [4]-[5] based on an approach that could be called
Gluttonous [6] have made it possible to obtain accurate results for simple elliptic problems in dimension 1 and 2. In this post-doc, we propose to extend these methods to parametric (i.e. higher-dimensional) temporal transport problems. This extension will be both numerical and theoretical. It is quite likely to require modifications of the previous method, following for example methods introduced
in [7].  Secondly, we propose to apply this to solve optimal closed-loop control problems by solving the Hamilton Jacobi Bellman equation [8], which describes the evolution of the value function that models the cost expectation over time for a given control. The Greedy method can be coupled with PINNs, but also with semi-Lagrangian methods as in [9] which could allows to reduce the time error
 
Practical information: The post-doctorate will last 18 months. It can begin between november 2024 and aptil 2025. The post-doc will take place at University of Strasbourg, within the Mathematics laboratory (downtown campus). The campus provides a very nice working environment with an excellent cafeteria.
 
Supervisors: E. Franck, L. Navoret, V. Michel Dansac (INRIA and IRMA)
Collaborators: V. Ehrlacher (Cermics Paris and INRIA), Y. Privat (Mines Nancy and INRIA), C. Courtès (INRIA and IRMA)
 
 
[1] Raissi, Maziar; Perdikaris, Paris; Karniadakis, George Em (2017-11-28). "Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations". 
[2] The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems, Y. Bing and al.
[3] Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations, Joan Bruna, Benjamin Peherstorfer, Eric Vanden-Eijnden
[4] Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control, Mark Ainsworth, Justin Dong
[5] Multi-stage Neural Networks: Function Approximator of Machine Precision Yongji Wang, Ching-Yao Lai
[6] Convergence of a greedy algorithm for high-dimensional convex nonlinear problems, Eric Cances, Virginie Ehrlacher, Tony Lelievre
[7] Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces, Virginie Ehrlacher, Damiano Lombardi, Olga Mula, François-Xavier Vialard
[8] Learning HJB Viscosity Solutions with PINNs for Continuous-Time Reinforcement Learning, Alena Shilova , Thomas Delliaux , Philippe Preux , Bruno Raffin
[9] Semi-Lagrangian schemes for linear and fully non-linear Hamilton-Jacobi-Bellman equations, Kristian Debrabant, Espen R. Jakobsen
 

Principales activités

Principales activés (5 maximum) :

Activités complémentaires (3 maximum) :

 

Exemples d'activités :

  • Analyser les besoins des {partenaires, clients, usagers}
  • Proposer des solutions **** pour ****
  • Développer des programmes/ des applications/ des interfaces de ****, ****
  • Concevoir des plateformes expérimentales ****
  • Rédiger la documentation
  • Rédiger les rapports
  • Rédiger ****
  • Tester, modifier jusqu’à valider
  • Diffuser le(s) **** vers **** via ****
  • Former à l’utilisation les principaux clients du service
  • Animer une communauté d’utilisateurs
  • Présenter l’avancée des travaux aux partenaires, **** devant un public de financiers ****
  • Autre ****

Compétences

 
Research profil: The candidate must have a PhD in applied mathematics. This could be an EDP profile with knowledge of ML or an ML profile with knowledge of EDP. A good knowledge of Python programming is important.

Avantages

  • Restauration subventionnée
  • Transports publics remboursés partiellement
  • Congés: 7 semaines de congés annuels + 10 jours de RTT (base temps plein) + possibilité d'autorisations d'absence exceptionnelle (ex : enfants malades, déménagement)
  • Possibilité de télétravail (après 6 mois d'ancienneté) et aménagement du temps de travail
  • Équipements professionnels à disposition (visioconférence, prêts de matériels informatiques, etc.)
  • Prestations sociales, culturelles et sportives (Association de gestion des œuvres sociales d'Inria)
  • Accès à la formation professionnelle
  • Sécurité sociale

Rémunération

2788 € gross/month