Asymptotic Self-Similar Blow-Up Profile for Three-Dimensional Axisymmetric Euler Equations Using Neural Networks
Asymptotic Self-Similar Blow-Up Profile for Three-Dimensional Axisymmetric Euler Equations Using Neural Networks
复制标题
使用神经网络的三维轴对称欧拉方程的渐近自相似放大剖面
DOI：
10.1103/physrevlett.130.244002
复制
发表时间：
2023
影响因子：
8.6
通讯作者：
Buckmaster, T.
中科院分区：
文献类型：
--
作者：
Wang, Y.;Lai, C.-Y.;Gómez-Serrano, J.;Buckmaster, T.
二维Boussinesq方程和三维Euler方程是否存在有限时间爆破解是流体力学领域的一个重要问题。我们开发了一个新的数值框架，采用物理信息的神经网络，发现，第一次，一个光滑的自相似爆破两个方程。这个解本身可以成为未来计算机辅助证明这两个方程爆破的基础。此外，我们证明了物理信息神经网络可以成功地应用于通过构建Córdoba-Córdoba-Fontelos方程的不稳定自相似解的第一个例子来找到流体方程的不稳定自相似解。我们表明，我们的数值框架是强大的，适应各种其他方程。
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.