  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.
