Self-similar blow-up profile for the Boussinesq equations via a physics-informed neural network

We develop a new numerical framework, employing physics-informed neural networks, to find a smooth self-similar solution for the Boussinesq equations. The solution in addition corresponds to an asymptotic self-similar profile for the 3-dimensional Euler equations in the presence of a cylindrical boundary. In particular, the solution represents a precise description of the Luo-Hou blow-up scenario [G. Luo, T. Hou, Proc. Natl. Acad. Sci. 111(36): 12968-12973, 2014] for 3-dimensional Euler. To the best of the authors' knowledge, the solution is the first truly multi-dimensional smooth backwards self-similar profile found for an equation from fluid mechanics. The new numerical framework is shown to be both robust and readily adaptable to other equations.