Physics-Informed Neural Network for electromagnetic wave propagation

January 08, 2023
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2022-05611 - Physics-Informed Neural Network for electromagnetic wave propagation

Contract type : Internship

Level of qualifications required : Graduate degree or equivalent

Fonction : Internship Research

Level of experience : Recently graduated

About the research centre or Inria department

The Inria Université Côte d'Azur center counts 36 research teams as well as 7 support departments. The center's staff (about 500 people including 320 Inria employees) is made up of scientists of different nationalities (250 foreigners of 50 nationalities), engineers, technicians and administrative staff. 1/3 of the staff are civil servants, the others are contractual agents. The majority of the center's research teams are located in Sophia Antipolis and Nice in the Alpes-Maritimes. Four teams are based in Montpellier and two teams are hosted in Bologna in Italy and Athens. The Center is a founding member of Université Côte d'Azur and partner of the I-site MUSE supported by the University of Montpellier.


Numerical simulations of electromagnetic wave propagation problems primarily rely on spatially and temporally discretization of the system of time-domain Maxwell equations using finite difference or finite element type methods. For complex and realistic three-dimensional situations, such a process can be computationally prohibitive, especially in view of many-query analyses (e.g., optimization design and uncertainty quantification). Therefore, developing cost-effective surrogate models is of great practical significance. Among the different possible approaches for building a surrogate model of a given PDE system in a non-intrusive way (i.e., with minimal modifications to an existing discretization-based simulation methodology), approaches based on neural networks and Deep Learning (DL) has recently shown new promises due to their capability of handling nonlinear or/and high dimensional problems. In the present study, we propose to focus on the particular case of Physics- Informed Neural Networks (PINNs) introduced in 1. PINNs are neural networks trained to solve supervised learning tasks while respecting any given laws of physics described by a general (possibly nonlinear) PDE system. They seamlessly integrate the information from both the measurements and partial differential equations (PDEs) by embedding the PDEs into the loss function of a neural network using automatic differentiation. Such PINNs have for instance been studied for inverse problems in nano-optics and metamaterials in 2.

At Inria, the Atlantis project-team 3 gathers applied mathematicians and computational scientists who are undertaking research activities aiming at the design, analysis, development and application of innovative numerical methods for systems of partial differential equations (PDEs) modeling nanoscale light-matter interaction problems. In this context, the team develops the DIOGENeS (DiscOntinuous GalErkin Nano Solvers) software suite 4. which implements several Discontinuous Galerkin (DG) type methods tailored to the systems of time- and frequency-domain Maxwell equations possibly coupled to differential equations modeling the behavior of propagation media at optical frequencies. DIOGENeS is a unique numerical framework leveraging the capabilities of DG techniques for the simulation of multiscale problems relevant to nanophotonics and nanoplasmonics.

The work carried out in this internship project will give first insights of the effectiveness of PINNs in the considered physical context, in view of a further study, which will aim at designing and developing a new component of the DIOGENeS software package dedicated to AI-based reduced-order modeling.

1 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)

2 Y. Chen, L. Lu, G.E. Karniadakis and L. Dal Negro. Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Opt. Expr., Vol. 28, No. 8, pp. 11618-11633 (2020)

3 https://

4 https: //


In this internship, we propose to conduct a study of the applicability of PINNs for building efficient surrogate models of the system Maxwell equations in 2D (two-dimensional case). This internship will proceed in two steps: first a detailed bibliographical review on PINNs and related physics- constrained neural networks approaches will be conducted and documented, with a focus on recent works for electromagnetic and other wave propagation models; then, PINNs will be formulated and developed for several configurations of interest to optical wave modeling. A particular attention will be given to the performance of the training phase. More precisely, we want to study in detail the influence of the network configuration (number of layers and neurons in each layer, activation function, etc.) as well as of the definition of the loss function on the behavior of the training phase. Finally, we will study possible routes for improving the performance of the training phase and the accuracy of the resulting surrogate model such as adaptive sampling of the training points, locally adaptive activation function, etc.).


This position is intended for master students in the field of applied mathematics and scientific conmputing with a strong background in numerical computing and related subjects. to AI.

Required Skills

  • Good knowledge of one or two DL frameworks (tensorflow, pytorch)
  • Good knowledge of Linear Algebra and one or more libraries (Eigen, blas/lapack, xtensor, ...)
  • Good knowledge of Python and associated Numpy-type libraries
  • Benefits package
  • Subsidized meals
  • Partial reimbursement of public transport costs
  • Leave: 7 weeks of annual leave + 10 extra days off due to RTT (statutory reduction in working hours) + possibility of exceptional leave (sick children, moving home, etc.)
  • Possibility of teleworking (after 6 months of employment) and flexible organization of working hours
  • Professional equipment available (videoconferencing, loan of computer equipment, etc.)
  • Social, cultural and sports events and activities
  • Access to vocational training
  • Social security coverage
  • General Information
  • Theme/Domain : Optimization, machine learning and statistical methods Scientific computing (BAP E)

  • Town/city : Sophia Antipolis

  • Inria Center : Centre Inria d'Université Côte d'Azur
  • Starting date : 2023-03-01
  • Duration of contract : 6 months
  • Deadline to apply : 2023-01-08
  • Contacts
  • Inria Team : ATLANTIS
  • Recruiter : Lanteri Stéphane /
  • The keys to success

    With a scientific background, you have an appetite for scientific fields and digital sciences, with a first experience in the fields of scientific computing and AI. Motivation, curiosity and the desire to learn is the main quality expected. Your ability to work in a team, to be proactive and to motivate a team will also be very important.

    About Inria

    Inria is the French national research institute dedicated to digital science and technology. It employs 2,600 people. Its 200 agile project teams, generally run jointly with academic partners, include more than 3,500 scientists and engineers working to meet the challenges of digital technology, often at the interface with other disciplines. The Institute also employs numerous talents in over forty different professions. 900 research support staff contribute to the preparation and development of scientific and entrepreneurial projects that have a worldwide impact.

    Instruction to apply

    Defence Security : This position is likely to be situated in a restricted area (ZRR), as defined in Decree No. 2011-1425 relating to the protection of national scientific and technical potential (PPST).Authorisation to enter an area is granted by the director of the unit, following a favourable Ministerial decision, as defined in the decree of 3 July 2012 relating to the PPST. An unfavourable Ministerial decision in respect of a position situated in a ZRR would result in the cancellation of the appointment.

    Recruitment Policy : As part of its diversity policy, all Inria positions are accessible to people with disabilities.

    Warning : you must enter your e-mail address in order to save your application to Inria. Applications must be submitted online on the Inria website. Processing of applications sent from other channels is not guaranteed.

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