使用基于物理信息的卷积神经网络求解Navier–Stokes方程的物理合理且守恒解  

Physically plausible and conservative solutions to Navier–Stokes equations using physics-informed CNNs

作　　者：李健枫 周良滢 孙经纬 孙广中[1] Jianfeng Li;Liangying Zhou;Jingwei Sun;Guangzhong Sun(School of Computer Science and Technology,University of Science and Technology of China,Hefei 230027,China)

机构地区：[1]中国科学技术大学计算机科学与技术学院,安徽合肥230027

出　　处：《中国科学技术大学学报》2024年第4期24-35,23,66,67,共15页JUSTC

基　　金：supported by the National Natural Science Foundation of China(12171451);Anhui Center for Applied Mathematics。

摘　　要：基于物理信息的神经网络方法(PINN)是一种使用神经网络有效求解偏微分方程(PDEs)的新兴方法。基于物理信息的卷积神经网络方法(PICNN)是一种由卷积神经网络(CNNs)增强的PINN的变体。由于卷积神经网络的参数共享特性可以有效地学习空间依赖关系,因此PICNN在一系列偏微分方程的求解问题上取得了更好的结果。然而,应用现有的基于PICNN的方法求解Navier–Stokes方程时会产生振荡的预测解,这违背了物理定律和守恒特性。为了解决这一问题,我们提出了一种将PICNN与有限体积法相结合的新方法,以获得Navier–Stokes方程的物理上合理且具有守恒特性的预测解。我们使用有限体积法推导了Navier–Stokes方程的二阶迎风差分格式。然后我们使用所推导的格式来计算偏导数并构造基于物理信息的损失函数。我们对以稳态Navier–Stokes方程作为控制方程的不同场景进行了实验以评估所提出的方法,包括对流传热问题和顶盖驱动流问题等。实验结果表明,我们的方法可以有效地提高PICNN预测解的物理合理性和准确性。The physics-informed neural network(PINN)is an emerging approach for efficiently solving partial differential equations(PDEs)using neural networks.The physics-informed convolutional neural network(PICNN),a variant of PINN enhanced by convolutional neural networks(CNNs),has achieved better results on a series of PDEs since the parameter-sharing property of CNNs is effective in learning spatial dependencies.However,applying existing PICNNbased methods to solve Navier–Stokes equations can generate oscillating predictions,which are inconsistent with the laws of physics and the conservation properties.To address this issue,we propose a novel method that combines PICNN with the finite volume method to obtain physically plausible and conservative solutions to Navier–Stokes equations.We derive the second-order upwind difference scheme of Navier–Stokes equations using the finite volume method.Then we use the derived scheme to calculate the partial derivatives and construct the physics-informed loss function.The proposed method is assessed by experiments on steady-state Navier–Stokes equations under different scenarios,including convective heat transfer and lid-driven cavity flow.The experimental results demonstrate that our method can effectively improve the plausibility and accuracy of the predicted solutions from PICNN.

关 键 词：有限体积法 纳维-斯托克斯方程 偏微分方程 基于物理信息的卷积神经网络 

分 类 号：TP183[自动化与计算机技术—控制理论与控制工程]