If I understand the paper [1] correctly, they used Convolution Neural Network (ConvNet) to speed up a Navier-Stokes Partial Differential Equation (PDE) solver. Specifically, they are solving "incompressible flow" Navier-Stokes equations.
In each time step, the solver performs an "advection" update followed by a "pressure projection" step. This "pressure projection" step involves solving a "Poisson equation," which is computationally intensive and usually requires an iterative solver. Instead of directly solving the Poisson equation, they train a ConvNet regression model to infer these "pressure projections."
They have to deal with lots of fiddly details, like choice of neural network training penalty function, dealing with PDE boundary conditions issues, and dealing with errors accumulating during simulation.
Training the ConvNet involves running an off-line solver to obtain statistical fluid dynamics data. The training step takes 48 hours, using GPU ConvNet training. As far as I can tell, the network is trained on a specific geometry and must be re-trained if the geometry changes. [2]
The final run-time improvement of their model is "competitive with state-of-the-art GPU implementations of Jacobi and Gauss Seidel solvers for pressure." They believe that run-time could be further improved.
[1] https://arxiv.org/pdf/1607.03597v3.pdf
[2] "When simulating smoke, we do not explicitly satisfy the Dirichlet boundary conditions in our offline simulation (for smoke we surround our simulation domain by an empty air region). Typically “ghost cells” are used to incorporate these boundary conditions into the linear system solution. In our system, we do not include ghost cells (or even mark voxels as boundary cells) and instead let the ConvNet learn to apply these boundary conditions by learning edge effects in the functional mapping of the input tensor to pressure output. This means that we must learn a new model if the external boundary changes. This is certainly a limitation of our approach, however it is one that we feel is not severe since most applications use a constant domain boundary."