Jihun Han – Bridging PDEs and machine learning

Abstract: This talk consists of two main parts. In the first part, I will discuss a numerical method for solving PDEs based on a stochastic representation of the solution. This approach captures the underlying particle dynamics associated with the physical processes described by the PDE. By aggregating information from the particles’ collective exploration, the method iteratively reinforces the approximation toward the solution. I will cover its analysis regarding the trainability and highlight its effectiveness across a broad class of problems, including elliptic equations with interfaces, multiscale structures, and perforated domains, as well as hyperbolic-type problems such as the Eikonal and Burgers equations.
In the second part, I will present a method for learning in-between imagery dynamics. This approach integrates PDE models within latent spaces to enhance both learning capability and interpretability. Notably, this method demonstrates robustness in capturing intricate dynamics, such as rotation and outflow, which pose significant challenges for current state-of-the-art optimal transport methods.