Post-Doctoral Research Visit F/M Operator learning for the time-harmonic Maxwell equations

The Inria centre at Université Côte d'Azur includes 42 research teams and 9 support services. The centre's staff (about 500 people) is made up of scientists of different nationalities, engineers, technicians and administrative staff. The teams are mainly located on the university campuses of Sophia Antipolis and Nice as well as Montpellier, in close collaboration with research and higher education laboratories and establishments (Université Côte d'Azur, CNRS, INRAE, INSERM ...), but also with the regiona economic players.

With a presence in the fields of computational neuroscience and biology, data science and modeling, software engineering and certification, as well as collaborative robotics, the Inria Centre at Université Côte d'Azur  is a major player in terms of scientific excellence through its results and collaborations at both European and international levels.

Atlantis is  a joint project-team  between Inria, CNRS and  Université Côte d'Azur, which gathers applied mathematicians and  computational scientists who are collaboratively undertaking  research activities aiming at the design, analysis, development and  application of innovative numerical methods for studying nanoscale light-matter interaction problems. In the recent years, the team has  developed the   DIOGENeS  [https://diogenes.inria.fr/]  software suite, which is organized around several numerical tools for the simulation  of physical problems related to the fields of nanophotonics and nanoplasmonics. In particular, this  software suite implements several high-fidelity fullwave solvers based on high-order Discontinuous  Galerkin  (DG)  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. Moreover, DIOGENeS also includes algorithms and workflows for the inverse design of nanostructures and nanophotonic devices for harvesting and shaping nanoscale light-matter interactions.  The numerical methods currently implemented in DIOGENeS are accurate and flexible but they are also time consuming. For this reason, the team has recently launched a line of research aiming at the design of novel AI-based methods by considering purely data-driven or model-driven modeling approaches.

Scientific Machine Learning (SciML) is a relatively new research field bridging machine learning (ML) and scientific computing. Its aim is the development of new methods to solve several kinds of problems, which can be forward solution of PDEs, identification of parameters, or inverse problems.  The methods that are investigated in this context must be robust, scalable, reliable and interpretable. Two main families of methods can be distinguished. On one hand, methods that approximate the solution function, i.e., the mapping from instances of the function variables to the function values, such as with Physics-Informed Neural Networks (PINNS) and their numerous variants. On the other hand, methods that approximate the solution operator, which are generally classified as Neural Operators (NOs). Each of these two families has advantages and drawbacks when one is willing to consider complex PDE models of realistic physical problems. NOs require data, and when that is limited or not available, they are unable to learn the solution operator faithfully. PINNs do not require data but are prone to failure, especially on multi-scale dynamic systems due to optimization challenges. In this postdpctoral  project, we will focus on NOs in the context of time-harmonic electromagnetics wave propagation in heterogenous domains involving irregularly-shaped geometrical features. The overarching goal will be to design NOs that can efficiently deal with the system of time-harmonic Maxwell equations for the complex-valued electric and magnetic fields with different types of boundary conditions and source terms in two- and three-dimensional settings, and data from unstructured mesh-based FEM (Finite Element Method) simulators.  In addiiton, these NOs shall ultimately be capable of generalization over different geometrical characteristics of scattering structures to serve as fast surrogates in inverse design strategies for finding optimal scatterer shapes driven by a performance objective.

The wok will start by a detailed bibliographical review of existing operator learning methods including DeepONet, FNO (Fourier NO), PINO, etc. by addressing their viability in relation to the physical problems considered in high-frequency electromagnetism. Initial developments and assessment activities will be performed in a two-dimensional setting and considering variaous problems of increasing complexity. Further invesigations in a three-dimensional setting will be realized for the most promising approaches.

This postdoctoral project is proposed in the contexte of the collaborative projet MAXINET between Inria and Thales Research & Technology.

This position requires French or EU citizenship.

• Bibliographical study on operator learning for wave propagation PDE models

• Development of operator learning approaches for the time-harmoinic Maxwell equations in 2D and 3D

• Collaborate with academic and industrial partners 

• Represent the team at workshops, conferences, and dissemination events

• Develop and maintain technical documentation

• Contribute to scientific publications and technical reports

Knowledge and skills:

• Sound knowledge of numerical methods for PDEs, numerical optimization, scientific machine learning
• Strong background and experience with physics-based NNs and Neural Operators for PDEs
• Basic knowledge of modeling and numerics for electromagnetic wave propagation

Software development skills : Python, Pytorch, parallel programming with MPI

Relational skills : team worker (verbal communication, active listening, motivation and commitment).Other valued appreciated : good level of spoken and written english

• 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
• Contribution to mutual insurance (subject to conditions)

Gross Salary: 2788 € per month.