使用神经网络的三维轴对称欧勒方程的非对称自相似膨胀配置文件
Y Wang1, C-Y Lai1, J Gómez-Serrano2,3,4
1Department of Geosciences, Princeton University, Princeton, New Jersey 08544, USA.
Physical review letters
|June 30, 2023
概括
基于物理学的神经网络发现了2D Boussinesq和3D Euler方程的光滑自相似膨胀解决方案. 这一突破为计算机辅助的在流体动力学中的膨胀证明提供了一条道路.
科学领域:
- 流体动力学 流体动力学
- 计算数学 计算数学 计算数学
- 应用物理 应用物理
背景情况:
- 对于2D Boussinesq和3D Euler方程的有限时间膨胀解决方案的存在仍然是流体力学中一个关键的未解决的问题.
- 了解膨胀现象对于预测流体系统中的流行为和不稳定性至关重要.
研究的目的:
- 开发一种新的数值框架,能够发现复杂的流体动力学方程的自相似膨胀解决方案.
- 研究物理信息神经网络 (PINNs) 在寻找非线性偏微分方程的稳定和不稳定的解决方案方面的潜力.
主要方法:
- 实施使用物理信息神经网络 (PINNs) 的新数值框架.
- 这些PINN受过训练,以发现2D Boussinesq和3D Euler方程的光滑,自相似的解决方案配置文件.
- 框架的应用,为科尔多瓦-科尔多瓦-Fontelos方程找到一个不稳定的自相似的解决方案.
主要成果:
- 成功发现了第一个平滑的自相似膨胀形状,用于2D Boussinesq方程和3D Euler方程.
- 证明了PINNs能够识别不稳定的自相似解决方案的能力,以Córdoba-Córdoba-Fontelos方程为例.
- 在不同的流体方程中验证了开发的数值框架的稳定性和适应性.
结论:
- 发现的自相似解决方案为在流体动力学中潜在的计算机辅助膨胀证明提供了基础.
- 基于物理学的神经网络是探索流体力学及其他领域复杂解决方案的强大而多功能工具.
- 数值框架提供了一个强大的方法,用于找到对非线性偏微分方程的具有挑战性的解决方案.
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