Approximation of PDE solutions with nonlinear parametric models: Application to high-dimensional advection-diffusion equations

Classically, partial differential equations (PDE) solutions are approximated using linear parametric models. The objective of this talk is to give an overview of recent approximation methods using nonlinear parametric models (namely, neural networks). We start by recalling classical methods, and comparing them to neural methods, on a simple case: for the approximation of a given function. Then, we move on to PDE solutions, where classical methods (space-time and semi-discrete Galerkin) are recalled, and contrasted with their neural counterparts (physics-informed neural networks and the neural Galerkin methods). Finally, in the specific case of advection-diffusion equations, we introduce a novel neural variant of the semi-Lagrangian method.