纳维-斯托克斯方程的神经网络多重网格解法
A neural network multigrid solver for the Navier-Stokes equations
摘要：
我们提出了我们为非定常纳维-斯托克斯方程开发的深度神经网络多重网格求解器（DNN-MG）。DNN-MG使用几何多重网格求解器和具有记忆的递归神经网络的明智组合来提高计算效率。DNN-MG使用多重网格方法在粗层次上进行经典求解，而神经网络则在细层次上校正插值解，从而避免了必须在那里执行的越来越昂贵的计算。这导致通过DNN-MG的高度紧凑的神经网络减少计算时间。紧凑性的结果，从它的局部补丁和可用的粗多重网格解决方案，提供了一个“指南”的修正设计。具有少量参数的紧凑型神经网络还可以减少训练时间和数据。此外，网络的局部性有助于推广，并允许人们使用在一个网格域上训练的DNN-MG，也可以在不同的网格域上训练。我们证明了DNN-MG对障碍物周围二维层流变化的有效性。对于这些，我们的方法显着改善的解决方案，以及升降机和阻力泛函，而只需要大约一半的计算时间的一个完整的多重网格解决方案。我们还表明，DNN-MG训练的配置与一个障碍物可以推广到其他时间相关的问题，可以有效地解决使用几何多重网格方法。
英文摘要：
We present the deep neural network multigrid solver (DNN-MG) that we develop for the instationary Navier-Stokes equations. DNN-MG improves computational efficiency using a judicious combination of a geometric multigrid solver and a recurrent neural network with memory. DNN-MG uses the multi-grid method to classically solve on coarse levels while the neural network corrects interpolated solutions on fine ones, thus avoiding the increasingly expensive computations that would have to be performed there. This results in a reduction in computation time through DNN-MG's highly compact neural network. The compactness results from its design for local patches and the available coarse multigrid solutions that provides a "guide" for the corrections. A compact neural network with a small number of parameters also reduces training time and data. Furthermore, the network's locality facilitates generalizability and allows one to use DNN-MG trained on one mesh domain also on different ones. We demonstrate the efficacy of DNN-MG for variations of the 2D laminar flow around an obstacle. For these, our method significantly improves the solutions as well as lift and drag functionals while requiring only about half the computation time of a full multigrid solution. We also show that DNN-MG trained for the configuration with one obstacle can be generalized to other time dependent problems that can be solved efficiently using a geometric multigrid method.
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