Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks
This addresses a fundamental problem in fluid mechanics for researchers studying turbulence and singularity formation, representing a novel application rather than an incremental improvement.
The researchers tackled the problem of finite-time blow-up solutions in fluid mechanics by developing a physics-informed neural network framework, which discovered a smooth self-similar blow-up profile for the 3-D Euler and 2-D Boussinesq equations for the first time, and also found an unstable self-similar solution to the Córdoba-Córdoba-Fontelos equation.
Whether there exist finite time blow-up solutions for the 2-D Boussinesq and the 3-D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks (PINNs), that discover, for the first time, a smooth self-similar blow-up profile for both equations. The solution itself could form the basis of a future computer-assisted proof of blow-up for both equations. In addition, we demonstrate PINNs could be successfully applied to find unstable self-similar solutions to fluid equations by constructing the first example of an unstable self-similar solution to the Córdoba-Córdoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations.