Error estimates and physics informed augmentation of neural networks for thermally coupled incompressible Navier Stokes equations.
Error estimates and physics informed augmentation of neural networks for thermally coupled incompressible Navier Stokes equations.
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DOI：
10.1007/s00466-023-02334-7
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发表时间：
2023-08
影响因子：
4.1
通讯作者：
中科院分区：
文献类型：
--
作者：
物理信息神经网络（PINNs）被证明是一种很有前途的方法来逼近偏微分方程（PDE）。PINN通过在给定域上最小化基于物理的损失函数来近似PDE解。尽管PINN的应用范围的问题类，调查的误差估计和收敛性能的PINN，这是很重要的建立背后的理由，他们的良好的经验表现，一直缺乏实质性的进展。本文对一类热耦合不可压Navier-Stokes方程的多物理场问题进行了PINNs的收敛性分析和误差估计。通过Beltrami流的模型问题，它表明，一个小的训练误差意味着一个小的泛化误差。给出了总误差关于训练残差和配置点的后验收敛速度。这对确定合适的训练参数和训练残差阈值，得到良好的热耦合稳态层流PINN预测具有实际意义。这些收敛率，然后推广到不同的空间几何形状，以及不同的流动参数，在于层流制度。在PINN的PDE残差中加入了压力Poisson方程形式的压力稳定项。这种物理信息增强被证明是提高精度的压力场的数量级相比，没有增强的情况下。从PINNs的结果进行了比较，从稳定的有限元方法和PINNs的良好性能突出。
Physics Informed Neural Networks (PINNs) are shown to be a promising method for the approximation of partial differential equations (PDEs). PINNs approximate the PDE solution by minimizing physics-based loss functions over a given domain. Despite substantial progress in the application of PINNs to a range of problem classes, investigation of error estimation and convergence properties of PINNs, which is important for establishing the rationale behind their good empirical performance, has been lacking. This paper presents convergence analysis and error estimates of PINNs for a multi-physics problem of thermally coupled incompressible Navier–Stokes equations. Through a model problem of Beltrami flow it is shown that a small training error implies a small generalization error. Posteriori convergence rates of total error with respect to the training residual and collocation points are presented. This is of practical significance in determining appropriate number of training parameters and training residual thresholds to get good PINNs prediction of thermally coupled steady state laminar flows. These convergence rates are then generalized to different spatial geometries as well as to different flow parameters that lie in the laminar regime. A pressure stabilization term in the form of pressure Poisson equation is added to the PDE residuals for PINNs. This physics informed augmentation is shown to improve accuracy of the pressure field by an order of magnitude as compared to the case without augmentation. Results from PINNs are compared to the ones obtained from stabilized finite element method and good properties of PINNs are highlighted.
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影响因子：
3.1
作者：
Costabal, Francisco Sahli;Yang, Yibo;Kuhl, Ellen
通讯作者：
Kuhl, Ellen
影响因子：
--
作者：
Lagaris, IE;Likas, A;Fotiadis, DI
通讯作者：
Fotiadis, DI
影响因子：
11.2
作者：
Garnier, C.;Currie, J.;Muneer, T.
通讯作者：
Muneer, T.
DOI：
10.1016/j.cma.2003.12.047
发表时间：
2004-01-01
影响因子：
7.2
作者：
Masud, A;Khurram, RA
通讯作者：
Khurram, RA
影响因子：
--
作者：
Cai, Shengze;Wang, Zhicheng;Karniadakis, George E. M.
通讯作者：
Karniadakis, George E. M.