Discovering self-similar blow-up solutions using physics-informed neural networks

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 (blow-ups) in finite time. This long-standing open problem is closely related to one of the seven Millennium Prize Problems which considers the Navier-Stokes equations, the viscous analogue to the Euler equations. In this talk, we present a novel numerical approach utilizing physics-informed neural networks (PINNs), that enables the discovery of self-similar blow-up solutions to various fluid equations, ranging from 1-D Burgers' equation to the 3-D Euler equations with a cylindrical boundary. Moreover, we introduce multi-stage neural networks that achieve machine precision accuracy for predicting blow-up solutions, forming the basis for rigorous computer-assisted proofs of them. This breakthrough sheds new light to the century-old mystery of capital importance in the field of mathematical fluid dynamics.