发表机构
CAPT, LMAM, and School of Mathematical Sciences, Peking University; Beijing Computational Science Research Center; School of Mathematical Sciences, Peking University（北京大学数学科学学院; 北京计算科学研究中心; 北京大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出数据驱动的广义纳维-斯托克斯模型，通过端到端学习扩展NS方程至过渡区域，并采用长期学习方法减少累积误差，数值实验验证其有效性。
AI 中文摘要
当克努森数较大时，纳维-斯托克斯（NS）方程在过渡区域会失去准确性，而玻尔兹曼方程提供了充分的动力学描述，但计算成本显著更高。在本工作中，提出了一种数据驱动的广义纳维-斯托克斯（GNS）模型，以将NS方程的适用性扩展到过渡区域。从玻尔兹曼方程的数值解形成的数据中，学习了非平衡变量（如应力张量和热通量）与平衡变量（包括密度、宏观速度和温度）以及克努森数之间的新关系。利用标准的端到端学习来构建损失函数，在此基础上，提出了一种长期端到端学习方法用于损失函数，以减少累积误差并提高长期预测准确性。研究了几个数值算例，包括一维波和黎曼问题以及二维等熵涡和泰勒-格林涡问题，以验证这种新GNS模型的有效性。
英文摘要
Navier-Stokes (NS) equations lose accuracy in the transition regime when the Knudsen number is large, while the Boltzmann equation provides an adequate kinetic description with a substantially higher computational cost. In this work, a data-driven generalized Navier-Stokes (GNS) model is proposed to extend the applicability of the NS equations to the transition regime. A new relationship between the non-equilibrium variables, such as the stress tensor and the heat flux, and the equilibrium variables, including the density, macroscopic velocity, and the temperature, together with the Knudsen number, is learned from the data formed by the numerical solution to the Boltzmann equation. The standard end-to-end learning is utilized to construct the loss function, based on which, a long-term end-to-end learning method is proposed for the loss function to reduce accumulated error and improve long-term predictive accuracy. Several numerical examples, including one-dimensional wave and Riemann problems as well as two-dimensional isentropic vortex and Taylor-Green vortex problems, are studied to validate the efficiency of this new GNS model.
