Mathematics & Machine Learning Seminar

In mathematical fluid dynamics, a challenging question persists regarding whether an inviscid incompressible fluid with initially smooth velocity and finite energy, governed by the 3-dimensional Euler equations, can develop singularities within a finite time. Here physics-informed deep learning allows us to numerically find the first smooth asymptotic self-similar blow-up profile of the 3D Euler equations for the Luo-Hou scenario, or equivalently, the self-similar blow-up solution of the 2D Boussinesq equation. This is the first use of deep learning to identify a self-similar blowup solution. Furthermore, our method numerically found the first example of an unstable self-similar solution to the Cordoba-Cordoba-Fontelos equation. This shows exciting promises to use this method to discover unstable solutions for other fluid equations.