Recent Progress in Physics-Informed Neural Networks for Self-Similar Blow-Up Solutions
Date:
One of the most challenging open questions in mathematical fluid dynamics is whether an inviscid, incompressible fluid, described by the 3-dimensional Euler equations, with initially smooth velocity and finite energy, can develop singularities in finite time. This long-standing problem is closely related to one of the seven Millennium Prize Problems, which considers the Navier-Stokes equations, the viscous counterpart to the Euler equations. In this talk, I will describe several advanced techniques associated with physics-informed neural networks (PINNs) for finding smooth self-similar blow-up solutions in various fluid equations, ranging from the 1-D CCF equation to the 3-D axisymmetric Euler equations in the presence of a cylindrical boundary. By integrating these techniques, we are able to find blow-up solutions with machine precision. This work sheds new light on a century-old mystery of fundamental importance in the field of mathematical fluid dynamics.