The team aims at investigating the coupling between machine learning and traditional approaches for the numerical approximation of PDEs, the construction of new reduced models mixing classical approaches (model order reduction mainly and asymptotic models) with machine learning as well as the construction of structure preserving numerical schemes for hyperbolic systems.
The team focuses its efforts on the construction of new numerical methods and models on plasma physics, compressible fluid mechanics, and on the construction of new approaches in optimal control and inverse problem on wave propagation. The team’s effort will converge towards a proof of concept of numerical codes which self specializes according to the simulations performed.
Scientific Machine Learning
SciML is a recent branch of scientific computing that focuses on the coupling between learning and PDEs. The team’s research is entirely within this framework. In particular, the team is working on the study and construction of PINNs and Neural Galerkin methods that preserve the properties of PDEs and are capable of accurately predicting solutions to high-dimensional PDEs (kinetic or parametric PDEs). We investigate also reduce order modeling and hybrid methods.
Numerical methods for hyperbolic and kinetic PDEs
The team works mainly on the construction of numerical methods for hyperbolic and kinetic PDEs. These include the construction of structure- and asymptotic-preserving (AP) schemes, the study of structural methods and the creation of hybrid methods between PDEs and learning. These methods are applied to hyperbolic systems of equations such as Euler equations, Magneto-Hydrodynamics, shallow water equations, multi-phase and multi-physics flows. We also study hybrid (grid and particles based) and pure ML numerical methods for high dimensional kinetic equations like Vlasov or radiative transfer.
Reduced models for plasma
Plasma physics is very simulation-intensive. It is therefore important to offer reduced models for these equations. The team is studying approaches based on dimension reduction methods, new closures for moment models and neural operators. The PDEs used in plasma physics have important geometric properties, essential for long time simulations that we want to preserve in the reduced models.
Direct and inverse problems for acoustic/electromagnetic waves
The team is also interested in wave propagation in heterogeneous media, with possible applications in plasma or acoustics. We are working on direct approaches based on PINNs or methods for inverse problems based on neural operators or parsimonious methods in signal processing. The latter are used in particular for applications in room acoustics.
Optimal control
The team works also on numerical methods for optimal control mixing classical approaches and machine learning methods to increase its performance. We also look at the interaction with reinforcement learning. Applications of interest contain quantum computing, ferromagnetism and shape optimization.
