  The goal of this work is to train a neural network which approximates
solutions to the Navier-Stokes equations across a region of parameter space, in
which the parameters define physical properties such as domain shape and
boundary conditions. The contributions of this work are threefold:
  1) To demonstrate that neural networks can be efficient aggregators of whole
families of parameteric solutions to physical problems, trained using data
created with traditional, trusted numerical methods such as finite elements.
Advantages include extremely fast evaluation of pressure and velocity at any
point in physical and parameter space (asymptotically, ~3 $\mu s$ / query), and
data compression (the network requires 99\% less storage space compared to its
own training data).
  2) To demonstrate that the neural networks can accurately interpolate between
finite element solutions in parameter space, allowing them to be instantly
queried for pressure and velocity field solutions to problems for which
traditional simulations have never been performed.
  3) To introduce an active learning algorithm, so that during training, a
finite element solver can automatically be queried to obtain additional
training data in locations where the neural network's predictions are in most
need of improvement, thus autonomously acquiring and efficiently distributing
training data throughout parameter space.
  In addition to the obvious utility of Item 2, above, we demonstrate an
application of the network in rapid parameter sweeping, very precisely
predicting the degree of narrowing in a tube which would result in a 50\%
increase in end-to-end pressure difference at a given flow rate. This
capability could have applications in both medical diagnosis of arterial
disease, and in computer-aided design.
