论文标题
DivetizationNet:使用有限卷离散化的Navier-Stokes方程的基于机器学习的求解器
DiscretizationNet: A Machine-Learning based solver for Navier-Stokes Equations using Finite Volume Discretization
论文作者
论文摘要
在过去的几十年中,现有的部分微分方程(PDE)求解器在求解复杂的非线性PDE方面取得了巨大成功。尽管准确,但这些PDE求解器的计算昂贵。随着机器学习(ML)技术的进步,使用ML来求解PDE的研究显着增加。这项工作的目的是开发基于ML的PDE求解器,该求解器将现有PDE求解器的重要特征与ML Technologies结合在一起。这项工作中采用的两个求解器特征是:1)使用基于离散化的方案近似于时空部分衍生物,以及2)使用迭代算法以离散形式求解线性化的PDE。在存在高度非线性,耦合PDE解决方案的情况下,这些策略对于实现良好的准确性,更好的稳定性和更快的收敛性可能非常重要。我们的ML-Solver DivetizationNet使用具有PDE变量作为输入和输出功能的基于生成CNN的编码器模型。在训练过程中,离散化方案在计算图内实施,以使PDE残差更快地计算gpu计算,这些计算用于更新导致融合解决方案的网络权重。在网络训练期间实现了一种新颖的迭代能力,以提高ML溶剂的稳定性和收敛性。证明ML溶剂可以解决3-D中稳定的,不可压缩的Navier-Stokes方程,例如,盖子驱动的腔,流过气缸和共轭热传递。
Over the last few decades, existing Partial Differential Equation (PDE) solvers have demonstrated a tremendous success in solving complex, non-linear PDEs. Although accurate, these PDE solvers are computationally costly. With the advances in Machine Learning (ML) technologies, there has been a significant increase in the research of using ML to solve PDEs. The goal of this work is to develop an ML-based PDE solver, that couples important characteristics of existing PDE solvers with ML technologies. The two solver characteristics that have been adopted in this work are: 1) the use of discretization-based schemes to approximate spatio-temporal partial derivatives and 2) the use of iterative algorithms to solve linearized PDEs in their discrete form. In the presence of highly non-linear, coupled PDE solutions, these strategies can be very important in achieving good accuracy, better stability and faster convergence. Our ML-solver, DiscretizationNet, employs a generative CNN-based encoder-decoder model with PDE variables as both input and output features. During training, the discretization schemes are implemented inside the computational graph to enable faster GPU computation of PDE residuals, which are used to update network weights that result into converged solutions. A novel iterative capability is implemented during the network training to improve the stability and convergence of the ML-solver. The ML-Solver is demonstrated to solve the steady, incompressible Navier-Stokes equations in 3-D for several cases such as, lid-driven cavity, flow past a cylinder and conjugate heat transfer.