Asymptotic Self-Similar Blow-Up Profile for Three-Dimensional Axisymmetric Euler Equations Using Neural Networks

Whether there exist finite-time blow-up solutions for the 2D Boussinesq and the 3D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks, 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 physics-informed neural networks 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.

Figure 1

Solution for Eq. (3) found by PINN. $f_i$ indicate the residues, which are of 5 orders of magnitude smaller than the solution. The inferred value of $\lambda$ is $1.917\pm 0.002$ after systematic test (see Supplemental Material [14]).

Figure 2

(a) Spatial distribution of the collocation points. Here ($z_1$, $z_2$) are the rescaled coordinates: $y_1=\sinh (z_1)$ and $y_2=\sinh (z_2)$. 10 000 collocation points in total are used for training. (b) Decrease of the total loss over the training iterations. The inset shows the loss curve in a log-log scale.

Figure 3

Comparison between stable and unstable solutions with associated $\lambda$ for CCF equation.