Department of Geosciences, Princeton University, Princeton, New Jersey 08544, USA.
Department of Mathematics, Brown University, Kassar House, 151 Thayer Street, Providence, Rhode Island 02912, USA.
Phys Rev Lett. 2023 Jun 16;130(24):244002. doi: 10.1103/PhysRevLett.130.244002.
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.
二维 Boussinesq 方程和三维 Euler 方程是否存在有限时间爆破解，这对流体力学领域具有重要意义。我们开发了一种新的数值框架，使用物理信息神经网络，首次发现了这两个方程的光滑自相似爆破解。该解本身可能成为未来这两个方程爆破证明的计算机辅助基础。此外，我们通过构建 Córdoba-Córdoba-Fontelos 方程第一个不稳定自相似解的例子，证明了物理信息神经网络可以成功应用于寻找流体方程的不稳定自相似解。我们表明，我们的数值框架既稳健又适用于各种其他方程。
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