This paper solves the discretised multiphase flow equations using tools and methods from machine-learning libraries. The idea comes from the observation that convolutional layers can be used to express a discretisation as a neural network whose weights are determined by the numerical method, rather than by training, and hence, we refer to this approach as Neural Networks for PDEs (NN4PDEs). To solve the discretised multiphase flow equations, a multigrid solver is implemented through a convolutional neural network with a U-Net architecture. Immiscible two-phase flow is modelled by the 3D incompressible Navier–Stokes equations with surface tension and advection of a volume fraction field, which describes the interface between the fluids. A new compressive algebraic volume-of-fluids method is introduced, based on a residual formulation using Petrov–Galerkin for accuracy and designed with NN4PDEs in mind. High-order finite-element based schemes are chosen to model a collapsing water column and a rising bubble. Results compare well with experimental data and other numerical results from the literature, demonstrating that, for the first time, finite element discretisations of multiphase flows can be solved using an approach based on (untrained) convolutional neural networks. A benefit of expressing numerical discretisations as neural networks is that the code can run, without modification, on CPUs, GPUs or the latest accelerators designed especially to run AI codes.

中文翻译：

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使用机器学习库在结构化网格上捕获界面来求解离散多相流方程

本文使用机器学习库中的工具和方法求解离散多相流方程。这个想法来自于这样的观察：卷积层可用于将离散化表示为神经网络，其权重由数值方法而不是训练确定，因此，我们将这种方法称为偏微分方程的神经网络（NN4PDE） 。为了求解离散多相流方程，通过具有 U-Net 架构的卷积神经网络实现了多重网格求解器。不混溶两相流通过具有表面张力和体积分数场平流的 3D 不可压缩纳维-斯托克斯方程进行建模，该方程描述了流体之间的界面。引入了一种新的压缩代数流体体积方法，该方法基于使用 Petrov-Galerkin 的残差公式以确保准确性，并在设计时考虑了 NN4PDE。选择基于高阶有限元的方案来模拟塌陷的水柱和上升的气泡。结果与实验数据和文献中的其他数值结果相比较，首次证明多相流的有限元离散可以使用基于（未经训练的）卷积神经网络的方法来求解。将数值离散表示为神经网络的一个好处是，代码无需修改即可在 CPU、GPU 或专为运行 AI 代码而设计的最新加速器上运行。