A propos du centre ou de la direction fonctionnelle

The Inria Saclay-Île-de-France Research Centre was established in 2008. It has developed as part of the Saclay site in partnership with Paris-Saclay University and with the Institut Polytechnique de Paris .

The centre has 40 project teams , 32 of which operate jointly with Paris-Saclay University and the Institut Polytechnique de Paris; Its activities occupy over 600 people, scientists and research and innovation support staff, including 44 different nationalities.

Contexte et atouts du poste

The research activity of the Inria href{https://uma.ensta-paris.fr/idefix/}{IDEFIX} team
is dedicated to the design, analysis and implementation of efficient numerical methods
for solving application-oriented inverse problems related to partial differential equations.
Practical applications include non-destructive testing, X-ray, electromagnetic (radar) and
ultrasound imaging, biomedical modeling and imaging, acoustics and sound modeling,
spatial audio simulation, invisibility, and metamaterial design.
A generic problem would consist in determining the geometry (with unknown topology)
or physical properties of unknown targets from indirect measurements.
In general, this type of problem is non-linear and severely ill-posed, thus requiring special
attention from the point of view of regularization
and non-trivial adaptations of classical optimization methods.
We are particularly interested in developing fast, data-driven methods suitable
for real-time applications or large-scale problems.
Given the complexity of these problems, there is an urgent call for the development
of innovative techniques capable of stabilizing or optimizing
the performance of current algorithms. Consequently, our overarching goal is to
delve into the capabilities of neural networks (NNs) as a means of effectively
tackling these pressing issues.

Mission confiée

Employing simple NN models such as fully connected or convolutional NNs for the complete
inverse scattering problem comes with certain limitations, including the requirement
for a significant amount of training data and the vulnerability to the inherent severe
ill-posed nature of the inverse problem, particularly when contrasted with the traditional
uses of NNs in imaging. Although initial investigations have shown the potential for
constructing a robust neural network on synthetic datasets to address the geometric inverse
problem within the constraints of simple geometry parameterizations,
we intend to take a different route. First, we would like to incorporate NNs into conventional
methods for handling more intricate geometries.
Second, we want to investigate more advanced NN architectures.

Principales activités

One of our primary algorithmic tool in this context is the Linear Sampling Method
(LSM) cite{colton1996}. A significant challenge associated with the LSM is the selection
of an appropriate regularization parameter, which has to be carefully chosen
for each sampling point. The efficient determination of this parameter is crucial
for enhancing efficiency, particularly in real-time imaging applications.
To address this issue, we are investigating the application of NNs for the automatic selection
of regularization parameters and strategies within the LSM.
Additionally, we are also actively working on iterative methods based on
forward solvers for the direct and adjoint problems. Hence, an approach we are considering
as well is the integration of NNs into the regularization of these iterative techniques. Next, we aim to explore more advanced architectures. Several approaches, including
so-called physics-informed neural networks, have successfully reduced the
need for extensive training data by incorporating physical principles. Other popular methods,
such as neural operators, utilize fully connected neural networks
with additional elements, making them particularly efficient for solving PDEs and inverse problems.
For inverse scattering problems, a promising research direction would be to add information
related to the operators into the loss function.
We propose incorporating information from the LSM or the generalized LSM
into the loss function. The latter seems to be a natural candidate as its formulation involves
minimizing a least-squares misfit functional with a carefully chosen penalty term.
This approach offers a complementary perspective to our earlier work, where we integrated neural
networks into traditional methods. Additionally, we aim to examine the potential of Kolmogorov--Arnold networks
as substitutes for fully connected NNs in these architectures.
A recent study has showcased their efficacy in solving PDEs,
though their potential for addressing inverse problems remains ambiguous.
There is also considerable scope for enhancing the representation of activation functions.
Initially, splines were proposed, followed by the suggestion of orthogonal
polynomials. An intriguing avenue for improvement lies in the exploration of
rational functions.

Compétences

Technical skills and level required :

Languages :

Relational skills :

Other valued appreciated :

Avantages

• 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 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

Rémunération

1st et 2nd year : 2 100 euros brut /mois

3st year : 2 190 euros brut / mois