Date: September 21, 23, 25, 28, 2026
Event: Mini Course
Organizer: FAU and UFF, CAPES/DAAD PROBRAL.
Mini-Course: Computational PDEs and Scientific Machine Learning
Speaker: Daniel Fernández Martínez, PhD student at FAU DCN-AvH, Chair for Dynamics, Control, Machine Learning and Numerics – Alexander von Humboldt Professorship. Friedrich-Alexander-Universität Erlangen-Nürnberg
Abstract. This graduate course explores classical and machine learning approaches to the numerical solution of partial differential equations (PDEs), combining mathematical foundations with practical implementation.
The first part develops the basic theory of the finite element method (FEM), emphasizing weak and variational formulations and their finite-dimensional discretization. We then translate these concepts into computational PDE solvers using FEniCS, establishing a foundation for understanding and evaluating computational methods.
The second part introduces the fundamentals of neural networks and their use in approximating PDE solutions, with a focus on the Deep Ritz method and physics-informed neural networks (PINNs). These approaches are examined through their mathematical formulations, training objectives, and numerical behavior. Particular attention is given to their theoretical limitations and practical caveats, especially the coercivity gap and the extent to which minimizing a training loss provides meaningful control of the solution error.
Throughout the course, benchmark PDEs and a selected applied problem provide a common framework for numerical experimentation and comparison between finite element and neural approaches. By connecting variational theory, numerical analysis, and scientific machine learning, the course allows participants to implement PDE solvers and critically assess the accuracy, stability, and reliability of their results.
WHEN
September 21, 23, 25, 28, 2026 at 16:00H -Berlin time (11:00H -local time)
WHERE
On-site
On-site: Room 401. Building H. Campus Gragoatá.
UFF. Brazil
