M2 Internship From individual data to population dynamics: inference in mixed-effects differential models

Internship objective The objective of this internship is to explore the different mathematical and numerical building blocks requiblack for the development of a future inference method combining mixed-effects models, variational inference and PINNs. The work will deliberately focus on simple differential models and simulated data in order to understand the properties and limitations of the different approaches before considering a more general framework. The internship will notably include: • a literature review on PINNs, weak PINNs and inference methods for mixed-effects models governed by differential equations; • the implementation of classical and weak PINNs on one or several simple differential models, starting from existing code; • the investigation of parameter estimation from noisy and sparse observations; • a first extension to a population of individuals exhibiting inter-individual variability; • a comparison of the approaches in terms of accuracy, stability, computational time and sensitivity to hyperparameters; • the identification of the main scientific and numerical challenges to be addressed in order to subsequently develop a complete framework combining PINNs, mixed effects and variational inference. Additional information The internship is intended for an M2 student in mathematics or applied mathematics, with an interest in differential equations, optimisation, statistics and/or machine learning. Python programming skills are mandatory. Previous expertise in neural networks or PINNs is not requiblack. Supervision: Annabelle Collin (PR, Nantes Université) et Quentin Clairon (ISFP, Inria Bordeaux)