Geometric methods for differential equations and learning applications

Timetable
November 18th 2024, 13:30-15:30 room G
November 19th 2024, 14:30-16:30 room G
November 20th 2024, 13:30-15:30 room F

Abstract
Differential equations are one of the main tools for describing physical phenomena and play a crucial role in many disciplines, including engineering, physics, economics, and biology. The exact solution to these equations is generally impossible to find; hence, numerical approximations are needed. Among these methods, one might distinguish those aiming not only to get a quantitatively accurate approximation of the solution but also a qualitatively accurate reproduced behaviour. These methods are generally called geometric or structure-preserving. In this mini-course, we present a selection of structure-preserving numerical methods for Ordinary Differential Equations. We will start with Symplectic numerical methods for Hamiltonian systems, showing their improved performance when considering long-time simulations of these systems. We will then analyse a few methodologies to preserve an energy function characterising the ODE. We will consider the more straightforward cases of linear and quadratic energy functions and more general ones. As a third class of methods, we will consider those reproducing the non-expansive nature of some ordinary differential equations, such as negative gradient flows of convex potentials. The course is both theoretical and practical. After theoretically introducing the numerical schemes, we will demonstrate their effectiveness with Python simulations, highlighting their ability to preserve a specific structure and their benefits over general-purpose numerical solvers.

Background
The student should be familiar with the basic notions of linear algebra and numerical methods for ordinary differential equations, like Runge-Kutta methods. Knowledge of differential geometry and experience with Python would help, but it is not necessary.

Reference material
[1] Hairer, E., Lubich, C., & Wanner, G. (2011). Geometric numerical integration. Geometric Numerical Integration.

[2] Dahlquist, G. (1979). Generalised disks of contractivity for explicit and implicit Runge-Kutta methods (No. TRITA-NA-79-06). CM-P00069451.

[3] McLachlan, R. I., Quispel, G. R. W., & Robidoux, N. (1999). Geometric integration using discrete gradients. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 357(1754), 1021-1045.