Abstract
This talk addresses numerical methods for gradient flows in Hilbert spaces based on neural network approximations. The central idea is to represent the solution on a neural network manifold and evolve its parameters over time. At first glance, this approach appears general, elegant, and easy to implement, and it has achieved notable empirical success in machine learning and scientific computing for PDEs. A closer look, however, reveals significant challenges. Developing a proper functional framework that ensures the existence of solutions and rigorously connects to practical algorithms raises subtle issues. In this talk, I will present a framework to address these challenges, and show why they are not merely technical obstacles, but rather reflect fundamental aspects of neural approximation.
