Post-Doctoral Research Visit F/M Inverse problems for nonlinear conservation laws and applications to cell physiology

INRIA | Saclay, France

Classification: Partial Differential Equations

This postdoctoral offer, co-supervized by Romain Yvinec (INRAE Tours / Inria Saclay) and Mauricio Sepúlveda (Universidad de Concepción), is within the framework of the associate team ANACONDA, whose focus is on the theoretical and numerical ANAlysis of CONservation laws for multicellular DynAmics. The consortium gather experts in mathematical biology, partial differental equations of conservation law type, and artificial intelligence, in order to bring new results in cell physiology. The recruited person will be in close connection with researchers of the ANACONDA associate team, and will spend part of her/his time in Chile. The postdoctoral fellow will develop innovative inverse problems strategies, and apply them in a synergistic way to various cell biology processes studied in the team, including ovarian folliculogenesis maturation within the female reproductive system, cell-size dynamics in the adipocyte system, host-microbiota dialog at the intestinal crypt level. These applications share in common equations of conservation laws with non-local nonlinear terms whose inference with snapshot type data is challenging. The research program includes (i) the study of the well-posedness of nonlinear non-local conservation laws developed in the team, and their associated adjoint formulations; (ii) the design of efficient numerical schemes for the direct and adjoint formulations; (iii) the resolution of the inverse problem on several test and application cases developed in the team. The latter will be based on two different approaches. A first one will consider optimization of a cost function thanks to gradient-based strategies derived from the adjoint formulation. This is a standard yet powerful approach, which has to be developed on a case by case study [1-4]. The second strategy will explore innovative physics-informed deep learning approaches, which take advantage of the underlying PDE system or its numerical scheme to construct the architecture of the neural networks that will learn model parameters [5-6]. With the help of the researchers in MUSCA, the post-doctoral fellow will finally tackle the identifiability issues and parameter value interpretation, in order to draw conclusions and predictions in the different physiological applications. D. Givoli, Dan, A tutorial on the adjoint method for inverse problems. Comput. Methods Appl. Mech. Engrg. 380 (2021),113810, 23 pp. A. Coronel, R. Lagos, P. Mulet, M. Sepúlveda, A numerical method for an inverse problem arising in two-phase fluid flow transport through a homogeneous porous medium. Numerical mathematics and advanced applications-ENUMATH, 2017, 615-623, Lect. Notes Comput. Sci. Eng., 126, Springer, Cham, 2019. R. Bürger, A. Coronel, M. Sepúlveda, Numerical solution of an inverse problem for a scalar conservation law modelling sedimentation. Hyperbolic problems: theory, numerics and applications, 445-454, Proc. Sympos. Appl. Math., 67, Part 2, Amer. Math. Soc., Providence, RI, 2009. A. Coronel, F. Huancas, M. Sepúlveda, Identification of space distributed coefficients in an indirectly transmitted diseases model. Inverse Problems 35 (2019), no. 11, 115001, 20 pp. Z. Cai, J. Chen, M. Liu, Least-squares neural network (LSNN) method for scalar nonlinear hyperbolic conservation laws: Discrete divergence operator, Journal of Comp. Appl. Maths., 2023, in press Z. Chen, A. Gelb, Y. Lee, Designing Neural Networks for Hyperbolic Conservation Laws, arXiv:2211.14375 2022

Last updated: 27 May 2023

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