My interests are varied and essentially cover the study of phenomena from physics or biology, and their applications via the use of methods derived from optimization (variational methods) or nonlinear analysis.

The latter, when studied on a sufficiently large scale (mesoscopic or macroscopic), are often modeled by partial differential equations, the structure of which can vary greatly depending on the nature of the problem studied. Taking the flow of an incompressible fluid as an example, depending on its viscosity, it can be described by the Euler equations (viscosity-free case) or the Navier-Stokes equations (some viscous cases), but also by equations with a non-linear term relating to a particular viscosity, known as non-Newtonian.
Parabolic, hyperbolic and elliptic partial differential equations, as well as coupled systems, can all be found in this framework, offering a wide variety of problems to study.

Working with Dimitri COBB

Variational inequality solutions and finite stopping time for a class of shear-thinning flows
L Chupin, N Cîndea, G Lacour Annali di Matematica Pura ed Applicata (2024).
Publisher access

RESEARCH

Null controllability of quasilinear parabolic equations with gradient dependent coefficients
N Cindea, G Lacour
arXiv preprint arXiv:2304.08022

Weak Solutions for a non-Newtonian Stokes-Transport System
D Cobb, G Lacour
arXiv preprint arXiv:2401.02599

Publications

Preprints

When it comes to interactions at smaller scales (say, microscopic), random principles naturally come into play. An important aspect here is to justify that from the microscopic modeling we can recover the macroscopic modeling in the form of partial differential equations, which also highlights the full consideration of the physical principles contained in these descriptions. Note that this microscopic framework is natural for many studies, for example to consider interactions in certain dilute gases.

All these methods can be used in Machine Learning problems. For example, the training process for large neural networks can be described in a similar way to a system of interacting particles. When the number of parameters is large, which is often the case, the relevant information is contained in their distribution, i.e. the training process is described by what might be considered the macroscopic behavior associated with the system under consideration. In some cases, it is then possible to return to the analysis of nonlinear optimization problems.