Description du sujet de thèse

The objective of this PhD thesis is to use machine learning and artificial intelligence tools such as constrained deconvolution, sensitivity analysis, and physics-informed neural networks to answer questions concerning the physics of materials ; more specifically, to characterize dielectric materials and simulate charge transport.

Background: Dielectrics constitute a weak point in all electrical insulation systems (power electronics components, energy transmission cables, etc.) in which they are used. These materials, subjected to elec- trical and thermal stresses, can accumulate charges that can lead to a loss of functionality of the system in which they are used. A first approach is based on the development of mathematical models capable of predicting their behavior under constraints, and particularly the spatio-temporal behavior of the net charge density. This type of model [7], for which about fifteen physicochemical parameters characterizing the material under study are necessary, is based on solving a system of partial differential equations (PDEs) composed of the advection-diffusion equation and Gauss-Maxwell equation linking the charge type, den- sity, sign, mobility, and the source term, for each charge type. They involve also the diffusion coefficient, the electric field, the vacuum permittivity, the relative permittivity of the material, and the net charge density.
Experimentally, charge measurement techniques can be used to obtain the net charge density. One of these techniques (the PEA - Pulsed Electro Acoustics method), developed at Laplace, allows to get the distribution of the electric charge by measuring the pressure waves generated by an electrical pulse on the charge using a piezoelectric sensor. The spatio-temporal distribution of the charge is recovered using inverse methods applied to the voltage signal measured by the sensor. It is therefore possible to compare experimental and simulation results regarding charge density.

Challenges: This research area lies at the interface of materials physics, electrical engineering, applied mathematics, statistics, and artificial intelligence (AI). Using powerful optimization techniques, the objec- tive is to improve the identification of model parameters and the processing of PEA data by combining the two approaches in a global model integrating charge transport and simulation of the voltage from the measurement bench. This original but computationally expensive approach will be implemented using innovative mathematical tools :
1. Constrained Deconvolution - The previously developed electroacoustic model of the PEA mea-
surement bench allows us to simulate the response of the measuring cell, as a function of the attenuation, dispersion, and reflection of acoustic waves [8]. The deconvolution of this measured signal then consists in finding the charge density in the volume of the material and on the electrodes to create a charge map and in recalculating the internal field and the applied potential. See [12] for more details on deconvolution. During this step, it is important to take into account the fact that the space charge in the volume of the sample, as a function of time, is the solution of a PDE.
2. Physics-informed Neural Networks (PINNs) - PINNS, introduced in [9, 10, 11] are a type of
universal function approximators that integrate knowledge of the physical laws governing a data set into the learning process and can in particular be described by PDEs. Prior knowledge of general physical laws acts in the training of neural networks as a regularization agent that limits the space of admissible solutions, thus increasing the accuracy and robustness of the function approximation. Their theoretical properties have been studied in particular in [4]. We wish to use PINNs to replace the charge transport model during optimization for parameter search.
3. Learning and Sensitivity Analysis (SA) - Based on the two previous points, we will be able to
create a dataset that will be the basis for learning and SA. The purpose of SA is to determine the model parameters that most influence the output, thus leading to the design of a simplified model, reducing computation time for the optimization process. See [6, 2] for the description and theoretical and numerical properties of the classic methods used in SA.

Detailed description of the thesis topic : To answer the questions previously raised, the candidate will begin by familiarizing with and developing statistical tools on models for which the Laplace Plasma and Energy Conversion Laboratory (Laplace) and Institut Clément Ader (ICA) teams have already explo- red some avenues related to AS [1]. This initial work will be supported by the Institut de Mathématiques de Toulouse (IMT) and the Laboratoire J.A. Dieudonné (LJAD) in Nice teams, which will propose im- plementing innovative AS techniques for the numerical simulation of appropriate indices, such as the "Pick-Freeze" [6], Chatterjee rank [5], and kernel [3] methods. The core of the thesis topic will consist of developing AI tools, such as the PINNs mentioned above. The AI expertise of the IMT/LJAD members, combined with that of the Laplace/ICA engineering team, will be essential for solving the specific problem of dielectric materials.

References:
[1] F. Baudoin, S. Le Roy, G. Teyssedre, C. Laurent, I. Alhossen, F. Bugarin, S. Segonds, and N. Bi- naud. Parameters sensitivity analysis in charge transport model using sobol indexes for optimization purpose. In 2016 IEEE International Conference on Dielectrics (ICD), volume 2, pages 832–835, 2016.
[2] S. Da Veiga, F. Gamboa, B. Iooss, and C. Prieur. Basics and Trends in Sensitivity Analysis : Theory and Practice in R. SIAM, 2021.
[3] S. Da Veiga, F. Gamboa, A. Lagnoux, T. Klein, and C. Prieur. Efficient estimation of Sobol' indices of any order froma single input/output sample. working paper or preprint, Oct. 2024.
[4] N. Doumèche, G. Biau, and C. Boyer. Convergence and error analysis of PINNs. arXiv preprint arXiv :2305.01240, 2023.
[5] F. Gamboa, P. Gremaud, T. Klein, and A. Lagnoux. Global sensitivity analysis : A novel generation of mighty estimators based on rank statistics. Bernoulli, 28(4) :2345–2374, 2022.
[6] F. Gamboa, A. Janon, T. Klein, A. Lagnoux, and C. Prieur. Statistical inference for Sobol Pick-Freeze Monte Carlo method. Statistics, 50(4) :881–902, 2016.
[7] S. Le Roy, F. Baudoin, C. Laurent, and G. Teyssèdre. Analysis of current–voltage characteristics in insulating polymers using a bipolar charge transport model. IEEE Transactions on Dielectrics and Electrical Insulation, 29(6) :2101–2109, 2022.
[8] A. Pujol, L. Berquez, F. Baudoin, and D. Payan. PSpice modeling of the pulsed electroacoustic method for dispersive polymer sample application. Review of Scientific Instruments, 91(10) :105112, 10 2020.
[9] M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics informed deep learning (part i) : Data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv :1711.10561, 2017.
[10] M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics informed deep learning (part ii) : Data-driven discovery of nonlinear partial differential equations. arxiv 2017. arXiv preprint arXiv :1711.10566, 2017.
[11] 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. Journal of Computational physics, 378 :686–707, 2019.
[12] W. Ren, J. Zhang, L. Ma, J. Pan, X. Cao, W. Zuo, W. Liu, and M.-H. Yang. Deep non-blind deconvolution via generalized low-rank approximation. Advances in neural information processing systems, 31, 2018.

Contexte de travail

The thesis will be conducted at the Toulouse Institute of Mathematics and will be co-supervised by Agnès Lagnoux (Institut de Mathématiques de Toulouse) et Fulbert Baudoin (Laboratoire plasma et conversion d'énergie - Laplace). The work will be carried out in collaboration between two teams, IMT/LJAD and Laplace/ICA.
The recruited person will also interact with the other members of the project : Laurent Berquez (Laplace), Florian Burgarin (Institut Clément Ader - ICA), Reda Chhaibi (Laboratoire J.A. Dieudonné - LJAD, Nice), Séverine Le Roy (Laplace), Clément Pellegrini (IMT), and Stéphane Segonds (ICA).
The thesis will be funded by the MITI 80 Prime - DISCO project for a duration of 3 years starting in September or October.

Contraintes et risques

The thesis will be conducted at the Toulouse Institute of Mathematics. Several trips to LJAD in Nice are planned as part of the project, including short and long stays. The work will be carried out in collaboration between two teams, IMT/LJAD and Laplace/ICA.

Informations complémentaires

The doctoral student will ideally have a Master's degree in probability/statistics and de- monstrate a strong interest in rigorous mathematics and its applications in physics. Computer skills will be a major asset.
Candidates must provide the following documents:
-a CV including transcripts of M1 and M2
-a cover letter (maximum 1 page) setting out their professional goals, their scientific interests, including the name and contact details of a reference person who can provide a recommendation.