论文标题
使用神经网络的三维轴对称Euler方程的渐近自相似爆炸轮廓
Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks
论文作者
论文摘要
2-D Boussinesq和3-D Euler方程是否存在有限的时间爆破解决方案对流体力学领域至关重要。我们开发了一个新的数值框架，该框架采用了物理知识的神经网络（PINN），该框架首次发现了两个方程式的平滑自相似爆炸配置文件。该解决方案本身可以构成两个方程式未来计算机辅助证明的基础。此外，我们可以通过构建对Córdoba-Córdoba-fontelos方程的不稳定自相似解的第一个示例来证明PINN可以成功应用于流体方程的不稳定自相似解。我们表明，我们的数值框架既健身又适合各种其他方程。
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.
