Neural networks and PDEs: from physics-informed learning to unstable solutions

In this talk, I will attempt to survey the recent applications of neural networks to PDEs with a focus on two approaches. I will start by introducing physics-informed neural networks and how they have been applied to discovering unstable singular solutions to fluid equations. Then, I will finish by discussing neural operators that are trained to learn solution maps of PDEs and how this lead to the creation of Poseidon, a foundation model for numerical PDE solving.