Post-doctoral Research Visit F/m Operator Learning for The Time-harmonic Maxwell Equations

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

**Type de contrat**: CDD

**Contrat renouvelable**: Oui

**Niveau de diplôme exigé**: Thèse ou équivalent

**Fonction**: Post-Doctorant

**Niveau d'expérience souhaité**: De 3 à 5 ans

**A propos du centre ou de la direction fonctionnelle**:
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.

**Contexte et atouts du poste**:

- 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.

**Mission confiée**:
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 position requires French or EU citizenship**.

**Principales activités**:

- 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

**Compétences**:
Knowledge and skills:

- Sound knowledge of numerical methods for PDEs, numerical optimization, scient