SHORT Course: Nonlinear approximations to high-dimensional evolution equations

Tuesday 24 Mar 2026, 09:00 → Friday 27 Mar 2026, 17:00 Europe/Rome

GSSI

Lecturer
Christian Lubich

Timetable and workload
Lectures: 8 hours

Course content
This series of lectures is about approximating the solution to initial value problems of potentially high-dimensional evolutionary partial differential equations such as  Schroedinger or Vlasov or Fokker-Planck equations. In the past decades, nonlinear parametric approximations have increasingly been used in the corresponding application areas in physics and chemistry. These nonlinear parametrizations include

- Gaussians, where parameters are given by position, momentum, width matrix and phase;
- tensor networks, where parameters are given by basis matrices and low-order connecting tensors; and
- neural networks, where parameters are given by weight matrices and bias vectors.

The Dirac-Frenkel time-dependent variational principle - or equivalently, defect minimization - is basic in providing a model reduction on the fly via evolution equations for the time-dependent parameters. These equations are, however, typically ill-posed and cannot be integrated numerically by standard time-marching methods. The lectures will present numerical approaches, developed in the past few years, which successfully address this ill-posedness problem in different ways, be it in a general setting or depending on the particular type of nonlinear parametrization.