Computational Partial Differential Equations

Partial Differential Equations (PDEs) are a powerful mathematical tool for describing the fundamental laws of nature. They play a crucial role in understanding countless phenomena, including key modern research areas such as the dynamics of machine learning algorithms or the behavior of quantum computers.

However, solving PDEs exactly is often impossible, so we must rely on computer-based approximations. Ensuring these computations are both accurate and applicable to real-world problems is the core focus of the research group. This, in turn, requires to bring together many other mathematical topics such as functional analysis, approximation theory, geometry, and probability.

Another important part of our research addresses how data can enhance the explanatory power of PDEs. We work on developing a unified mathematical and algorithmic framework that combines data with physical models in an optimal way. The ultimate goal is to create explainable, data-driven approximations that are more efficient --requiring less data than traditional black-box machine learning methods-- while leveraging the deep insights provided by physics.

Some selected topics in which we have worked

(In contruction)