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 open problem is closely related to one of the seven Millennium Prize Problems which considers the problem the viscous analogue to the Euler equations (the Navier-Stokes equations). In this talk, I will describe how we leverage the power of deep learning, using deep neural networks with equation constraints, namely physics-informed neural networks (PINNs), to find a smooth self-similar blow-up solution for the 3-dimensional Euler equations in the presence of a cylindrical boundary. To the best of our knowledge, the solution represents the first example of a truly 2-D or higher dimensional backwards self-similar solution. This new numerical framework based on PINNs is shown to be robust and readily adaptable to other fluid equations, which sheds new light to the century-old mystery of capital importance in the field of mathematical fluid dynamics.

Based on the paper

Yongji Wang, Ching-Yao Lai, Javier Gomez-Serrano, Tristan Buckmaster, Asymptotic self-similar blow up profile for 3-D Euler via physics-informed neural networks