Fluid Singularities, Unstable PDE Solutions and Computer-assisted proofs

Abstract: This talk presents recent work on understanding certain solutions of PDE by combining modern mathematics with classical analysis. Machine learning, particularly Physics-Informed Neural Networks (PINNs), is being applied to discover new solutions to nonlinear PDEs with high accuracy. A key aspect is the interplay between these methods to uncover the full spectrum of solutions, particularly, unstable solutions that are challenging to find otherwise.

We will also demonstrate how computer-assisted methods can transform these numerical discoveries into rigorous mathematical proofs. While motivated by fluid mechanics equations such as Euler and Navier-Stokes, the methods discussed have broader applicability to other PDEs.