Internship Position F/m Multilevel and Distributed

il y a 2 semaines

Sophia Antipolis, France Inria Temps plein

Le descriptif de l’offre ci-dessous est en Anglais_

**Type de contrat **:Stage

**Niveau de diplôme exigé **:Bac + 5 ou équivalent

**Autre diplôme apprécié **:Master in applied mathematics or scientific computing

**Fonction **:Stagiaire de la recherche

**A propos du centre ou de la direction fonctionnelle**:
Inria is a national research institute dedicated to digital sciences that promotes scientific excellence and transfer. Inria employs 2,400 collaborators organised in research project teams, usually in collaboration with its academic partners.
This agility allows its scientists, from the best universities in the world, to meet the challenges of computer science and mathematics, either through multidisciplinarity or with industrial partners.
A precursor to the creation of Deep Tech companies, Inria has also supported the creation of more than 150 start-ups from its research teams. Inria effectively faces the challenges of the digital transformation of science, society and the economy.

**Contexte et atouts du poste**:
Numerical simulations of electromagnetic wave propagation problems primarily rely on a space discretization of the system of Maxwell’s equations using methods such as finite differences or finite elements. For complex and realistic three-dimensional situations, such a process can be computationally prohibitive, especially when the end goal consists in many-query analyses (e.g., optimization design and uncertainty quantification). Therefore, developing cost-effective _surrogate models _is of great practical significance.

The concept of Physics-Informed Neural Networks (PINNs) introduced in [1], and later revisited in [2], is one typical example. PINNs are neural networks trained to solve supervised learning tasks while respecting some given physical laws, described by a (possibly nonlinear) PDE system. PINNs can be seen as a continuous approximation of the solution to the PDE. They seamlessly integrate information from both data and PDEs by embedding the PDEs into the loss function of a neural network. Automatic differentiation is then used to actually differentiate the network and compute the loss function.

This interniship is a preliminary to the PhD project described in the offer 2023-06769.

**Mission confiée**:
Recently, in [3], multilevel distributed (domain decomposition-based) PINNs architectures have been proposed to address the above-mentioned issues. In this internship, we propose to conduct a more detailed study of such multilevel distributed PINNs for frequency-domain acoustic wave propagation problems, considering various scenarios of boundary conditions, source field and heterogeneity of the propagation medium.

[1] I.E. Lagaris, A. Likas and D.I. Fotiadis. Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans. Neur. Netw., Vol. 9, No. 5, pp. 987-1000 (1998)

[2] M. Raissi, P. Perdikaris and G.E. Karniadakis. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comp. Phys., Vol. 378, pp. 686-707 (2019)

**Principales activités**:

- Bibliographical study for a review of (1) physics-based DNNs for wave propagation type models and (2) strategies for designing multilevel and distributed physics-based DNNs.
- Study in 2d case by considering wave propgation modeled by a Helmholtz-type PDE
- Software development activities
- Numerical assessment of the proposed NN-based physics-based multilevel surrogate models
- Publications

**Compétences**:
Technical skills and level required
- Sound knowledge of numerical analysis for PDEs
- Sound knowledge of Machine Learning / Deep Learning with Artificial Neural Networks
- Basic knowledge of PDE models of wave propagation

Software development skills : Python programming, TensorFlow, Pytorch

Relational skills : team worker (verbal communication, active listening, motivation and commitment)

Other valued appreciated : good level of spoken and written english

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